Step | Hyp | Ref
| Expression |
1 | | poimir.0 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℕ) |
2 | 1 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑁 ∈ ℕ) |
3 | | oveq1 6657 |
. . . . . . . . . . . . . 14
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) = (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘}))) |
4 | 3 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘}))) = (𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))) |
5 | 4 | fveq1d 6193 |
. . . . . . . . . . . 12
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏)) |
6 | 5 | breq2d 4665 |
. . . . . . . . . . 11
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏))) |
7 | | fveq1 6190 |
. . . . . . . . . . . 12
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝑝‘𝑏) = (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏)) |
8 | 7 | neeq1d 2853 |
. . . . . . . . . . 11
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝑝‘𝑏) ≠ 0 ↔ (((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)) |
9 | 6, 8 | anbi12d 747 |
. . . . . . . . . 10
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → ((0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))) |
10 | 9 | ralbidv 2986 |
. . . . . . . . 9
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))) |
11 | 10 | rabbidv 3189 |
. . . . . . . 8
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} = {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) |
12 | 11 | uneq2d 3767 |
. . . . . . 7
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) = ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)})) |
13 | 12 | supeq1d 8352 |
. . . . . 6
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → sup(({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) = sup(({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)) |
14 | 1 | nnnn0d 11351 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
15 | | 0elfz 12436 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ0
→ 0 ∈ (0...𝑁)) |
16 | 14, 15 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ∈ (0...𝑁)) |
17 | 16 | snssd 4340 |
. . . . . . . . 9
⊢ (𝜑 → {0} ⊆ (0...𝑁)) |
18 | | ssrab2 3687 |
. . . . . . . . . . 11
⊢ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ⊆ (1...𝑁) |
19 | | 1eluzge0 11732 |
. . . . . . . . . . . 12
⊢ 1 ∈
(ℤ≥‘0) |
20 | | fzss1 12380 |
. . . . . . . . . . . 12
⊢ (1 ∈
(ℤ≥‘0) → (1...𝑁) ⊆ (0...𝑁)) |
21 | 19, 20 | ax-mp 5 |
. . . . . . . . . . 11
⊢
(1...𝑁) ⊆
(0...𝑁) |
22 | 18, 21 | sstri 3612 |
. . . . . . . . . 10
⊢ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ⊆ (0...𝑁) |
23 | 22 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ⊆ (0...𝑁)) |
24 | 17, 23 | unssd 3789 |
. . . . . . . 8
⊢ (𝜑 → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ⊆ (0...𝑁)) |
25 | | ltso 10118 |
. . . . . . . . 9
⊢ < Or
ℝ |
26 | | snfi 8038 |
. . . . . . . . . . 11
⊢ {0}
∈ Fin |
27 | | fzfi 12771 |
. . . . . . . . . . . 12
⊢
(1...𝑁) ∈
Fin |
28 | | rabfi 8185 |
. . . . . . . . . . . 12
⊢
((1...𝑁) ∈ Fin
→ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ∈ Fin) |
29 | 27, 28 | ax-mp 5 |
. . . . . . . . . . 11
⊢ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ∈ Fin |
30 | | unfi 8227 |
. . . . . . . . . . 11
⊢ (({0}
∈ Fin ∧ {𝑎 ∈
(1...𝑁) ∣
∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ∈ Fin) → ({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ∈ Fin) |
31 | 26, 29, 30 | mp2an 708 |
. . . . . . . . . 10
⊢ ({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ∈ Fin |
32 | | c0ex 10034 |
. . . . . . . . . . . 12
⊢ 0 ∈
V |
33 | 32 | snid 4208 |
. . . . . . . . . . 11
⊢ 0 ∈
{0} |
34 | | elun1 3780 |
. . . . . . . . . . 11
⊢ (0 ∈
{0} → 0 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})) |
35 | | ne0i 3921 |
. . . . . . . . . . 11
⊢ (0 ∈
({0} ∪ {𝑎 ∈
(1...𝑁) ∣
∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ≠ ∅) |
36 | 33, 34, 35 | mp2b 10 |
. . . . . . . . . 10
⊢ ({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ≠ ∅ |
37 | | 0red 10041 |
. . . . . . . . . . . . 13
⊢ ((𝜑 → 𝑁 ∈ ℕ) → 0 ∈
ℝ) |
38 | 37 | snssd 4340 |
. . . . . . . . . . . 12
⊢ ((𝜑 → 𝑁 ∈ ℕ) → {0} ⊆
ℝ) |
39 | 1, 38 | ax-mp 5 |
. . . . . . . . . . 11
⊢ {0}
⊆ ℝ |
40 | | elfzelz 12342 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ) |
41 | 40 | ssriv 3607 |
. . . . . . . . . . . . 13
⊢
(1...𝑁) ⊆
ℤ |
42 | | zssre 11384 |
. . . . . . . . . . . . 13
⊢ ℤ
⊆ ℝ |
43 | 41, 42 | sstri 3612 |
. . . . . . . . . . . 12
⊢
(1...𝑁) ⊆
ℝ |
44 | 18, 43 | sstri 3612 |
. . . . . . . . . . 11
⊢ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ⊆ ℝ |
45 | 39, 44 | unssi 3788 |
. . . . . . . . . 10
⊢ ({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ⊆ ℝ |
46 | 31, 36, 45 | 3pm3.2i 1239 |
. . . . . . . . 9
⊢ (({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ∈ Fin ∧ ({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ≠ ∅ ∧ ({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ⊆ ℝ) |
47 | | fisupcl 8375 |
. . . . . . . . 9
⊢ (( <
Or ℝ ∧ (({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ∈ Fin ∧ ({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ≠ ∅ ∧ ({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ⊆ ℝ)) → sup(({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈ ({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})) |
48 | 25, 46, 47 | mp2an 708 |
. . . . . . . 8
⊢ sup(({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈ ({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) |
49 | | ssel 3597 |
. . . . . . . 8
⊢ (({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ⊆ (0...𝑁) → (sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈ ({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈
(0...𝑁))) |
50 | 24, 48, 49 | mpisyl 21 |
. . . . . . 7
⊢ (𝜑 → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈
(0...𝑁)) |
51 | 50 | ad2antrr 762 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑝:(1...𝑁)⟶(0...𝑘)) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈
(0...𝑁)) |
52 | | elfznn 12370 |
. . . . . . . . 9
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ) |
53 | | nngt0 11049 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 0 <
𝑛) |
54 | 53 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) → 0 < 𝑛) |
55 | | simpr 477 |
. . . . . . . . . . . . . 14
⊢ ((0 ≤
((𝐹‘(𝑝 ∘𝑓 /
((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → (𝑝‘𝑏) ≠ 0) |
56 | 55 | ralimi 2952 |
. . . . . . . . . . . . 13
⊢
(∀𝑏 ∈
(1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0) |
57 | | elfznn 12370 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ (1...𝑁) → 𝑠 ∈ ℕ) |
58 | | nnre 11027 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) |
59 | | nnre 11027 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ ℕ → 𝑠 ∈
ℝ) |
60 | | lenlt 10116 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (𝑛 ≤ 𝑠 ↔ ¬ 𝑠 < 𝑛)) |
61 | 58, 59, 60 | syl2an 494 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑛 ≤ 𝑠 ↔ ¬ 𝑠 < 𝑛)) |
62 | | elfz1b 12409 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ (1...𝑠) ↔ (𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ∧ 𝑛 ≤ 𝑠)) |
63 | 62 | biimpri 218 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ∧ 𝑛 ≤ 𝑠) → 𝑛 ∈ (1...𝑠)) |
64 | 63 | 3expia 1267 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑛 ≤ 𝑠 → 𝑛 ∈ (1...𝑠))) |
65 | 61, 64 | sylbird 250 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (¬
𝑠 < 𝑛 → 𝑛 ∈ (1...𝑠))) |
66 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑏 = 𝑛 → (𝑝‘𝑏) = (𝑝‘𝑛)) |
67 | 66 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑏 = 𝑛 → ((𝑝‘𝑏) = 0 ↔ (𝑝‘𝑛) = 0)) |
68 | 67 | rspcev 3309 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ (1...𝑠) ∧ (𝑝‘𝑛) = 0) → ∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0) |
69 | 68 | expcom 451 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑝‘𝑛) = 0 → (𝑛 ∈ (1...𝑠) → ∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0)) |
70 | 65, 69 | sylan9 689 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) ∧ (𝑝‘𝑛) = 0) → (¬ 𝑠 < 𝑛 → ∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0)) |
71 | 70 | an32s 846 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) ∧ 𝑠 ∈ ℕ) → (¬ 𝑠 < 𝑛 → ∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0)) |
72 | | nne 2798 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬
(𝑝‘𝑏) ≠ 0 ↔ (𝑝‘𝑏) = 0) |
73 | 72 | rexbii 3041 |
. . . . . . . . . . . . . . . . 17
⊢
(∃𝑏 ∈
(1...𝑠) ¬ (𝑝‘𝑏) ≠ 0 ↔ ∃𝑏 ∈ (1...𝑠)(𝑝‘𝑏) = 0) |
74 | | rexnal 2995 |
. . . . . . . . . . . . . . . . 17
⊢
(∃𝑏 ∈
(1...𝑠) ¬ (𝑝‘𝑏) ≠ 0 ↔ ¬ ∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0) |
75 | 73, 74 | bitr3i 266 |
. . . . . . . . . . . . . . . 16
⊢
(∃𝑏 ∈
(1...𝑠)(𝑝‘𝑏) = 0 ↔ ¬ ∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0) |
76 | 71, 75 | syl6ib 241 |
. . . . . . . . . . . . . . 15
⊢ (((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) ∧ 𝑠 ∈ ℕ) → (¬ 𝑠 < 𝑛 → ¬ ∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0)) |
77 | 76 | con4d 114 |
. . . . . . . . . . . . . 14
⊢ (((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) ∧ 𝑠 ∈ ℕ) → (∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0 → 𝑠 < 𝑛)) |
78 | 57, 77 | sylan2 491 |
. . . . . . . . . . . . 13
⊢ (((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) ∧ 𝑠 ∈ (1...𝑁)) → (∀𝑏 ∈ (1...𝑠)(𝑝‘𝑏) ≠ 0 → 𝑠 < 𝑛)) |
79 | 56, 78 | syl5 34 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) ∧ 𝑠 ∈ (1...𝑁)) → (∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → 𝑠 < 𝑛)) |
80 | 79 | ralrimiva 2966 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) → ∀𝑠 ∈ (1...𝑁)(∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → 𝑠 < 𝑛)) |
81 | | ralunb 3794 |
. . . . . . . . . . . 12
⊢
(∀𝑠 ∈
({0} ∪ {𝑎 ∈
(1...𝑁) ∣
∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑠 < 𝑛 ↔ (∀𝑠 ∈ {0}𝑠 < 𝑛 ∧ ∀𝑠 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}𝑠 < 𝑛)) |
82 | | breq1 4656 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 0 → (𝑠 < 𝑛 ↔ 0 < 𝑛)) |
83 | 32, 82 | ralsn 4222 |
. . . . . . . . . . . . 13
⊢
(∀𝑠 ∈
{0}𝑠 < 𝑛 ↔ 0 < 𝑛) |
84 | | oveq2 6658 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑠 → (1...𝑎) = (1...𝑠)) |
85 | 84 | raleqdv 3144 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑠 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) |
86 | 85 | ralrab 3368 |
. . . . . . . . . . . . 13
⊢
(∀𝑠 ∈
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}𝑠 < 𝑛 ↔ ∀𝑠 ∈ (1...𝑁)(∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → 𝑠 < 𝑛)) |
87 | 83, 86 | anbi12i 733 |
. . . . . . . . . . . 12
⊢
((∀𝑠 ∈
{0}𝑠 < 𝑛 ∧ ∀𝑠 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}𝑠 < 𝑛) ↔ (0 < 𝑛 ∧ ∀𝑠 ∈ (1...𝑁)(∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → 𝑠 < 𝑛))) |
88 | 81, 87 | bitri 264 |
. . . . . . . . . . 11
⊢
(∀𝑠 ∈
({0} ∪ {𝑎 ∈
(1...𝑁) ∣
∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑠 < 𝑛 ↔ (0 < 𝑛 ∧ ∀𝑠 ∈ (1...𝑁)(∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → 𝑠 < 𝑛))) |
89 | 54, 80, 88 | sylanbrc 698 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) → ∀𝑠 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑠 < 𝑛) |
90 | | breq1 4656 |
. . . . . . . . . . 11
⊢ (𝑠 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) → (𝑠 < 𝑛 ↔ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < 𝑛)) |
91 | 90 | rspcva 3307 |
. . . . . . . . . 10
⊢
((sup(({0} ∪ {𝑎
∈ (1...𝑁) ∣
∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈ ({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ∧ ∀𝑠 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑠 < 𝑛) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < 𝑛) |
92 | 48, 89, 91 | sylancr 695 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ (𝑝‘𝑛) = 0) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < 𝑛) |
93 | 52, 92 | sylan 488 |
. . . . . . . 8
⊢ ((𝑛 ∈ (1...𝑁) ∧ (𝑝‘𝑛) = 0) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < 𝑛) |
94 | 93 | 3adant2 1080 |
. . . . . . 7
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 0) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < 𝑛) |
95 | 94 | adantl 482 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 0)) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < 𝑛) |
96 | 40 | zred 11482 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ) |
97 | 96 | 3ad2ant1 1082 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) → 𝑛 ∈ ℝ) |
98 | 97 | adantl 482 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 𝑛 ∈ ℝ) |
99 | | simpr1 1067 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 𝑛 ∈ (1...𝑁)) |
100 | | simpll 790 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 𝜑) |
101 | | simplr 792 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → 𝑘 ∈ ℕ) |
102 | | elfzelz 12342 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0...𝑘) → 𝑖 ∈ ℤ) |
103 | 102 | zred 11482 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0...𝑘) → 𝑖 ∈ ℝ) |
104 | | nndivre 11056 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ) |
105 | 103, 104 | sylan 488 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ) |
106 | | elfzle1 12344 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0...𝑘) → 0 ≤ 𝑖) |
107 | 103, 106 | jca 554 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0...𝑘) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖)) |
108 | | nnrp 11842 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℝ+) |
109 | 108 | rpregt0d 11878 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 <
𝑘)) |
110 | | divge0 10892 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑖 ∈ ℝ ∧ 0 ≤
𝑖) ∧ (𝑘 ∈ ℝ ∧ 0 <
𝑘)) → 0 ≤ (𝑖 / 𝑘)) |
111 | 107, 109,
110 | syl2an 494 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑖 / 𝑘)) |
112 | | elfzle2 12345 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0...𝑘) → 𝑖 ≤ 𝑘) |
113 | 112 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ≤ 𝑘) |
114 | 103 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ∈ ℝ) |
115 | | 1red 10055 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈
ℝ) |
116 | 108 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+) |
117 | 114, 115,
116 | ledivmuld 11925 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ (𝑘 · 1))) |
118 | | nncn 11028 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℂ) |
119 | 118 | mulid1d 10057 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘) |
120 | 119 | breq2d 4665 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ ℕ → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖 ≤ 𝑘)) |
121 | 120 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖 ≤ 𝑘)) |
122 | 117, 121 | bitrd 268 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ 𝑘)) |
123 | 113, 122 | mpbird 247 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ≤ 1) |
124 | | 0re 10040 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 0 ∈
ℝ |
125 | | 1re 10039 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 1 ∈
ℝ |
126 | 124, 125 | elicc2i 12239 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 / 𝑘) ∈ (0[,]1) ↔ ((𝑖 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑖 / 𝑘) ∧ (𝑖 / 𝑘) ≤ 1)) |
127 | 105, 111,
123, 126 | syl3anbrc 1246 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ (0[,]1)) |
128 | 127 | ancoms 469 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑘)) → (𝑖 / 𝑘) ∈ (0[,]1)) |
129 | | elsni 4194 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ {𝑘} → 𝑗 = 𝑘) |
130 | 129 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) = (𝑖 / 𝑘)) |
131 | 130 | eleq1d 2686 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ {𝑘} → ((𝑖 / 𝑗) ∈ (0[,]1) ↔ (𝑖 / 𝑘) ∈ (0[,]1))) |
132 | 128, 131 | syl5ibrcom 237 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑘)) → (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) ∈ (0[,]1))) |
133 | 132 | impr 649 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ ℕ ∧ (𝑖 ∈ (0...𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1)) |
134 | 101, 133 | sylan 488 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) ∧ (𝑖 ∈ (0...𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1)) |
135 | | simprr 796 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → 𝑝:(1...𝑁)⟶(0...𝑘)) |
136 | | vex 3203 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑘 ∈ V |
137 | 136 | fconst 6091 |
. . . . . . . . . . . . . . . . 17
⊢
((1...𝑁) ×
{𝑘}):(1...𝑁)⟶{𝑘} |
138 | 137 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘}) |
139 | | fzfid 12772 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → (1...𝑁) ∈ Fin) |
140 | | inidm 3822 |
. . . . . . . . . . . . . . . 16
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
141 | 134, 135,
138, 139, 139, 140 | off 6912 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1)) |
142 | | poimir.i |
. . . . . . . . . . . . . . . . 17
⊢ 𝐼 = ((0[,]1)
↑𝑚 (1...𝑁)) |
143 | 142 | eleq2i 2693 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑝 ∘𝑓 /
((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ ((0[,]1)
↑𝑚 (1...𝑁))) |
144 | | ovex 6678 |
. . . . . . . . . . . . . . . . 17
⊢ (0[,]1)
∈ V |
145 | | ovex 6678 |
. . . . . . . . . . . . . . . . 17
⊢
(1...𝑁) ∈
V |
146 | 144, 145 | elmap 7886 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑝 ∘𝑓 /
((1...𝑁) × {𝑘})) ∈ ((0[,]1)
↑𝑚 (1...𝑁)) ↔ (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1)) |
147 | 143, 146 | bitri 264 |
. . . . . . . . . . . . . . 15
⊢ ((𝑝 ∘𝑓 /
((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1)) |
148 | 141, 147 | sylibr 224 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼) |
149 | 148 | 3adantr3 1222 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼) |
150 | | 3anass 1042 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘))) |
151 | | ancom 466 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ (1...𝑁) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) ↔ ((𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) ∧ 𝑛 ∈ (1...𝑁))) |
152 | 150, 151 | bitri 264 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) ↔ ((𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) ∧ 𝑛 ∈ (1...𝑁))) |
153 | | ffn 6045 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝:(1...𝑁)⟶(0...𝑘) → 𝑝 Fn (1...𝑁)) |
154 | 153 | ad2antrl 764 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 𝑝 Fn (1...𝑁)) |
155 | | fnconstg 6093 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ V → ((1...𝑁) × {𝑘}) Fn (1...𝑁)) |
156 | 136, 155 | mp1i 13 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((1...𝑁) × {𝑘}) Fn (1...𝑁)) |
157 | | fzfid 12772 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → (1...𝑁) ∈ Fin) |
158 | | simplrr 801 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑝‘𝑛) = 𝑘) |
159 | 136 | fvconst2 6469 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘) |
160 | 159 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘) |
161 | 154, 156,
157, 157, 140, 158, 160 | ofval 6906 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑝 ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = (𝑘 / 𝑘)) |
162 | 161 | anasss 679 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ((𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) ∧ 𝑛 ∈ (1...𝑁))) → ((𝑝 ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = (𝑘 / 𝑘)) |
163 | 152, 162 | sylan2b 492 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((𝑝 ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = (𝑘 / 𝑘)) |
164 | | nnne0 11053 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0) |
165 | 118, 164 | dividd 10799 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ → (𝑘 / 𝑘) = 1) |
166 | 165 | ad2antlr 763 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → (𝑘 / 𝑘) = 1) |
167 | 163, 166 | eqtrd 2656 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((𝑝 ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 1) |
168 | | ovex 6678 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∘𝑓 /
((1...𝑁) × {𝑘})) ∈ V |
169 | | eleq1 2689 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) → (𝑧 ∈ 𝐼 ↔ (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼)) |
170 | | fveq1 6190 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) → (𝑧‘𝑛) = ((𝑝 ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑛)) |
171 | 170 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) → ((𝑧‘𝑛) = 1 ↔ ((𝑝 ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 1)) |
172 | 169, 171 | 3anbi23d 1402 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ∧ ((𝑝 ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 1))) |
173 | 172 | anbi2d 740 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ∧ ((𝑝 ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 1)))) |
174 | | fveq2 6191 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) → (𝐹‘𝑧) = (𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))) |
175 | 174 | fveq1d 6193 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) → ((𝐹‘𝑧)‘𝑛) = ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)) |
176 | 175 | breq2d 4665 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) → (0 ≤ ((𝐹‘𝑧)‘𝑛) ↔ 0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))) |
177 | 173, 176 | imbi12d 334 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛)) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ∧ ((𝑝 ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 1)) → 0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))) |
178 | | poimir.3 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛)) |
179 | 168, 177,
178 | vtocl 3259 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ∧ ((𝑝 ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 1)) → 0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)) |
180 | 100, 99, 149, 167, 179 | syl13anc 1328 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → 0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)) |
181 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) |
182 | | simp3 1063 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘) → (𝑝‘𝑛) = 𝑘) |
183 | | neeq1 2856 |
. . . . . . . . . . . . . . 15
⊢ ((𝑝‘𝑛) = 𝑘 → ((𝑝‘𝑛) ≠ 0 ↔ 𝑘 ≠ 0)) |
184 | 164, 183 | syl5ibrcom 237 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → ((𝑝‘𝑛) = 𝑘 → (𝑝‘𝑛) ≠ 0)) |
185 | 184 | imp 445 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ ∧ (𝑝‘𝑛) = 𝑘) → (𝑝‘𝑛) ≠ 0) |
186 | 181, 182,
185 | syl2an 494 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → (𝑝‘𝑛) ≠ 0) |
187 | | vex 3203 |
. . . . . . . . . . . . 13
⊢ 𝑛 ∈ V |
188 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑛 → ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)) |
189 | 188 | breq2d 4665 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = 𝑛 → (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))) |
190 | 66 | neeq1d 2853 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = 𝑛 → ((𝑝‘𝑏) ≠ 0 ↔ (𝑝‘𝑛) ≠ 0)) |
191 | 189, 190 | anbi12d 747 |
. . . . . . . . . . . . 13
⊢ (𝑏 = 𝑛 → ((0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑝‘𝑛) ≠ 0))) |
192 | 187, 191 | ralsn 4222 |
. . . . . . . . . . . 12
⊢
(∀𝑏 ∈
{𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑝‘𝑛) ≠ 0)) |
193 | 180, 186,
192 | sylanbrc 698 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) |
194 | 40 | zcnd 11483 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ) |
195 | | 1cnd 10056 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ (1...𝑁) → 1 ∈ ℂ) |
196 | 194, 195 | subeq0ad 10402 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1)) |
197 | 196 | biimpcd 239 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 − 1) = 0 → (𝑛 ∈ (1...𝑁) → 𝑛 = 1)) |
198 | | 1z 11407 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 1 ∈
ℤ |
199 | | fzsn 12383 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (1 ∈
ℤ → (1...1) = {1}) |
200 | 198, 199 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (1...1) =
{1} |
201 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 1 → (1...𝑛) = (1...1)) |
202 | | sneq 4187 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 1 → {𝑛} = {1}) |
203 | 200, 201,
202 | 3eqtr4a 2682 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 1 → (1...𝑛) = {𝑛}) |
204 | 203 | raleqdv 3144 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 1 → (∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) |
205 | 204 | biimprd 238 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 1 → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) |
206 | 197, 205 | syl6 35 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 − 1) = 0 → (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)))) |
207 | | ralun 3795 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑏 ∈
(1...(𝑛 − 1))(0 ≤
((𝐹‘(𝑝 ∘𝑓 /
((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) → ∀𝑏 ∈ ((1...(𝑛 − 1)) ∪ {𝑛})(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) |
208 | | npcan1 10455 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛) |
209 | 194, 208 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) = 𝑛) |
210 | | elfzuz 12338 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈
(ℤ≥‘1)) |
211 | 209, 210 | eqeltrd 2701 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈
(ℤ≥‘1)) |
212 | | peano2zm 11420 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 ∈ ℤ → (𝑛 − 1) ∈
ℤ) |
213 | | uzid 11702 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑛 − 1) ∈ ℤ
→ (𝑛 − 1) ∈
(ℤ≥‘(𝑛 − 1))) |
214 | | peano2uz 11741 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑛 − 1) ∈
(ℤ≥‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈
(ℤ≥‘(𝑛 − 1))) |
215 | 40, 212, 213, 214 | 4syl 19 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈
(ℤ≥‘(𝑛 − 1))) |
216 | 209, 215 | eqeltrrd 2702 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ≥‘(𝑛 − 1))) |
217 | | fzsplit2 12366 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑛 − 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑛 ∈ (ℤ≥‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛))) |
218 | 211, 216,
217 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛))) |
219 | 209 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛)) |
220 | | fzsn 12383 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛}) |
221 | 40, 220 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 ∈ (1...𝑁) → (𝑛...𝑛) = {𝑛}) |
222 | 219, 221 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = {𝑛}) |
223 | 222 | uneq2d 3767 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 ∈ (1...𝑁) → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛})) |
224 | 218, 223 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛})) |
225 | 224 | raleqdv 3144 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ ((1...(𝑛 − 1)) ∪ {𝑛})(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) |
226 | 207, 225 | syl5ibr 236 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ (1...𝑁) → ((∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) |
227 | 226 | expd 452 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)))) |
228 | 227 | com12 32 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑏 ∈
(1...(𝑛 − 1))(0 ≤
((𝐹‘(𝑝 ∘𝑓 /
((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)))) |
229 | 228 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) → (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)))) |
230 | 206, 229 | jaoi 394 |
. . . . . . . . . . . . . . 15
⊢ (((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) → (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)))) |
231 | 230 | imdistand 728 |
. . . . . . . . . . . . . 14
⊢ (((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) → ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) → (𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)))) |
232 | 231 | com12 32 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) → (((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) → (𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)))) |
233 | | elun 3753 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 − 1) ∈ ({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ↔ ((𝑛 − 1) ∈ {0} ∨ (𝑛 − 1) ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})) |
234 | | ovex 6678 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 − 1) ∈
V |
235 | 234 | elsn 4192 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 − 1) ∈ {0} ↔
(𝑛 − 1) =
0) |
236 | | oveq2 6658 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = (𝑛 − 1) → (1...𝑎) = (1...(𝑛 − 1))) |
237 | 236 | raleqdv 3144 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = (𝑛 − 1) → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) |
238 | 237 | elrab 3363 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 − 1) ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ↔ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) |
239 | 235, 238 | orbi12i 543 |
. . . . . . . . . . . . . 14
⊢ (((𝑛 − 1) ∈ {0} ∨
(𝑛 − 1) ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ↔ ((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)))) |
240 | 233, 239 | bitri 264 |
. . . . . . . . . . . . 13
⊢ ((𝑛 − 1) ∈ ({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ↔ ((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)))) |
241 | | oveq2 6658 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑛 → (1...𝑎) = (1...𝑛)) |
242 | 241 | raleqdv 3144 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑛 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) |
243 | 242 | elrab 3363 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} ↔ (𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0))) |
244 | 232, 240,
243 | 3imtr4g 285 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) → 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})) |
245 | | elun2 3781 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)} → 𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})) |
246 | 244, 245 | syl6 35 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) → 𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}))) |
247 | 99, 193, 246 | syl2anc 693 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) → 𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}))) |
248 | | fimaxre2 10969 |
. . . . . . . . . . . . 13
⊢ ((({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ⊆ ℝ ∧ ({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ∈ Fin) → ∃𝑖 ∈ ℝ ∀𝑗 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑗 ≤ 𝑖) |
249 | 45, 31, 248 | mp2an 708 |
. . . . . . . . . . . 12
⊢
∃𝑖 ∈
ℝ ∀𝑗 ∈
({0} ∪ {𝑎 ∈
(1...𝑁) ∣
∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑗 ≤ 𝑖 |
250 | 45, 36, 249 | 3pm3.2i 1239 |
. . . . . . . . . . 11
⊢ (({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ⊆ ℝ ∧ ({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ≠ ∅ ∧ ∃𝑖 ∈ ℝ ∀𝑗 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})𝑗 ≤ 𝑖) |
251 | 250 | suprubii 10998 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) → 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, <
)) |
252 | 247, 251 | syl6 35 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) → 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, <
))) |
253 | | ltm1 10863 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛) |
254 | | peano2rem 10348 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈
ℝ) |
255 | 45, 48 | sselii 3600 |
. . . . . . . . . . . 12
⊢ sup(({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈
ℝ |
256 | | ltletr 10129 |
. . . . . . . . . . . 12
⊢ (((𝑛 − 1) ∈ ℝ ∧
𝑛 ∈ ℝ ∧
sup(({0} ∪ {𝑎 ∈
(1...𝑁) ∣
∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈ ℝ)
→ (((𝑛 − 1) <
𝑛 ∧ 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < )) → (𝑛 − 1) < sup(({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, <
))) |
257 | 255, 256 | mp3an3 1413 |
. . . . . . . . . . 11
⊢ (((𝑛 − 1) ∈ ℝ ∧
𝑛 ∈ ℝ) →
(((𝑛 − 1) < 𝑛 ∧ 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < )) → (𝑛 − 1) < sup(({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, <
))) |
258 | 254, 257 | mpancom 703 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℝ → (((𝑛 − 1) < 𝑛 ∧ 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < )) → (𝑛 − 1) < sup(({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, <
))) |
259 | 253, 258 | mpand 711 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℝ → (𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) → (𝑛 − 1) < sup(({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, <
))) |
260 | 98, 252, 259 | sylsyld 61 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) → (𝑛 − 1) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, <
))) |
261 | 255 | ltnri 10146 |
. . . . . . . . . 10
⊢ ¬
sup(({0} ∪ {𝑎 ∈
(1...𝑁) ∣
∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < sup(({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) |
262 | | breq1 4656 |
. . . . . . . . . 10
⊢ (sup(({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) = (𝑛 − 1) → (sup(({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) < sup(({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ↔ (𝑛 − 1) < sup(({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, <
))) |
263 | 261, 262 | mtbii 316 |
. . . . . . . . 9
⊢ (sup(({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) = (𝑛 − 1) → ¬ (𝑛 − 1) < sup(({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, <
)) |
264 | 263 | necon2ai 2823 |
. . . . . . . 8
⊢ ((𝑛 − 1) < sup(({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) → sup(({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ≠ (𝑛 − 1)) |
265 | 260, 264 | syl6 35 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ≠ (𝑛 − 1))) |
266 | | eleq1 2689 |
. . . . . . . . 9
⊢ (sup(({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) = (𝑛 − 1) → (sup(({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ∈ ({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) ↔ (𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}))) |
267 | 48, 266 | mpbii 223 |
. . . . . . . 8
⊢ (sup(({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) = (𝑛 − 1) → (𝑛 − 1) ∈ ({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)})) |
268 | 267 | necon3bi 2820 |
. . . . . . 7
⊢ (¬
(𝑛 − 1) ∈ ({0}
∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ≠ (𝑛 − 1)) |
269 | 265, 268 | pm2.61d1 171 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝‘𝑛) = 𝑘)) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝 ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝‘𝑏) ≠ 0)}), ℝ, < ) ≠ (𝑛 − 1)) |
270 | 2, 13, 51, 95, 269, 181 | poimirlem28 33437 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∃𝑠 ∈ (((0..^𝑘) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)) |
271 | | nn0ex 11298 |
. . . . . . . . . . . 12
⊢
ℕ0 ∈ V |
272 | | fzo0ssnn0 12548 |
. . . . . . . . . . . 12
⊢
(0..^𝑘) ⊆
ℕ0 |
273 | | mapss 7900 |
. . . . . . . . . . . 12
⊢
((ℕ0 ∈ V ∧ (0..^𝑘) ⊆ ℕ0) →
((0..^𝑘)
↑𝑚 (1...𝑁)) ⊆ (ℕ0
↑𝑚 (1...𝑁))) |
274 | 271, 272,
273 | mp2an 708 |
. . . . . . . . . . 11
⊢
((0..^𝑘)
↑𝑚 (1...𝑁)) ⊆ (ℕ0
↑𝑚 (1...𝑁)) |
275 | | xpss1 5228 |
. . . . . . . . . . 11
⊢
(((0..^𝑘)
↑𝑚 (1...𝑁)) ⊆ (ℕ0
↑𝑚 (1...𝑁)) → (((0..^𝑘) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ⊆ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
276 | 274, 275 | ax-mp 5 |
. . . . . . . . . 10
⊢
(((0..^𝑘)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ⊆ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
277 | 276 | sseli 3599 |
. . . . . . . . 9
⊢ (𝑠 ∈ (((0..^𝑘) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → 𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
278 | | xp1st 7198 |
. . . . . . . . . 10
⊢ (𝑠 ∈ (((0..^𝑘) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘𝑠) ∈ ((0..^𝑘) ↑𝑚
(1...𝑁))) |
279 | | elmapi 7879 |
. . . . . . . . . 10
⊢
((1st ‘𝑠) ∈ ((0..^𝑘) ↑𝑚 (1...𝑁)) → (1st
‘𝑠):(1...𝑁)⟶(0..^𝑘)) |
280 | | frn 6053 |
. . . . . . . . . 10
⊢
((1st ‘𝑠):(1...𝑁)⟶(0..^𝑘) → ran (1st ‘𝑠) ⊆ (0..^𝑘)) |
281 | 278, 279,
280 | 3syl 18 |
. . . . . . . . 9
⊢ (𝑠 ∈ (((0..^𝑘) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ran (1st ‘𝑠) ⊆ (0..^𝑘)) |
282 | 277, 281 | jca 554 |
. . . . . . . 8
⊢ (𝑠 ∈ (((0..^𝑘) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ran (1st ‘𝑠) ⊆ (0..^𝑘))) |
283 | 282 | anim1i 592 |
. . . . . . 7
⊢ ((𝑠 ∈ (((0..^𝑘) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ((𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ran (1st ‘𝑠) ⊆ (0..^𝑘)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
284 | | anass 681 |
. . . . . . 7
⊢ (((𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ran (1st ‘𝑠) ⊆ (0..^𝑘)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) ↔ (𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (ran (1st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)))) |
285 | 283, 284 | sylib 208 |
. . . . . 6
⊢ ((𝑠 ∈ (((0..^𝑘) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → (𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (ran (1st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)))) |
286 | 285 | reximi2 3010 |
. . . . 5
⊢
(∃𝑠 ∈
(((0..^𝑘)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) →
∃𝑠 ∈
((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
287 | 270, 286 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∃𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
288 | 287 | ralrimiva 2966 |
. . 3
⊢ (𝜑 → ∀𝑘 ∈ ℕ ∃𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
289 | | nnex 11026 |
. . . 4
⊢ ℕ
∈ V |
290 | 145, 271 | ixpconst 7918 |
. . . . . . 7
⊢ X𝑛 ∈
(1...𝑁)ℕ0
= (ℕ0 ↑𝑚 (1...𝑁)) |
291 | | omelon 8543 |
. . . . . . . . . 10
⊢ ω
∈ On |
292 | | nn0ennn 12778 |
. . . . . . . . . . 11
⊢
ℕ0 ≈ ℕ |
293 | | nnenom 12779 |
. . . . . . . . . . 11
⊢ ℕ
≈ ω |
294 | 292, 293 | entr2i 8011 |
. . . . . . . . . 10
⊢ ω
≈ ℕ0 |
295 | | isnumi 8772 |
. . . . . . . . . 10
⊢ ((ω
∈ On ∧ ω ≈ ℕ0) → ℕ0
∈ dom card) |
296 | 291, 294,
295 | mp2an 708 |
. . . . . . . . 9
⊢
ℕ0 ∈ dom card |
297 | 296 | rgenw 2924 |
. . . . . . . 8
⊢
∀𝑛 ∈
(1...𝑁)ℕ0
∈ dom card |
298 | | finixpnum 33394 |
. . . . . . . 8
⊢
(((1...𝑁) ∈ Fin
∧ ∀𝑛 ∈
(1...𝑁)ℕ0
∈ dom card) → X𝑛 ∈ (1...𝑁)ℕ0 ∈ dom
card) |
299 | 27, 297, 298 | mp2an 708 |
. . . . . . 7
⊢ X𝑛 ∈
(1...𝑁)ℕ0
∈ dom card |
300 | 290, 299 | eqeltrri 2698 |
. . . . . 6
⊢
(ℕ0 ↑𝑚 (1...𝑁)) ∈ dom card |
301 | 145, 145 | mapval 7869 |
. . . . . . . . 9
⊢
((1...𝑁)
↑𝑚 (1...𝑁)) = {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} |
302 | | mapfi 8262 |
. . . . . . . . . 10
⊢
(((1...𝑁) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((1...𝑁)
↑𝑚 (1...𝑁)) ∈ Fin) |
303 | 27, 27, 302 | mp2an 708 |
. . . . . . . . 9
⊢
((1...𝑁)
↑𝑚 (1...𝑁)) ∈ Fin |
304 | 301, 303 | eqeltrri 2698 |
. . . . . . . 8
⊢ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} ∈ Fin |
305 | | f1of 6137 |
. . . . . . . . 9
⊢ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁)) |
306 | 305 | ss2abi 3674 |
. . . . . . . 8
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} |
307 | | ssfi 8180 |
. . . . . . . 8
⊢ (({𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}) → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) |
308 | 304, 306,
307 | mp2an 708 |
. . . . . . 7
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin |
309 | | finnum 8774 |
. . . . . . 7
⊢ ({𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ dom card) |
310 | 308, 309 | ax-mp 5 |
. . . . . 6
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ dom card |
311 | | xpnum 8777 |
. . . . . 6
⊢
(((ℕ0 ↑𝑚 (1...𝑁)) ∈ dom card ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ dom card) →
((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ dom card) |
312 | 300, 310,
311 | mp2an 708 |
. . . . 5
⊢
((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ dom card |
313 | | ssrab2 3687 |
. . . . . . . 8
⊢ {𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st
‘𝑠) ⊆
(0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆
((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
314 | 313 | rgenw 2924 |
. . . . . . 7
⊢
∀𝑘 ∈
ℕ {𝑠 ∈
((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st
‘𝑠) ⊆
(0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆
((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
315 | | ss2iun 4536 |
. . . . . . 7
⊢
(∀𝑘 ∈
ℕ {𝑠 ∈
((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st
‘𝑠) ⊆
(0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆
((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st
‘𝑠) ⊆
(0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆
∪ 𝑘 ∈ ℕ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
316 | 314, 315 | ax-mp 5 |
. . . . . 6
⊢ ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st
‘𝑠) ⊆
(0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆
∪ 𝑘 ∈ ℕ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
317 | | 1nn 11031 |
. . . . . . 7
⊢ 1 ∈
ℕ |
318 | | ne0i 3921 |
. . . . . . 7
⊢ (1 ∈
ℕ → ℕ ≠ ∅) |
319 | | iunconst 4529 |
. . . . . . 7
⊢ (ℕ
≠ ∅ → ∪ 𝑘 ∈ ℕ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
320 | 317, 318,
319 | mp2b 10 |
. . . . . 6
⊢ ∪ 𝑘 ∈ ℕ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
321 | 316, 320 | sseqtri 3637 |
. . . . 5
⊢ ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st
‘𝑠) ⊆
(0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆
((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
322 | | ssnum 8862 |
. . . . 5
⊢
((((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ dom card ∧ ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st
‘𝑠) ⊆
(0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆
((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st
‘𝑠) ⊆
(0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ∈ dom
card) |
323 | 312, 321,
322 | mp2an 708 |
. . . 4
⊢ ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st
‘𝑠) ⊆
(0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ∈ dom
card |
324 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑠 = (𝑔‘𝑘) → (1st ‘𝑠) = (1st
‘(𝑔‘𝑘))) |
325 | 324 | rneqd 5353 |
. . . . . . 7
⊢ (𝑠 = (𝑔‘𝑘) → ran (1st ‘𝑠) = ran (1st
‘(𝑔‘𝑘))) |
326 | 325 | sseq1d 3632 |
. . . . . 6
⊢ (𝑠 = (𝑔‘𝑘) → (ran (1st ‘𝑠) ⊆ (0..^𝑘) ↔ ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘))) |
327 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 = (𝑔‘𝑘) → (2nd ‘𝑠) = (2nd
‘(𝑔‘𝑘))) |
328 | 327 | imaeq1d 5465 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 = (𝑔‘𝑘) → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd
‘(𝑔‘𝑘)) “ (1...𝑗))) |
329 | 328 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = (𝑔‘𝑘) → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) =
(((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1})) |
330 | 327 | imaeq1d 5465 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 = (𝑔‘𝑘) → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁))) |
331 | 330 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = (𝑔‘𝑘) → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})) |
332 | 329, 331 | uneq12d 3768 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 = (𝑔‘𝑘) → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑁)) × {0})) =
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
333 | 324, 332 | oveq12d 6668 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 = (𝑔‘𝑘) → ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
334 | 333 | oveq1d 6665 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 = (𝑔‘𝑘) → (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})) = (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘}))) |
335 | 334 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 = (𝑔‘𝑘) → (𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘}))) = (𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))) |
336 | 335 | fveq1d 6193 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = (𝑔‘𝑘) → ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏)) |
337 | 336 | breq2d 4665 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = (𝑔‘𝑘) → (0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏))) |
338 | 333 | fveq1d 6193 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = (𝑔‘𝑘) → (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) = (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏)) |
339 | 338 | neeq1d 2853 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = (𝑔‘𝑘) → ((((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0 ↔ (((1st
‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)) |
340 | 337, 339 | anbi12d 747 |
. . . . . . . . . . . . 13
⊢ (𝑠 = (𝑔‘𝑘) → ((0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))) |
341 | 340 | ralbidv 2986 |
. . . . . . . . . . . 12
⊢ (𝑠 = (𝑔‘𝑘) → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))) |
342 | 341 | rabbidv 3189 |
. . . . . . . . . . 11
⊢ (𝑠 = (𝑔‘𝑘) → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)} = {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) |
343 | 342 | uneq2d 3767 |
. . . . . . . . . 10
⊢ (𝑠 = (𝑔‘𝑘) → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) = ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)})) |
344 | 343 | supeq1d 8352 |
. . . . . . . . 9
⊢ (𝑠 = (𝑔‘𝑘) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) = sup(({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)) |
345 | 344 | eqeq2d 2632 |
. . . . . . . 8
⊢ (𝑠 = (𝑔‘𝑘) → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
346 | 345 | rexbidv 3052 |
. . . . . . 7
⊢ (𝑠 = (𝑔‘𝑘) → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔
∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
347 | 346 | ralbidv 2986 |
. . . . . 6
⊢ (𝑠 = (𝑔‘𝑘) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔
∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
348 | 326, 347 | anbi12d 747 |
. . . . 5
⊢ (𝑠 = (𝑔‘𝑘) → ((ran (1st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) ↔ (ran
(1st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)))) |
349 | 348 | ac6num 9301 |
. . . 4
⊢ ((ℕ
∈ V ∧ ∪ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st
‘𝑠) ⊆
(0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ∈ dom
card ∧ ∀𝑘 ∈
ℕ ∃𝑠 ∈
((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) →
∃𝑔(𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)))) |
350 | 289, 323,
349 | mp3an12 1414 |
. . 3
⊢
(∀𝑘 ∈
ℕ ∃𝑠 ∈
((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1st ‘𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) →
∃𝑔(𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)))) |
351 | 288, 350 | syl 17 |
. 2
⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)))) |
352 | 1 | ad2antrr 762 |
. . . 4
⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → 𝑁 ∈
ℕ) |
353 | | poimir.r |
. . . 4
⊢ 𝑅 =
(∏t‘((1...𝑁) × {(topGen‘ran
(,))})) |
354 | | poimir.1 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅)) |
355 | 354 | ad2antrr 762 |
. . . 4
⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅)) |
356 | | eqid 2622 |
. . . 4
⊢ ((𝐹‘(((1st
‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑛) = ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑛) |
357 | | simplr 792 |
. . . 4
⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → 𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
358 | | simpl 473 |
. . . . . . 7
⊢ ((ran
(1st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ran
(1st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘)) |
359 | 358 | ralimi 2952 |
. . . . . 6
⊢
(∀𝑘 ∈
ℕ (ran (1st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) →
∀𝑘 ∈ ℕ
ran (1st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘)) |
360 | 359 | adantl 482 |
. . . . 5
⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) →
∀𝑘 ∈ ℕ
ran (1st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘)) |
361 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑘 = 𝑝 → (𝑔‘𝑘) = (𝑔‘𝑝)) |
362 | 361 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑘 = 𝑝 → (1st ‘(𝑔‘𝑘)) = (1st ‘(𝑔‘𝑝))) |
363 | 362 | rneqd 5353 |
. . . . . . 7
⊢ (𝑘 = 𝑝 → ran (1st ‘(𝑔‘𝑘)) = ran (1st ‘(𝑔‘𝑝))) |
364 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑘 = 𝑝 → (0..^𝑘) = (0..^𝑝)) |
365 | 363, 364 | sseq12d 3634 |
. . . . . 6
⊢ (𝑘 = 𝑝 → (ran (1st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ↔ ran (1st ‘(𝑔‘𝑝)) ⊆ (0..^𝑝))) |
366 | 365 | rspccva 3308 |
. . . . 5
⊢
((∀𝑘 ∈
ℕ ran (1st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ 𝑝 ∈ ℕ) → ran (1st
‘(𝑔‘𝑝)) ⊆ (0..^𝑝)) |
367 | 360, 366 | sylan 488 |
. . . 4
⊢ ((((𝜑 ∧ 𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) ∧ 𝑝 ∈ ℕ) → ran
(1st ‘(𝑔‘𝑝)) ⊆ (0..^𝑝)) |
368 | | simpll 790 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → 𝜑) |
369 | | poimir.2 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0) |
370 | 368, 369 | sylan 488 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0) |
371 | | eqid 2622 |
. . . . 5
⊢
((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) = ((1st
‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) |
372 | | simpr 477 |
. . . . . . . 8
⊢ ((ran
(1st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) →
∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)) |
373 | 372 | ralimi 2952 |
. . . . . . 7
⊢
(∀𝑘 ∈
ℕ (ran (1st ‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) →
∀𝑘 ∈ ℕ
∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)) |
374 | 373 | adantl 482 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) →
∀𝑘 ∈ ℕ
∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)) |
375 | 361 | fveq2d 6195 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑝 → (2nd ‘(𝑔‘𝑘)) = (2nd ‘(𝑔‘𝑝))) |
376 | 375 | imaeq1d 5465 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑝 → ((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) = ((2nd ‘(𝑔‘𝑝)) “ (1...𝑗))) |
377 | 376 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑝 → (((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) = (((2nd
‘(𝑔‘𝑝)) “ (1...𝑗)) ×
{1})) |
378 | 375 | imaeq1d 5465 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑝 → ((2nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁))) |
379 | 378 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑝 → (((2nd ‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})) |
380 | 377, 379 | uneq12d 3768 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑝 → ((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪
(((2nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
381 | 362, 380 | oveq12d 6668 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑝 → ((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
382 | | sneq 4187 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑝 → {𝑘} = {𝑝}) |
383 | 382 | xpeq2d 5139 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑝 → ((1...𝑁) × {𝑘}) = ((1...𝑁) × {𝑝})) |
384 | 381, 383 | oveq12d 6668 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑝 → (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})) = (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝}))) |
385 | 384 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑝 → (𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘}))) = (𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))) |
386 | 385 | fveq1d 6193 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑝 → ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏)) |
387 | 386 | breq2d 4665 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑝 → (0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏))) |
388 | 381 | fveq1d 6193 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑝 → (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) = (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏)) |
389 | 388 | neeq1d 2853 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑝 → ((((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0 ↔ (((1st
‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)) |
390 | 387, 389 | anbi12d 747 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑝 → ((0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))) |
391 | 390 | ralbidv 2986 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑝 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))) |
392 | 391 | rabbidv 3189 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑝 → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)} = {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) |
393 | 392 | uneq2d 3767 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑝 → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) = ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)})) |
394 | 393 | supeq1d 8352 |
. . . . . . . . 9
⊢ (𝑘 = 𝑝 → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) = sup(({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)) |
395 | 394 | eqeq2d 2632 |
. . . . . . . 8
⊢ (𝑘 = 𝑝 → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
396 | 395 | rexbidv 3052 |
. . . . . . 7
⊢ (𝑘 = 𝑝 → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔
∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
397 | | eqeq1 2626 |
. . . . . . . . 9
⊢ (𝑖 = 𝑞 → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
398 | 397 | rexbidv 3052 |
. . . . . . . 8
⊢ (𝑖 = 𝑞 → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔
∃𝑗 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
399 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = 𝑚 → (1...𝑗) = (1...𝑚)) |
400 | 399 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑚 → ((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) = ((2nd ‘(𝑔‘𝑝)) “ (1...𝑚))) |
401 | 400 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑚 → (((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) = (((2nd
‘(𝑔‘𝑝)) “ (1...𝑚)) ×
{1})) |
402 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = 𝑚 → (𝑗 + 1) = (𝑚 + 1)) |
403 | 402 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = 𝑚 → ((𝑗 + 1)...𝑁) = ((𝑚 + 1)...𝑁)) |
404 | 403 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑚 → ((2nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁))) |
405 | 404 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑚 → (((2nd ‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})) |
406 | 401, 405 | uneq12d 3768 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑚 → ((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪
(((2nd ‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) |
407 | 406 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑚 → ((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))) |
408 | 407 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑚 → (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})) = (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝}))) |
409 | 408 | fveq2d 6195 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑚 → (𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝}))) = (𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))) |
410 | 409 | fveq1d 6193 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑚 → ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) = ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏)) |
411 | 410 | breq2d 4665 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑚 → (0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ↔ 0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏))) |
412 | 407 | fveq1d 6193 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑚 → (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) = (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏)) |
413 | 412 | neeq1d 2853 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑚 → ((((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0 ↔ (((1st
‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)) |
414 | 411, 413 | anbi12d 747 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑚 → ((0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))) |
415 | 414 | ralbidv 2986 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑚 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))) |
416 | 415 | rabbidv 3189 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑚 → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)} = {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) |
417 | 416 | uneq2d 3767 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑚 → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) = ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)})) |
418 | 417 | supeq1d 8352 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑚 → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) = sup(({0} ∪
{𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)) |
419 | 418 | eqeq2d 2632 |
. . . . . . . . 9
⊢ (𝑗 = 𝑚 → (𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
420 | 419 | cbvrexv 3172 |
. . . . . . . 8
⊢
(∃𝑗 ∈
(0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔
∃𝑚 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)) |
421 | 398, 420 | syl6bb 276 |
. . . . . . 7
⊢ (𝑖 = 𝑞 → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔
∃𝑚 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
422 | 396, 421 | rspc2v 3322 |
. . . . . 6
⊢ ((𝑝 ∈ ℕ ∧ 𝑞 ∈ (0...𝑁)) → (∀𝑘 ∈ ℕ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) →
∃𝑚 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
))) |
423 | 374, 422 | mpan9 486 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) ∧ (𝑝 ∈ ℕ ∧ 𝑞 ∈ (0...𝑁))) → ∃𝑚 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, <
)) |
424 | 352, 142,
353, 355, 370, 371, 357, 367, 423 | poimirlem31 33440 |
. . . 4
⊢ ((((𝜑 ∧ 𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) ∧ (𝑝 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ◡ ≤ })) → ∃𝑚 ∈ (0...𝑁)0𝑟((𝐹‘(((1st ‘(𝑔‘𝑝)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd
‘(𝑔‘𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑛)) |
425 | 352, 142,
353, 355, 356, 357, 367, 424 | poimirlem30 33439 |
. . 3
⊢ (((𝜑 ∧ 𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) →
∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))) |
426 | 425 | anasss 679 |
. 2
⊢ ((𝜑 ∧ (𝑔:ℕ⟶((ℕ0
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑘 ∈ ℕ (ran (1st
‘(𝑔‘𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔‘𝑘)) ∘𝑓 +
((((2nd ‘(𝑔‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(𝑔‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))) →
∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))) |
427 | 351, 426 | exlimddv 1863 |
1
⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))) |