Step | Hyp | Ref
| Expression |
1 | | poimir.0 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℕ) |
2 | 1 | nnnn0d 11351 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
3 | 1 | nnred 11035 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℝ) |
4 | 3 | leidd 10594 |
. . . . 5
⊢ (𝜑 → 𝑁 ≤ 𝑁) |
5 | 2, 2, 4 | 3jca 1242 |
. . . 4
⊢ (𝜑 → (𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0
∧ 𝑁 ≤ 𝑁)) |
6 | | oveq2 6658 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → (1...𝑘) = (1...0)) |
7 | | fz10 12362 |
. . . . . . . . . . . . . . . 16
⊢ (1...0) =
∅ |
8 | 6, 7 | syl6eq 2672 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → (1...𝑘) = ∅) |
9 | 8 | oveq2d 6666 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → ((0..^𝐾) ↑𝑚
(1...𝑘)) = ((0..^𝐾) ↑𝑚
∅)) |
10 | | fzofi 12773 |
. . . . . . . . . . . . . . . 16
⊢
(0..^𝐾) ∈
Fin |
11 | | map0e 7895 |
. . . . . . . . . . . . . . . 16
⊢
((0..^𝐾) ∈ Fin
→ ((0..^𝐾)
↑𝑚 ∅) = 1𝑜) |
12 | 10, 11 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
((0..^𝐾)
↑𝑚 ∅) = 1𝑜 |
13 | | df1o2 7572 |
. . . . . . . . . . . . . . 15
⊢
1𝑜 = {∅} |
14 | 12, 13 | eqtri 2644 |
. . . . . . . . . . . . . 14
⊢
((0..^𝐾)
↑𝑚 ∅) = {∅} |
15 | 9, 14 | syl6eq 2672 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 0 → ((0..^𝐾) ↑𝑚
(1...𝑘)) =
{∅}) |
16 | | eqidd 2623 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → 𝑓 = 𝑓) |
17 | 16, 8, 8 | f1oeq123d 6133 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:∅–1-1-onto→∅)) |
18 | | eqid 2622 |
. . . . . . . . . . . . . . . . 17
⊢ ∅ =
∅ |
19 | | f1o00 6171 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:∅–1-1-onto→∅ ↔ (𝑓 = ∅ ∧ ∅ =
∅)) |
20 | 18, 19 | mpbiran2 954 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:∅–1-1-onto→∅ ↔ 𝑓 = ∅) |
21 | 17, 20 | syl6bb 276 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓 = ∅)) |
22 | 21 | abbidv 2741 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓 = ∅}) |
23 | | df-sn 4178 |
. . . . . . . . . . . . . 14
⊢ {∅}
= {𝑓 ∣ 𝑓 = ∅} |
24 | 22, 23 | syl6eqr 2674 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 0 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {∅}) |
25 | 15, 24 | xpeq12d 5140 |
. . . . . . . . . . . 12
⊢ (𝑘 = 0 → (((0..^𝐾) ↑𝑚
(1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = ({∅} ×
{∅})) |
26 | | 0ex 4790 |
. . . . . . . . . . . . 13
⊢ ∅
∈ V |
27 | 26, 26 | xpsn 6407 |
. . . . . . . . . . . 12
⊢
({∅} × {∅}) = {〈∅,
∅〉} |
28 | 25, 27 | syl6req 2673 |
. . . . . . . . . . 11
⊢ (𝑘 = 0 → {〈∅,
∅〉} = (((0..^𝐾)
↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)})) |
29 | | elsni 4194 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ {〈∅,
∅〉} → 𝑠 =
〈∅, ∅〉) |
30 | 26, 26 | op1std 7178 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 = 〈∅, ∅〉
→ (1st ‘𝑠) = ∅) |
31 | 29, 30 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ {〈∅,
∅〉} → (1st ‘𝑠) = ∅) |
32 | 31 | oveq1d 6665 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ {〈∅,
∅〉} → ((1st ‘𝑠) ∘𝑓 + ∅) =
(∅ ∘𝑓 + ∅)) |
33 | | f0 6086 |
. . . . . . . . . . . . . . . . . . . 20
⊢
∅:∅⟶∅ |
34 | | ffn 6045 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∅:∅⟶∅ → ∅ Fn
∅) |
35 | 33, 34 | mp1i 13 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ {〈∅,
∅〉} → ∅ Fn ∅) |
36 | 26 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ {〈∅,
∅〉} → ∅ ∈ V) |
37 | | inidm 3822 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∅
∩ ∅) = ∅ |
38 | | 0fv 6227 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∅‘𝑛) =
∅ |
39 | 38 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ {〈∅,
∅〉} ∧ 𝑛
∈ ∅) → (∅‘𝑛) = ∅) |
40 | 35, 35, 36, 36, 37, 39, 39 | offval 6904 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ {〈∅,
∅〉} → (∅ ∘𝑓 + ∅) = (𝑛 ∈ ∅ ↦ (∅
+ ∅))) |
41 | | mpt0 6021 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ∅ ↦ (∅
+ ∅)) = ∅ |
42 | 40, 41 | syl6eq 2672 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ {〈∅,
∅〉} → (∅ ∘𝑓 + ∅) =
∅) |
43 | 32, 42 | eqtrd 2656 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ {〈∅,
∅〉} → ((1st ‘𝑠) ∘𝑓 + ∅) =
∅) |
44 | 43 | uneq1d 3766 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈ {〈∅,
∅〉} → (((1st ‘𝑠) ∘𝑓 + ∅)
∪ ((1...𝑁) ×
{0})) = (∅ ∪ ((1...𝑁) × {0}))) |
45 | | uncom 3757 |
. . . . . . . . . . . . . . . 16
⊢ (∅
∪ ((1...𝑁) ×
{0})) = (((1...𝑁) ×
{0}) ∪ ∅) |
46 | | un0 3967 |
. . . . . . . . . . . . . . . 16
⊢
(((1...𝑁) ×
{0}) ∪ ∅) = ((1...𝑁) × {0}) |
47 | 45, 46 | eqtri 2644 |
. . . . . . . . . . . . . . 15
⊢ (∅
∪ ((1...𝑁) ×
{0})) = ((1...𝑁) ×
{0}) |
48 | 44, 47 | syl6req 2673 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ {〈∅,
∅〉} → ((1...𝑁) × {0}) = (((1st
‘𝑠)
∘𝑓 + ∅) ∪ ((1...𝑁) × {0}))) |
49 | 48 | csbeq1d 3540 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ {〈∅,
∅〉} → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵) |
50 | 49 | eqeq2d 2632 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ {〈∅,
∅〉} → (0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st
‘𝑠)
∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)) |
51 | | oveq2 6658 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → (0...𝑘) = (0...0)) |
52 | | 0z 11388 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
ℤ |
53 | | fzsn 12383 |
. . . . . . . . . . . . . . . 16
⊢ (0 ∈
ℤ → (0...0) = {0}) |
54 | 52, 53 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (0...0) =
{0} |
55 | 51, 54 | syl6eq 2672 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → (0...𝑘) = {0}) |
56 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = 0 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...0)) |
57 | 56 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 0 → ((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...0))) |
58 | 57 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 0 → (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) ×
{0})) |
59 | 58 | uneq2d 3767 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 0 → ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑘)) × {0})) =
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) ×
{0}))) |
60 | 59 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → ((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) ×
{0})))) |
61 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 0 → (𝑘 + 1) = (0 + 1)) |
62 | | 0p1e1 11132 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (0 + 1) =
1 |
63 | 61, 62 | syl6eq 2672 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 0 → (𝑘 + 1) = 1) |
64 | 63 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 0 → ((𝑘 + 1)...𝑁) = (1...𝑁)) |
65 | 64 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (((𝑘 + 1)...𝑁) × {0}) = ((1...𝑁) × {0})) |
66 | 60, 65 | uneq12d 3768 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → (((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) ×
{0}))) |
67 | 66 | csbeq1d 3540 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 →
⦋(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵) |
68 | 67 | eqeq2d 2632 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → (𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵)) |
69 | 55, 68 | rexeqbidv 3153 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ {0}𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵)) |
70 | | c0ex 10034 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
V |
71 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 = 0 → (1...𝑗) = (1...0)) |
72 | 71, 7 | syl6eq 2672 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 = 0 → (1...𝑗) = ∅) |
73 | 72 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 = 0 → ((2nd
‘𝑠) “
(1...𝑗)) = ((2nd
‘𝑠) “
∅)) |
74 | | ima0 5481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((2nd ‘𝑠) “ ∅) = ∅ |
75 | 73, 74 | syl6eq 2672 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = 0 → ((2nd
‘𝑠) “
(1...𝑗)) =
∅) |
76 | 75 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = 0 → (((2nd
‘𝑠) “
(1...𝑗)) × {1}) =
(∅ × {1})) |
77 | | 0xp 5199 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (∅
× {1}) = ∅ |
78 | 76, 77 | syl6eq 2672 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 0 → (((2nd
‘𝑠) “
(1...𝑗)) × {1}) =
∅) |
79 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1)) |
80 | 79, 62 | syl6eq 2672 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 = 0 → (𝑗 + 1) = 1) |
81 | 80 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 = 0 → ((𝑗 + 1)...0) = (1...0)) |
82 | 81, 7 | syl6eq 2672 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 = 0 → ((𝑗 + 1)...0) = ∅) |
83 | 82 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 = 0 → ((2nd
‘𝑠) “ ((𝑗 + 1)...0)) = ((2nd
‘𝑠) “
∅)) |
84 | 83, 74 | syl6eq 2672 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = 0 → ((2nd
‘𝑠) “ ((𝑗 + 1)...0)) =
∅) |
85 | 84 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = 0 → (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}) =
(∅ × {0})) |
86 | | 0xp 5199 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (∅
× {0}) = ∅ |
87 | 85, 86 | syl6eq 2672 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 0 → (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}) =
∅) |
88 | 78, 87 | uneq12d 3768 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 0 → ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...0))
× {0})) = (∅ ∪ ∅)) |
89 | | un0 3967 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (∅
∪ ∅) = ∅ |
90 | 88, 89 | syl6eq 2672 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 0 → ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...0))
× {0})) = ∅) |
91 | 90 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 0 → ((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) =
((1st ‘𝑠)
∘𝑓 + ∅)) |
92 | 91 | uneq1d 3766 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 0 → (((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) × {0})) =
(((1st ‘𝑠)
∘𝑓 + ∅) ∪ ((1...𝑁) × {0}))) |
93 | 92 | csbeq1d 3540 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 0 →
⦋(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵 =
⦋(((1st ‘𝑠) ∘𝑓 + ∅)
∪ ((1...𝑁) ×
{0})) / 𝑝⦌𝐵) |
94 | 93 | eqeq2d 2632 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 0 → (𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)) |
95 | 70, 94 | rexsn 4223 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑗 ∈
{0}𝑖 =
⦋(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵) |
96 | 69, 95 | syl6bb 276 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)) |
97 | 55, 96 | raleqbidv 3152 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 0 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ {0}𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)) |
98 | | eqeq1 2626 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 0 → (𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st
‘𝑠)
∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)) |
99 | 70, 98 | ralsn 4222 |
. . . . . . . . . . . . 13
⊢
(∀𝑖 ∈
{0}𝑖 =
⦋(((1st ‘𝑠) ∘𝑓 + ∅)
∪ ((1...𝑁) ×
{0})) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st
‘𝑠)
∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵) |
100 | 97, 99 | syl6rbb 277 |
. . . . . . . . . . . 12
⊢ (𝑘 = 0 → (0 =
⦋(((1st ‘𝑠) ∘𝑓 + ∅)
∪ ((1...𝑁) ×
{0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
101 | 50, 100 | sylan9bbr 737 |
. . . . . . . . . . 11
⊢ ((𝑘 = 0 ∧ 𝑠 ∈ {〈∅, ∅〉})
→ (0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
102 | 28, 101 | rabeqbidva 3196 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → {𝑠 ∈ {〈∅, ∅〉}
∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) |
103 | 102 | eqcomd 2628 |
. . . . . . . . 9
⊢ (𝑘 = 0 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ {〈∅, ∅〉}
∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}) |
104 | 103 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑘 = 0 → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ {〈∅, ∅〉}
∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})) |
105 | 104 | breq2d 4665 |
. . . . . . 7
⊢ (𝑘 = 0 → (2 ∥
(#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ {〈∅,
∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))) |
106 | 105 | notbid 308 |
. . . . . 6
⊢ (𝑘 = 0 → (¬ 2 ∥
(#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥ (#‘{𝑠 ∈ {〈∅,
∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))) |
107 | 106 | imbi2d 330 |
. . . . 5
⊢ (𝑘 = 0 → ((𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ {〈∅,
∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})))) |
108 | | oveq2 6658 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑚 → (1...𝑘) = (1...𝑚)) |
109 | 108 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑚 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...𝑚))) |
110 | | eqidd 2623 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑚 → 𝑓 = 𝑓) |
111 | 110, 108,
108 | f1oeq123d 6133 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑚 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚))) |
112 | 111 | abbidv 2741 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑚 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) |
113 | 109, 112 | xpeq12d 5140 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑚 → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)})) |
114 | | oveq2 6658 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑚 → (0...𝑘) = (0...𝑚)) |
115 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑚 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑚)) |
116 | 115 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑚 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚))) |
117 | 116 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑚 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0})) |
118 | 117 | uneq2d 3767 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑚 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑘)) × {0})) =
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) |
119 | 118 | oveq2d 6666 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑚 → ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0})))) |
120 | | oveq1 6657 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1)) |
121 | 120 | oveq1d 6665 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑚 → ((𝑘 + 1)...𝑁) = ((𝑚 + 1)...𝑁)) |
122 | 121 | xpeq1d 5138 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑚 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑚 + 1)...𝑁) × {0})) |
123 | 119, 122 | uneq12d 3768 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑚 → (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0}))) |
124 | 123 | csbeq1d 3540 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑚 → ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
125 | 124 | eqeq2d 2632 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑚 → (𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
126 | 114, 125 | rexeqbidv 3153 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑚 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
127 | 114, 126 | raleqbidv 3152 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑚 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
128 | 113, 127 | rabeqbidv 3195 |
. . . . . . . . 9
⊢ (𝑘 = 𝑚 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) |
129 | 128 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑘 = 𝑚 → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) |
130 | 129 | breq2d 4665 |
. . . . . . 7
⊢ (𝑘 = 𝑚 → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
131 | 130 | notbid 308 |
. . . . . 6
⊢ (𝑘 = 𝑚 → (¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
132 | 131 | imbi2d 330 |
. . . . 5
⊢ (𝑘 = 𝑚 → ((𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))) |
133 | | oveq2 6658 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑚 + 1) → (1...𝑘) = (1...(𝑚 + 1))) |
134 | 133 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑚 + 1) → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1)))) |
135 | | eqidd 2623 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (𝑚 + 1) → 𝑓 = 𝑓) |
136 | 135, 133,
133 | f1oeq123d 6133 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑚 + 1) → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1)))) |
137 | 136 | abbidv 2741 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑚 + 1) → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) |
138 | 134, 137 | xpeq12d 5140 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑚 + 1) → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))})) |
139 | | oveq2 6658 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑚 + 1) → (0...𝑘) = (0...(𝑚 + 1))) |
140 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = (𝑚 + 1) → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...(𝑚 + 1))) |
141 | 140 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = (𝑚 + 1) → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1)))) |
142 | 141 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = (𝑚 + 1) → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0})) |
143 | 142 | uneq2d 3767 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (𝑚 + 1) → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑘)) × {0})) =
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) |
144 | 143 | oveq2d 6666 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (𝑚 + 1) → ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0})))) |
145 | | oveq1 6657 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = (𝑚 + 1) → (𝑘 + 1) = ((𝑚 + 1) + 1)) |
146 | 145 | oveq1d 6665 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (𝑚 + 1) → ((𝑘 + 1)...𝑁) = (((𝑚 + 1) + 1)...𝑁)) |
147 | 146 | xpeq1d 5138 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (𝑚 + 1) → (((𝑘 + 1)...𝑁) × {0}) = ((((𝑚 + 1) + 1)...𝑁) × {0})) |
148 | 144, 147 | uneq12d 3768 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (𝑚 + 1) → (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))) |
149 | 148 | csbeq1d 3540 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (𝑚 + 1) → ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
150 | 149 | eqeq2d 2632 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑚 + 1) → (𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
151 | 139, 150 | rexeqbidv 3153 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑚 + 1) → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
152 | 139, 151 | raleqbidv 3152 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑚 + 1) → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
153 | 138, 152 | rabeqbidv 3195 |
. . . . . . . . 9
⊢ (𝑘 = (𝑚 + 1) → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) |
154 | 153 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑘 = (𝑚 + 1) → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) |
155 | 154 | breq2d 4665 |
. . . . . . 7
⊢ (𝑘 = (𝑚 + 1) → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
156 | 155 | notbid 308 |
. . . . . 6
⊢ (𝑘 = (𝑚 + 1) → (¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
157 | 156 | imbi2d 330 |
. . . . 5
⊢ (𝑘 = (𝑚 + 1) → ((𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))) |
158 | | oveq2 6658 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑁 → (1...𝑘) = (1...𝑁)) |
159 | 158 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑁 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...𝑁))) |
160 | | eqidd 2623 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑁 → 𝑓 = 𝑓) |
161 | 160, 158,
158 | f1oeq123d 6133 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑁 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))) |
162 | 161 | abbidv 2741 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑁 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
163 | 159, 162 | xpeq12d 5140 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑁 → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
164 | | oveq2 6658 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑁 → (0...𝑘) = (0...𝑁)) |
165 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑁 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑁)) |
166 | 165 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑁 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁))) |
167 | 166 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑁 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})) |
168 | 167 | uneq2d 3767 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑁 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑘)) × {0})) =
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) |
169 | 168 | oveq2d 6666 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑁 → ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))) |
170 | | oveq1 6657 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑁 → (𝑘 + 1) = (𝑁 + 1)) |
171 | 170 | oveq1d 6665 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑁 → ((𝑘 + 1)...𝑁) = ((𝑁 + 1)...𝑁)) |
172 | 171 | xpeq1d 5138 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑁 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑁 + 1)...𝑁) × {0})) |
173 | 169, 172 | uneq12d 3768 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑁 → (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0}))) |
174 | 173 | csbeq1d 3540 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑁 → ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
175 | 174 | eqeq2d 2632 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑁 → (𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
176 | 164, 175 | rexeqbidv 3153 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑁 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
177 | 164, 176 | raleqbidv 3152 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑁 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
178 | 163, 177 | rabeqbidv 3195 |
. . . . . . . . 9
⊢ (𝑘 = 𝑁 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) |
179 | 178 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑘 = 𝑁 → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) |
180 | 179 | breq2d 4665 |
. . . . . . 7
⊢ (𝑘 = 𝑁 → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
181 | 180 | notbid 308 |
. . . . . 6
⊢ (𝑘 = 𝑁 → (¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
182 | 181 | imbi2d 330 |
. . . . 5
⊢ (𝑘 = 𝑁 → ((𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))) |
183 | | n2dvds1 15104 |
. . . . . . 7
⊢ ¬ 2
∥ 1 |
184 | | opex 4932 |
. . . . . . . . . 10
⊢
〈∅, ∅〉 ∈ V |
185 | | hashsng 13159 |
. . . . . . . . . 10
⊢
(〈∅, ∅〉 ∈ V → (#‘{〈∅,
∅〉}) = 1) |
186 | 184, 185 | ax-mp 5 |
. . . . . . . . 9
⊢
(#‘{〈∅, ∅〉}) = 1 |
187 | | nnuz 11723 |
. . . . . . . . . . . . . . . . 17
⊢ ℕ =
(ℤ≥‘1) |
188 | 1, 187 | syl6eleq 2711 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘1)) |
189 | | eluzfz1 12348 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘1) → 1 ∈ (1...𝑁)) |
190 | 188, 189 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 1 ∈ (1...𝑁)) |
191 | | poimirlem28.5 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐾 ∈ ℕ) |
192 | 191 | nnnn0d 11351 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐾 ∈
ℕ0) |
193 | | 0elfz 12436 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ ℕ0
→ 0 ∈ (0...𝐾)) |
194 | | fconst6g 6094 |
. . . . . . . . . . . . . . . 16
⊢ (0 ∈
(0...𝐾) → ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾)) |
195 | 192, 193,
194 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾)) |
196 | 70 | fvconst2 6469 |
. . . . . . . . . . . . . . . 16
⊢ (1 ∈
(1...𝑁) → (((1...𝑁) × {0})‘1) =
0) |
197 | 190, 196 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((1...𝑁) × {0})‘1) =
0) |
198 | 190, 195,
197 | 3jca 1242 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) =
0)) |
199 | | nfv 1843 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑝(𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) =
0)) |
200 | | nfcsb1v 3549 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑝⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 |
201 | 200 | nfeq1 2778 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑝⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0 |
202 | 199, 201 | nfim 1825 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑝((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) →
⦋((1...𝑁)
× {0}) / 𝑝⦌𝐵 = 0) |
203 | | ovex 6678 |
. . . . . . . . . . . . . . . 16
⊢
(1...𝑁) ∈
V |
204 | | snex 4908 |
. . . . . . . . . . . . . . . 16
⊢ {0}
∈ V |
205 | 203, 204 | xpex 6962 |
. . . . . . . . . . . . . . 15
⊢
((1...𝑁) ×
{0}) ∈ V |
206 | | feq1 6026 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = ((1...𝑁) × {0}) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾))) |
207 | | fveq1 6190 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = ((1...𝑁) × {0}) → (𝑝‘1) = (((1...𝑁) × {0})‘1)) |
208 | 207 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = ((1...𝑁) × {0}) → ((𝑝‘1) = 0 ↔ (((1...𝑁) × {0})‘1) =
0)) |
209 | 206, 208 | 3anbi23d 1402 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 = ((1...𝑁) × {0}) → ((1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0) ↔ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) =
0))) |
210 | 209 | anbi2d 740 |
. . . . . . . . . . . . . . . 16
⊢ (𝑝 = ((1...𝑁) × {0}) → ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) ↔ (𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) =
0)))) |
211 | | csbeq1a 3542 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 = ((1...𝑁) × {0}) → 𝐵 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵) |
212 | 211 | eqeq1d 2624 |
. . . . . . . . . . . . . . . 16
⊢ (𝑝 = ((1...𝑁) × {0}) → (𝐵 = 0 ↔ ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0)) |
213 | 210, 212 | imbi12d 334 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 = ((1...𝑁) × {0}) → (((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0) ↔ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) →
⦋((1...𝑁)
× {0}) / 𝑝⦌𝐵 = 0))) |
214 | | 1ex 10035 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
V |
215 | | eleq1 2689 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 1 → (𝑛 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁))) |
216 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 1 → (𝑝‘𝑛) = (𝑝‘1)) |
217 | 216 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 1 → ((𝑝‘𝑛) = 0 ↔ (𝑝‘1) = 0)) |
218 | 215, 217 | 3anbi13d 1401 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 1 → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) ↔ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0))) |
219 | 218 | anbi2d 740 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 1 → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) ↔ (𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)))) |
220 | | breq2 4657 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 1 → (𝐵 < 𝑛 ↔ 𝐵 < 1)) |
221 | 219, 220 | imbi12d 334 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 1 → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 < 1))) |
222 | | poimirlem28.3 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) |
223 | 214, 221,
222 | vtocl 3259 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 < 1) |
224 | | poimirlem28.2 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) |
225 | | elfznn0 12433 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ∈
ℕ0) |
226 | | nn0lt10b 11439 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈ ℕ0
→ (𝐵 < 1 ↔
𝐵 = 0)) |
227 | 224, 225,
226 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝐵 < 1 ↔ 𝐵 = 0)) |
228 | 227 | 3ad2antr2 1227 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → (𝐵 < 1 ↔ 𝐵 = 0)) |
229 | 223, 228 | mpbid 222 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0) |
230 | 202, 205,
213, 229 | vtoclf 3258 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) →
⦋((1...𝑁)
× {0}) / 𝑝⦌𝐵 = 0) |
231 | 198, 230 | mpdan 702 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0) |
232 | 231 | eqcomd 2628 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 =
⦋((1...𝑁)
× {0}) / 𝑝⦌𝐵) |
233 | 232 | ralrimivw 2967 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑠 ∈ {〈∅, ∅〉}0 =
⦋((1...𝑁)
× {0}) / 𝑝⦌𝐵) |
234 | | rabid2 3118 |
. . . . . . . . . . 11
⊢
({〈∅, ∅〉} = {𝑠 ∈ {〈∅, ∅〉}
∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵} ↔ ∀𝑠 ∈ {〈∅, ∅〉}0 =
⦋((1...𝑁)
× {0}) / 𝑝⦌𝐵) |
235 | 233, 234 | sylibr 224 |
. . . . . . . . . 10
⊢ (𝜑 → {〈∅,
∅〉} = {𝑠 ∈
{〈∅, ∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}) |
236 | 235 | fveq2d 6195 |
. . . . . . . . 9
⊢ (𝜑 → (#‘{〈∅,
∅〉}) = (#‘{𝑠 ∈ {〈∅, ∅〉}
∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})) |
237 | 186, 236 | syl5eqr 2670 |
. . . . . . . 8
⊢ (𝜑 → 1 = (#‘{𝑠 ∈ {〈∅,
∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})) |
238 | 237 | breq2d 4665 |
. . . . . . 7
⊢ (𝜑 → (2 ∥ 1 ↔ 2
∥ (#‘{𝑠 ∈
{〈∅, ∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))) |
239 | 183, 238 | mtbii 316 |
. . . . . 6
⊢ (𝜑 → ¬ 2 ∥
(#‘{𝑠 ∈
{〈∅, ∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})) |
240 | 239 | a1i 11 |
. . . . 5
⊢ (𝑁 ∈ ℕ0
→ (𝜑 → ¬ 2
∥ (#‘{𝑠 ∈
{〈∅, ∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))) |
241 | | 2z 11409 |
. . . . . . . . . . . . 13
⊢ 2 ∈
ℤ |
242 | | fzfi 12771 |
. . . . . . . . . . . . . . . . 17
⊢
(1...(𝑚 + 1)) ∈
Fin |
243 | | mapfi 8262 |
. . . . . . . . . . . . . . . . 17
⊢
(((0..^𝐾) ∈ Fin
∧ (1...(𝑚 + 1)) ∈
Fin) → ((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) ∈ Fin) |
244 | 10, 242, 243 | mp2an 708 |
. . . . . . . . . . . . . . . 16
⊢
((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) ∈ Fin |
245 | | ovex 6678 |
. . . . . . . . . . . . . . . . . . 19
⊢
(1...(𝑚 + 1)) ∈
V |
246 | 245, 245 | mapval 7869 |
. . . . . . . . . . . . . . . . . 18
⊢
((1...(𝑚 + 1))
↑𝑚 (1...(𝑚 + 1))) = {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} |
247 | | mapfi 8262 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1...(𝑚 + 1))
∈ Fin ∧ (1...(𝑚 +
1)) ∈ Fin) → ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) ∈
Fin) |
248 | 242, 242,
247 | mp2an 708 |
. . . . . . . . . . . . . . . . . 18
⊢
((1...(𝑚 + 1))
↑𝑚 (1...(𝑚 + 1))) ∈ Fin |
249 | 246, 248 | eqeltrri 2698 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin |
250 | | f1of 6137 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1)) → 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))) |
251 | 250 | ss2abi 3674 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ⊆ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} |
252 | | ssfi 8180 |
. . . . . . . . . . . . . . . . 17
⊢ (({𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ⊆ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}) → {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin) |
253 | 249, 251,
252 | mp2an 708 |
. . . . . . . . . . . . . . . 16
⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin |
254 | | xpfi 8231 |
. . . . . . . . . . . . . . . 16
⊢
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin) → (((0..^𝐾) ↑𝑚
(1...(𝑚 + 1))) ×
{𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin) |
255 | 244, 253,
254 | mp2an 708 |
. . . . . . . . . . . . . . 15
⊢
(((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin |
256 | | rabfi 8185 |
. . . . . . . . . . . . . . 15
⊢
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin) |
257 | | hashcl 13147 |
. . . . . . . . . . . . . . 15
⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈
ℕ0) |
258 | 255, 256,
257 | mp2b 10 |
. . . . . . . . . . . . . 14
⊢
(#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈
ℕ0 |
259 | 258 | nn0zi 11402 |
. . . . . . . . . . . . 13
⊢
(#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ |
260 | | rabfi 8185 |
. . . . . . . . . . . . . . 15
⊢
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin) |
261 | | hashcl 13147 |
. . . . . . . . . . . . . . 15
⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈
ℕ0) |
262 | 255, 260,
261 | mp2b 10 |
. . . . . . . . . . . . . 14
⊢
(#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈
ℕ0 |
263 | 262 | nn0zi 11402 |
. . . . . . . . . . . . 13
⊢
(#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ |
264 | 241, 259,
263 | 3pm3.2i 1239 |
. . . . . . . . . . . 12
⊢ (2 ∈
ℤ ∧ (#‘{𝑠
∈ (((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ) |
265 | | nn0p1nn 11332 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ0
→ (𝑚 + 1) ∈
ℕ) |
266 | 265 | ad2antrl 764 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑚 + 1) ∈ ℕ) |
267 | | uneq1 3760 |
. . . . . . . . . . . . . . . 16
⊢ (𝑞 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) = (((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))) |
268 | 267 | csbeq1d 3540 |
. . . . . . . . . . . . . . 15
⊢ (𝑞 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) →
⦋(𝑞 ∪
((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
269 | 70 | fconst 6091 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0} |
270 | 269 | jctr 565 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0})) |
271 | 265 | nnred 11035 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℕ0
→ (𝑚 + 1) ∈
ℝ) |
272 | 271 | ltp1d 10954 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑚 ∈ ℕ0
→ (𝑚 + 1) < ((𝑚 + 1) + 1)) |
273 | | fzdisj 12368 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑚 + 1) < ((𝑚 + 1) + 1) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅) |
274 | 272, 273 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 ∈ ℕ0
→ ((1...(𝑚 + 1)) ∩
(((𝑚 + 1) + 1)...𝑁)) = ∅) |
275 | | fun 6066 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}) ∧ ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0})) |
276 | 270, 274,
275 | syl2anr 495 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑚 ∈ ℕ0
∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0})) |
277 | 276 | adantlr 751 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑚 ∈ ℕ0
∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0})) |
278 | 277 | adantl 482 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0})) |
279 | 265 | peano2nnd 11037 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑚 ∈ ℕ0
→ ((𝑚 + 1) + 1) ∈
ℕ) |
280 | 279, 187 | syl6eleq 2711 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℕ0
→ ((𝑚 + 1) + 1) ∈
(ℤ≥‘1)) |
281 | 280 | ad2antrl 764 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝑚 + 1) + 1) ∈
(ℤ≥‘1)) |
282 | | nn0z 11400 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑚 ∈ ℕ0
→ 𝑚 ∈
ℤ) |
283 | 1 | nnzd 11481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → 𝑁 ∈ ℤ) |
284 | | zltp1le 11427 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁)) |
285 | 282, 283,
284 | syl2anr 495 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁)) |
286 | 285 | biimpa 501 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → (𝑚 + 1) ≤ 𝑁) |
287 | 286 | anasss 679 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑚 + 1) ≤ 𝑁) |
288 | 282 | peano2zd 11485 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑚 ∈ ℕ0
→ (𝑚 + 1) ∈
ℤ) |
289 | 288 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑚 ∈ ℕ0
∧ 𝑚 < 𝑁) → (𝑚 + 1) ∈ ℤ) |
290 | | eluz 11701 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈
(ℤ≥‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁)) |
291 | 289, 283,
290 | syl2anr 495 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑁 ∈ (ℤ≥‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁)) |
292 | 287, 291 | mpbird 247 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑁 ∈ (ℤ≥‘(𝑚 + 1))) |
293 | | fzsplit2 12366 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑚 + 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑚 + 1))) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))) |
294 | 281, 292,
293 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))) |
295 | 294 | eqcomd 2628 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)) = (1...𝑁)) |
296 | 192, 193 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 0 ∈ (0...𝐾)) |
297 | 296 | snssd 4340 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → {0} ⊆ (0...𝐾)) |
298 | | ssequn2 3786 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({0}
⊆ (0...𝐾) ↔
((0...𝐾) ∪ {0}) =
(0...𝐾)) |
299 | 297, 298 | sylib 208 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((0...𝐾) ∪ {0}) = (0...𝐾)) |
300 | 299 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((0...𝐾) ∪ {0}) = (0...𝐾)) |
301 | 295, 300 | feq23d 6040 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))) |
302 | 301 | adantrr 753 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))) |
303 | 278, 302 | mpbid 222 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
304 | | nfv 1843 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑝(𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
305 | | nfcsb1v 3549 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 |
306 | 305 | nfel1 2779 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁) |
307 | 304, 306 | nfim 1825 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑝((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)) |
308 | | vex 3203 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑞 ∈ V |
309 | | ovex 6678 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑚 + 1) + 1)...𝑁) ∈ V |
310 | 309, 204 | xpex 6962 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}) ∈ V |
311 | 308, 310 | unex 6956 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) ∈ V |
312 | | feq1 6026 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))) |
313 | 312 | anbi2d 740 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) ↔ (𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))) |
314 | | csbeq1a 3542 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → 𝐵 = ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
315 | 314 | eleq1d 2686 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ∈ (0...𝑁) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))) |
316 | 313, 315 | imbi12d 334 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) ↔ ((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)))) |
317 | 307, 311,
316, 224 | vtoclf 3258 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)) |
318 | 303, 317 | syldan 487 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)) |
319 | 318 | anassrs 680 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)) |
320 | | elfznn0 12433 |
. . . . . . . . . . . . . . . . 17
⊢
(⦋(𝑞
∪ ((((𝑚 + 1) +
1)...𝑁) × {0})) /
𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈
ℕ0) |
321 | 319, 320 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈
ℕ0) |
322 | 265 | nnnn0d 11351 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 ∈ ℕ0
→ (𝑚 + 1) ∈
ℕ0) |
323 | 322 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑚 ∈ ℕ0
∧ 𝑚 < 𝑁) → (𝑚 + 1) ∈
ℕ0) |
324 | 323 | ad2antlr 763 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑚 + 1) ∈
ℕ0) |
325 | | leloe 10124 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑚 + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁))) |
326 | 271, 3, 325 | syl2anr 495 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁))) |
327 | 285, 326 | bitrd 268 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁))) |
328 | 327 | biimpd 219 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 → ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁))) |
329 | 328 | imdistani 726 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → ((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁))) |
330 | 329 | anasss 679 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁))) |
331 | | simplll 798 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝜑) |
332 | 279 | nnge1d 11063 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑚 ∈ ℕ0
→ 1 ≤ ((𝑚 + 1) +
1)) |
333 | 332 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 1 ≤ ((𝑚 + 1) + 1)) |
334 | | zltp1le 11427 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁)) |
335 | 288, 283,
334 | syl2anr 495 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁)) |
336 | 335 | biimpa 501 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ≤ 𝑁) |
337 | 288 | peano2zd 11485 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℕ0
→ ((𝑚 + 1) + 1) ∈
ℤ) |
338 | | 1z 11407 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 1 ∈
ℤ |
339 | | elfz 12332 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧
1 ∈ ℤ ∧ 𝑁
∈ ℤ) → (((𝑚
+ 1) + 1) ∈ (1...𝑁)
↔ (1 ≤ ((𝑚 + 1) +
1) ∧ ((𝑚 + 1) + 1) ≤
𝑁))) |
340 | 338, 339 | mp3an2 1412 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧
𝑁 ∈ ℤ) →
(((𝑚 + 1) + 1) ∈
(1...𝑁) ↔ (1 ≤
((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁))) |
341 | 337, 283,
340 | syl2anr 495 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁))) |
342 | 341 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁))) |
343 | 333, 336,
342 | mpbir2and 957 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁)) |
344 | 343 | adantlr 751 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁)) |
345 | | nn0re 11301 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℕ0
→ 𝑚 ∈
ℝ) |
346 | 345 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 ∈ ℝ) |
347 | 271 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℝ) |
348 | 3 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ ℝ) |
349 | 345 | ltp1d 10954 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℕ0
→ 𝑚 < (𝑚 + 1)) |
350 | 349 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < (𝑚 + 1)) |
351 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) < 𝑁) |
352 | 346, 347,
348, 350, 351 | lttrd 10198 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁) |
353 | 352 | adantlr 751 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁) |
354 | | anass 681 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ↔ (𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁))) |
355 | 303 | anassrs 680 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
356 | 354, 355 | sylanb 489 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
357 | 356 | an32s 846 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
358 | 353, 357 | syldan 487 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
359 | | ffn 6045 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → 𝑞 Fn (1...(𝑚 + 1))) |
360 | 359 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑞 Fn (1...(𝑚 + 1))) |
361 | 274 | ad3antlr 767 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅) |
362 | | eluz 11701 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧
𝑁 ∈ ℤ) →
(𝑁 ∈
(ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁)) |
363 | 337, 283,
362 | syl2anr 495 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑁 ∈
(ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁)) |
364 | 363 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑁 ∈
(ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁)) |
365 | 336, 364 | mpbird 247 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈
(ℤ≥‘((𝑚 + 1) + 1))) |
366 | | eluzfz1 12348 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 ∈
(ℤ≥‘((𝑚 + 1) + 1)) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁)) |
367 | 365, 366 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁)) |
368 | 367 | adantlr 751 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁)) |
369 | | fnconstg 6093 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (0 ∈
V → ((((𝑚 + 1) +
1)...𝑁) × {0}) Fn
(((𝑚 + 1) + 1)...𝑁)) |
370 | 70, 369 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁) |
371 | | fvun2 6270 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1))) |
372 | 370, 371 | mp3an2 1412 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1))) |
373 | 360, 361,
368, 372 | syl12anc 1324 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1))) |
374 | 70 | fvconst2 6469 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0) |
375 | 368, 374 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0) |
376 | 373, 375 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0) |
377 | | nfv 1843 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑝(𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) |
378 | | nfcv 2764 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑝
< |
379 | | nfcv 2764 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑝((𝑚 + 1) + 1) |
380 | 305, 378,
379 | nfbr 4699 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1) |
381 | 377, 380 | nfim 1825 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑝((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) →
⦋(𝑞 ∪
((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)) |
382 | | fveq1 6190 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘((𝑚 + 1) + 1)) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1))) |
383 | 382 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘((𝑚 + 1) + 1)) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) |
384 | 312, 383 | 3anbi23d 1402 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0) ↔ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0))) |
385 | 384 | anbi2d 740 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) ↔ (𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)))) |
386 | 314 | breq1d 4663 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < ((𝑚 + 1) + 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))) |
387 | 385, 386 | imbi12d 334 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1)) ↔ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) →
⦋(𝑞 ∪
((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)))) |
388 | | ovex 6678 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑚 + 1) + 1) ∈
V |
389 | | eleq1 2689 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = ((𝑚 + 1) + 1) → (𝑛 ∈ (1...𝑁) ↔ ((𝑚 + 1) + 1) ∈ (1...𝑁))) |
390 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 = ((𝑚 + 1) + 1) → (𝑝‘𝑛) = (𝑝‘((𝑚 + 1) + 1))) |
391 | 390 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝑝‘𝑛) = 0 ↔ (𝑝‘((𝑚 + 1) + 1)) = 0)) |
392 | 389, 391 | 3anbi13d 1401 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) ↔ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0))) |
393 | 392 | anbi2d 740 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) ↔ (𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)))) |
394 | | breq2 4657 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = ((𝑚 + 1) + 1) → (𝐵 < 𝑛 ↔ 𝐵 < ((𝑚 + 1) + 1))) |
395 | 393, 394 | imbi12d 334 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = ((𝑚 + 1) + 1) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1)))) |
396 | 388, 395,
222 | vtocl 3259 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1)) |
397 | 381, 311,
387, 396 | vtoclf 3258 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) →
⦋(𝑞 ∪
((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)) |
398 | 331, 344,
358, 376, 397 | syl13anc 1328 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)) |
399 | 354, 319 | sylanb 489 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)) |
400 | 399 | an32s 846 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)) |
401 | | elfzelz 12342 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(⦋(𝑞
∪ ((((𝑚 + 1) +
1)...𝑁) × {0})) /
𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ) |
402 | 400, 401 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ) |
403 | 353, 402 | syldan 487 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ) |
404 | 288 | ad3antlr 767 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℤ) |
405 | | zleltp1 11428 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((⦋(𝑞
∪ ((((𝑚 + 1) +
1)...𝑁) × {0})) /
𝑝⦌𝐵 ∈ ℤ ∧ (𝑚 + 1) ∈ ℤ) →
(⦋(𝑞 ∪
((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))) |
406 | 403, 404,
405 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))) |
407 | 398, 406 | mpbird 247 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1)) |
408 | 349 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → 𝑚 < (𝑚 + 1)) |
409 | | breq2 4657 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑚 + 1) = 𝑁 → (𝑚 < (𝑚 + 1) ↔ 𝑚 < 𝑁)) |
410 | 409 | biimpac 503 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑚 < (𝑚 + 1) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁) |
411 | 408, 410 | sylan 488 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁) |
412 | | elfzle2 12345 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(⦋(𝑞
∪ ((((𝑚 + 1) +
1)...𝑁) × {0})) /
𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁) |
413 | 400, 412 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁) |
414 | 411, 413 | syldan 487 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁) |
415 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑚 + 1) = 𝑁) |
416 | 414, 415 | breqtrrd 4681 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1)) |
417 | 407, 416 | jaodan 826 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1)) |
418 | 417 | an32s 846 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1)) |
419 | 330, 418 | sylan 488 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1)) |
420 | | elfz2nn0 12431 |
. . . . . . . . . . . . . . . 16
⊢
(⦋(𝑞
∪ ((((𝑚 + 1) +
1)...𝑁) × {0})) /
𝑝⦌𝐵 ∈ (0...(𝑚 + 1)) ↔ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℕ0 ∧ (𝑚 + 1) ∈ ℕ0
∧ ⦋(𝑞
∪ ((((𝑚 + 1) +
1)...𝑁) × {0})) /
𝑝⦌𝐵 ≤ (𝑚 + 1))) |
421 | 321, 324,
419, 420 | syl3anbrc 1246 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...(𝑚 + 1))) |
422 | | fzss2 12381 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈
(ℤ≥‘(𝑚 + 1)) → (1...(𝑚 + 1)) ⊆ (1...𝑁)) |
423 | 292, 422 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (1...(𝑚 + 1)) ⊆ (1...𝑁)) |
424 | 423 | sselda 3603 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑛 ∈ (1...(𝑚 + 1))) → 𝑛 ∈ (1...𝑁)) |
425 | 424 | 3ad2antr1 1226 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → 𝑛 ∈ (1...𝑁)) |
426 | 355 | 3ad2antr2 1227 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
427 | 359 | ad2antll 765 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑞 Fn (1...(𝑚 + 1))) |
428 | 274 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅) |
429 | | simprl 794 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑛 ∈ (1...(𝑚 + 1))) |
430 | | fvun1 6269 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛)) |
431 | 370, 430 | mp3an2 1412 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛)) |
432 | 427, 428,
429, 431 | syl12anc 1324 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛)) |
433 | 432 | adantlrr 757 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛)) |
434 | 433 | 3adantr3 1222 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛)) |
435 | | simpr3 1069 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑞‘𝑛) = 0) |
436 | 434, 435 | eqtrd 2656 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0) |
437 | 425, 426,
436 | 3jca 1242 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) |
438 | | nfv 1843 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) |
439 | | nfcv 2764 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑝𝑛 |
440 | 305, 378,
439 | nfbr 4699 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛 |
441 | 438, 440 | nfim 1825 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑝((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛) |
442 | | fveq1 6190 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘𝑛) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛)) |
443 | 442 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘𝑛) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) |
444 | 312, 443 | 3anbi23d 1402 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))) |
445 | 444 | anbi2d 740 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)))) |
446 | 314 | breq1d 4663 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < 𝑛 ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛)) |
447 | 445, 446 | imbi12d 334 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛))) |
448 | 441, 311,
447, 222 | vtoclf 3258 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛) |
449 | 448 | adantlr 751 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛) |
450 | 437, 449 | syldan 487 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛) |
451 | | simp1 1061 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾) → 𝑛 ∈ (1...(𝑚 + 1))) |
452 | 424 | anasss 679 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → 𝑛 ∈ (1...𝑁)) |
453 | 451, 452 | sylanr2 685 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → 𝑛 ∈ (1...𝑁)) |
454 | | simp2 1062 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾) → 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) |
455 | 454, 303 | sylanr2 685 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
456 | 432 | 3adantr3 1222 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛)) |
457 | | simpr3 1069 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → (𝑞‘𝑛) = 𝐾) |
458 | 456, 457 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾) |
459 | 458 | anasss 679 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾) |
460 | 459 | adantrlr 759 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾) |
461 | 453, 455,
460 | 3jca 1242 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) |
462 | | nfv 1843 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) |
463 | | nfcv 2764 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑝(𝑛 − 1) |
464 | 305, 463 | nfne 2894 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1) |
465 | 462, 464 | nfim 1825 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑝((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)) |
466 | 442 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘𝑛) = 𝐾 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) |
467 | 312, 466 | 3anbi23d 1402 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))) |
468 | 467 | anbi2d 740 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)))) |
469 | 314 | neeq1d 2853 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ≠ (𝑛 − 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1))) |
470 | 468, 469 | imbi12d 334 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)))) |
471 | | poimirlem28.4 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) |
472 | 465, 311,
470, 471 | vtoclf 3258 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)) |
473 | 461, 472 | syldan 487 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)) |
474 | 473 | anassrs 680 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)) |
475 | 266, 268,
421, 450, 474 | poimirlem27 33436 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...(𝑚 + 1))) ×
{𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))) |
476 | 266, 268,
421 | poimirlem26 33435 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...(𝑚 + 1))) ×
{𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
477 | | fzfi 12771 |
. . . . . . . . . . . . . . . . . . 19
⊢
(0...(𝑚 + 1)) ∈
Fin |
478 | | xpfi 8231 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin ∧ (0...(𝑚 + 1)) ∈ Fin) →
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin) |
479 | 255, 477,
478 | mp2an 708 |
. . . . . . . . . . . . . . . . . 18
⊢
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin |
480 | | rabfi 8185 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin) |
481 | | hashcl 13147 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...(𝑚 + 1))) ×
{𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈
ℕ0) |
482 | 479, 480,
481 | mp2b 10 |
. . . . . . . . . . . . . . . . 17
⊢
(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℕ0 |
483 | 482 | nn0zi 11402 |
. . . . . . . . . . . . . . . 16
⊢
(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ |
484 | | zsubcl 11419 |
. . . . . . . . . . . . . . . 16
⊢
(((#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ) →
((#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∈ ℤ) |
485 | 483, 263,
484 | mp2an 708 |
. . . . . . . . . . . . . . 15
⊢
((#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∈ ℤ |
486 | | zsubcl 11419 |
. . . . . . . . . . . . . . . 16
⊢
(((#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ) → ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ∈ ℤ) |
487 | 483, 259,
486 | mp2an 708 |
. . . . . . . . . . . . . . 15
⊢
((#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ∈ ℤ |
488 | | dvds2sub 15016 |
. . . . . . . . . . . . . . 15
⊢ ((2
∈ ℤ ∧ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∈ ℤ ∧ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ∈ ℤ) → ((2 ∥
((#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∧ 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) → 2 ∥ (((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))) |
489 | 241, 485,
487, 488 | mp3an 1424 |
. . . . . . . . . . . . . 14
⊢ ((2
∥ ((#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∧ 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) → 2 ∥ (((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))) |
490 | 475, 476,
489 | syl2anc 693 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ (((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...(𝑚 + 1))) ×
{𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))) |
491 | 482 | nn0cni 11304 |
. . . . . . . . . . . . . 14
⊢
(#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ |
492 | 262 | nn0cni 11304 |
. . . . . . . . . . . . . 14
⊢
(#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℂ |
493 | 258 | nn0cni 11304 |
. . . . . . . . . . . . . 14
⊢
(#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ |
494 | | nnncan1 10317 |
. . . . . . . . . . . . . 14
⊢
(((#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℂ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ) → (((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) = ((#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))) |
495 | 491, 492,
493, 494 | mp3an 1424 |
. . . . . . . . . . . . 13
⊢
(((#‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) = ((#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) |
496 | 490, 495 | syl6breq 4694 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))) |
497 | | dvdssub2 15023 |
. . . . . . . . . . . 12
⊢ (((2
∈ ℤ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ) ∧ 2 ∥
((#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))) → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))) |
498 | 264, 496,
497 | sylancr 695 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))) |
499 | | nn0cn 11302 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 ∈ ℕ0
→ 𝑚 ∈
ℂ) |
500 | | pncan1 10454 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 ∈ ℂ → ((𝑚 + 1) − 1) = 𝑚) |
501 | 499, 500 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈ ℕ0
→ ((𝑚 + 1) − 1)
= 𝑚) |
502 | 501 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 ∈ ℕ0
→ (0...((𝑚 + 1)
− 1)) = (0...𝑚)) |
503 | 502 | rexeqdv 3145 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 ∈ ℕ0
→ (∃𝑗 ∈
(0...((𝑚 + 1) −
1))𝑖 =
⦋(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
504 | 502, 503 | raleqbidv 3152 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ ℕ0
→ (∀𝑖 ∈
(0...((𝑚 + 1) −
1))∃𝑗 ∈
(0...((𝑚 + 1) −
1))𝑖 =
⦋(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
505 | 504 | 3anbi1d 1403 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ0
→ ((∀𝑖 ∈
(0...((𝑚 + 1) −
1))∃𝑗 ∈
(0...((𝑚 + 1) −
1))𝑖 =
⦋(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1)) ↔ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1)))) |
506 | 505 | rabbidv 3189 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ0
→ {𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) |
507 | 506 | fveq2d 6195 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℕ0
→ (#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) |
508 | 507 | ad2antrl 764 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) |
509 | 1 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑁 ∈ ℕ) |
510 | 191 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝐾 ∈ ℕ) |
511 | | simprl 794 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑚 ∈ ℕ0) |
512 | | simprr 796 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑚 < 𝑁) |
513 | 509, 510,
511, 512 | poimirlem4 33413 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) |
514 | | fzfi 12771 |
. . . . . . . . . . . . . . . . . 18
⊢
(1...𝑚) ∈
Fin |
515 | | mapfi 8262 |
. . . . . . . . . . . . . . . . . 18
⊢
(((0..^𝐾) ∈ Fin
∧ (1...𝑚) ∈ Fin)
→ ((0..^𝐾)
↑𝑚 (1...𝑚)) ∈ Fin) |
516 | 10, 514, 515 | mp2an 708 |
. . . . . . . . . . . . . . . . 17
⊢
((0..^𝐾)
↑𝑚 (1...𝑚)) ∈ Fin |
517 | | ovex 6678 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(1...𝑚) ∈
V |
518 | 517, 517 | mapval 7869 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1...𝑚)
↑𝑚 (1...𝑚)) = {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} |
519 | | mapfi 8262 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1...𝑚) ∈ Fin
∧ (1...𝑚) ∈ Fin)
→ ((1...𝑚)
↑𝑚 (1...𝑚)) ∈ Fin) |
520 | 514, 514,
519 | mp2an 708 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1...𝑚)
↑𝑚 (1...𝑚)) ∈ Fin |
521 | 518, 520 | eqeltrri 2698 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin |
522 | | f1of 6137 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓:(1...𝑚)–1-1-onto→(1...𝑚) → 𝑓:(1...𝑚)⟶(1...𝑚)) |
523 | 522 | ss2abi 3674 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ⊆ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} |
524 | | ssfi 8180 |
. . . . . . . . . . . . . . . . . 18
⊢ (({𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ⊆ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)}) → {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin) |
525 | 521, 523,
524 | mp2an 708 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin |
526 | | xpfi 8231 |
. . . . . . . . . . . . . . . . 17
⊢
((((0..^𝐾)
↑𝑚 (1...𝑚)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin) → (((0..^𝐾) ↑𝑚
(1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin) |
527 | 516, 525,
526 | mp2an 708 |
. . . . . . . . . . . . . . . 16
⊢
(((0..^𝐾)
↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin |
528 | | rabfi 8185 |
. . . . . . . . . . . . . . . 16
⊢
((((0..^𝐾)
↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin) |
529 | 527, 528 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin |
530 | | rabfi 8185 |
. . . . . . . . . . . . . . . 16
⊢
((((0..^𝐾)
↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin) |
531 | 255, 530 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin |
532 | | hashen 13135 |
. . . . . . . . . . . . . . 15
⊢ (({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin ∧ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin) → ((#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ↔ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) |
533 | 529, 531,
532 | mp2an 708 |
. . . . . . . . . . . . . 14
⊢
((#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ↔ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) |
534 | 513, 533 | sylibr 224 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) |
535 | 508, 534 | eqtr4d 2659 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) |
536 | 535 | breq2d 4665 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
537 | 498, 536 | bitrd 268 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
538 | 537 | biimpd 219 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
539 | 538 | con3d 148 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
540 | 539 | expcom 451 |
. . . . . . 7
⊢ ((𝑚 ∈ ℕ0
∧ 𝑚 < 𝑁) → (𝜑 → (¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))) |
541 | 540 | a2d 29 |
. . . . . 6
⊢ ((𝑚 ∈ ℕ0
∧ 𝑚 < 𝑁) → ((𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) → (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))) |
542 | 541 | 3adant1 1079 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑚 ∈
ℕ0 ∧ 𝑚
< 𝑁) → ((𝜑 → ¬ 2 ∥
(#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) → (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))) |
543 | 107, 132,
157, 182, 240, 542 | fnn0ind 11476 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑁
≤ 𝑁) → (𝜑 → ¬ 2 ∥
(#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
544 | 5, 543 | mpcom 38 |
. . 3
⊢ (𝜑 → ¬ 2 ∥
(#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) |
545 | | dvds0 14997 |
. . . . . . . 8
⊢ (2 ∈
ℤ → 2 ∥ 0) |
546 | 241, 545 | ax-mp 5 |
. . . . . . 7
⊢ 2 ∥
0 |
547 | | hash0 13158 |
. . . . . . 7
⊢
(#‘∅) = 0 |
548 | 546, 547 | breqtrri 4680 |
. . . . . 6
⊢ 2 ∥
(#‘∅) |
549 | | fveq2 6191 |
. . . . . 6
⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) = (#‘∅)) |
550 | 548, 549 | syl5breqr 4691 |
. . . . 5
⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})) |
551 | 3 | ltp1d 10954 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 < (𝑁 + 1)) |
552 | 283 | peano2zd 11485 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) |
553 | | fzn 12357 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅)) |
554 | 552, 283,
553 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅)) |
555 | 551, 554 | mpbid 222 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑁 + 1)...𝑁) = ∅) |
556 | 555 | xpeq1d 5138 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = (∅ ×
{0})) |
557 | 556, 86 | syl6eq 2672 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = ∅) |
558 | 557 | uneq2d 3767 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) = (((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪
∅)) |
559 | | un0 3967 |
. . . . . . . . . . . . . 14
⊢
(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ ∅) =
((1st ‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) |
560 | 558, 559 | syl6eq 2672 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) = ((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))) |
561 | 560 | csbeq1d 3540 |
. . . . . . . . . . . 12
⊢ (𝜑 →
⦋(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵) |
562 | | ovex 6678 |
. . . . . . . . . . . . 13
⊢
((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V |
563 | | poimirlem28.1 |
. . . . . . . . . . . . 13
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) |
564 | 562, 563 | csbie 3559 |
. . . . . . . . . . . 12
⊢
⦋((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = 𝐶 |
565 | 561, 564 | syl6eq 2672 |
. . . . . . . . . . 11
⊢ (𝜑 →
⦋(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = 𝐶) |
566 | 565 | eqeq2d 2632 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = 𝐶)) |
567 | 566 | rexbidv 3052 |
. . . . . . . . 9
⊢ (𝜑 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) |
568 | 567 | ralbidv 2986 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) |
569 | 568 | rabbidv 3189 |
. . . . . . 7
⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) |
570 | 569 | fveq2d 6195 |
. . . . . 6
⊢ (𝜑 → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})) |
571 | 570 | breq2d 4665 |
. . . . 5
⊢ (𝜑 → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))) |
572 | 550, 571 | syl5ibr 236 |
. . . 4
⊢ (𝜑 → ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
573 | 572 | necon3bd 2808 |
. . 3
⊢ (𝜑 → (¬ 2 ∥
(#‘{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅)) |
574 | 544, 573 | mpd 15 |
. 2
⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅) |
575 | | rabn0 3958 |
. 2
⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅ ↔ ∃𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) |
576 | 574, 575 | sylib 208 |
1
⊢ (𝜑 → ∃𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) |