Step | Hyp | Ref
| Expression |
1 | | fourierdlem25.i |
. . 3
⊢ 𝐼 = sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < ) |
2 | | ssrab2 3687 |
. . . 4
⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ (0..^𝑀) |
3 | | ltso 10118 |
. . . . . 6
⊢ < Or
ℝ |
4 | 3 | a1i 11 |
. . . . 5
⊢ (𝜑 → < Or
ℝ) |
5 | | fzofi 12773 |
. . . . . . 7
⊢
(0..^𝑀) ∈
Fin |
6 | | ssfi 8180 |
. . . . . . 7
⊢
(((0..^𝑀) ∈ Fin
∧ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ (0..^𝑀)) → {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ∈ Fin) |
7 | 5, 2, 6 | mp2an 708 |
. . . . . 6
⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ∈ Fin |
8 | 7 | a1i 11 |
. . . . 5
⊢ (𝜑 → {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ∈ Fin) |
9 | | 0zd 11389 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℤ) |
10 | | fourierdlem25.m |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ ℕ) |
11 | 10 | nnzd 11481 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℤ) |
12 | 10 | nngt0d 11064 |
. . . . . . . 8
⊢ (𝜑 → 0 < 𝑀) |
13 | | fzolb 12476 |
. . . . . . . 8
⊢ (0 ∈
(0..^𝑀) ↔ (0 ∈
ℤ ∧ 𝑀 ∈
ℤ ∧ 0 < 𝑀)) |
14 | 9, 11, 12, 13 | syl3anbrc 1246 |
. . . . . . 7
⊢ (𝜑 → 0 ∈ (0..^𝑀)) |
15 | | fourierdlem25.qf |
. . . . . . . . 9
⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ) |
16 | | elfzofz 12485 |
. . . . . . . . . 10
⊢ (0 ∈
(0..^𝑀) → 0 ∈
(0...𝑀)) |
17 | 14, 16 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 0 ∈ (0...𝑀)) |
18 | 15, 17 | ffvelrnd 6360 |
. . . . . . . 8
⊢ (𝜑 → (𝑄‘0) ∈ ℝ) |
19 | 10 | nnnn0d 11351 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
20 | | nn0uz 11722 |
. . . . . . . . . . . . 13
⊢
ℕ0 = (ℤ≥‘0) |
21 | 19, 20 | syl6eleq 2711 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘0)) |
22 | | eluzfz2 12349 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈
(ℤ≥‘0) → 𝑀 ∈ (0...𝑀)) |
23 | 21, 22 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ (0...𝑀)) |
24 | 15, 23 | ffvelrnd 6360 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ) |
25 | 18, 24 | iccssred 39727 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑄‘0)[,](𝑄‘𝑀)) ⊆ ℝ) |
26 | | fourierdlem25.cel |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) |
27 | 25, 26 | sseldd 3604 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ ℝ) |
28 | 18 | rexrd 10089 |
. . . . . . . . 9
⊢ (𝜑 → (𝑄‘0) ∈
ℝ*) |
29 | 24 | rexrd 10089 |
. . . . . . . . 9
⊢ (𝜑 → (𝑄‘𝑀) ∈
ℝ*) |
30 | | iccgelb 12230 |
. . . . . . . . 9
⊢ (((𝑄‘0) ∈
ℝ* ∧ (𝑄‘𝑀) ∈ ℝ* ∧ 𝐶 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → (𝑄‘0) ≤ 𝐶) |
31 | 28, 29, 26, 30 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝜑 → (𝑄‘0) ≤ 𝐶) |
32 | | fourierdlem25.cnel |
. . . . . . . . . 10
⊢ (𝜑 → ¬ 𝐶 ∈ ran 𝑄) |
33 | | simpr 477 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐶 = (𝑄‘0)) → 𝐶 = (𝑄‘0)) |
34 | | ffn 6045 |
. . . . . . . . . . . . . 14
⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀)) |
35 | 15, 34 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑄 Fn (0...𝑀)) |
36 | 35 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐶 = (𝑄‘0)) → 𝑄 Fn (0...𝑀)) |
37 | 17 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐶 = (𝑄‘0)) → 0 ∈ (0...𝑀)) |
38 | | fnfvelrn 6356 |
. . . . . . . . . . . 12
⊢ ((𝑄 Fn (0...𝑀) ∧ 0 ∈ (0...𝑀)) → (𝑄‘0) ∈ ran 𝑄) |
39 | 36, 37, 38 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐶 = (𝑄‘0)) → (𝑄‘0) ∈ ran 𝑄) |
40 | 33, 39 | eqeltrd 2701 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐶 = (𝑄‘0)) → 𝐶 ∈ ran 𝑄) |
41 | 32, 40 | mtand 691 |
. . . . . . . . 9
⊢ (𝜑 → ¬ 𝐶 = (𝑄‘0)) |
42 | 41 | neqned 2801 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ≠ (𝑄‘0)) |
43 | 18, 27, 31, 42 | leneltd 10191 |
. . . . . . 7
⊢ (𝜑 → (𝑄‘0) < 𝐶) |
44 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑘 = 0 → (𝑄‘𝑘) = (𝑄‘0)) |
45 | 44 | breq1d 4663 |
. . . . . . . 8
⊢ (𝑘 = 0 → ((𝑄‘𝑘) < 𝐶 ↔ (𝑄‘0) < 𝐶)) |
46 | 45 | elrab 3363 |
. . . . . . 7
⊢ (0 ∈
{𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ↔ (0 ∈ (0..^𝑀) ∧ (𝑄‘0) < 𝐶)) |
47 | 14, 43, 46 | sylanbrc 698 |
. . . . . 6
⊢ (𝜑 → 0 ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}) |
48 | | ne0i 3921 |
. . . . . 6
⊢ (0 ∈
{𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} → {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ≠ ∅) |
49 | 47, 48 | syl 17 |
. . . . 5
⊢ (𝜑 → {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ≠ ∅) |
50 | | fzossfz 12488 |
. . . . . . . 8
⊢
(0..^𝑀) ⊆
(0...𝑀) |
51 | | fzssz 12343 |
. . . . . . . . 9
⊢
(0...𝑀) ⊆
ℤ |
52 | | zssre 11384 |
. . . . . . . . 9
⊢ ℤ
⊆ ℝ |
53 | 51, 52 | sstri 3612 |
. . . . . . . 8
⊢
(0...𝑀) ⊆
ℝ |
54 | 50, 53 | sstri 3612 |
. . . . . . 7
⊢
(0..^𝑀) ⊆
ℝ |
55 | 2, 54 | sstri 3612 |
. . . . . 6
⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ ℝ |
56 | 55 | a1i 11 |
. . . . 5
⊢ (𝜑 → {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ ℝ) |
57 | | fisupcl 8375 |
. . . . 5
⊢ (( <
Or ℝ ∧ ({𝑘 ∈
(0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ∈ Fin ∧ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ≠ ∅ ∧ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ ℝ)) → sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < ) ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}) |
58 | 4, 8, 49, 56, 57 | syl13anc 1328 |
. . . 4
⊢ (𝜑 → sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < ) ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}) |
59 | 2, 58 | sseldi 3601 |
. . 3
⊢ (𝜑 → sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < ) ∈ (0..^𝑀)) |
60 | 1, 59 | syl5eqel 2705 |
. 2
⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) |
61 | 50, 60 | sseldi 3601 |
. . . . 5
⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
62 | 15, 61 | ffvelrnd 6360 |
. . . 4
⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ) |
63 | 62 | rexrd 10089 |
. . 3
⊢ (𝜑 → (𝑄‘𝐼) ∈
ℝ*) |
64 | | fzofzp1 12565 |
. . . . . 6
⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) |
65 | 60, 64 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
66 | 15, 65 | ffvelrnd 6360 |
. . . 4
⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ) |
67 | 66 | rexrd 10089 |
. . 3
⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈
ℝ*) |
68 | 1, 58 | syl5eqel 2705 |
. . . . 5
⊢ (𝜑 → 𝐼 ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}) |
69 | | fveq2 6191 |
. . . . . . 7
⊢ (𝑘 = 𝐼 → (𝑄‘𝑘) = (𝑄‘𝐼)) |
70 | 69 | breq1d 4663 |
. . . . . 6
⊢ (𝑘 = 𝐼 → ((𝑄‘𝑘) < 𝐶 ↔ (𝑄‘𝐼) < 𝐶)) |
71 | 70 | elrab 3363 |
. . . . 5
⊢ (𝐼 ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ↔ (𝐼 ∈ (0..^𝑀) ∧ (𝑄‘𝐼) < 𝐶)) |
72 | 68, 71 | sylib 208 |
. . . 4
⊢ (𝜑 → (𝐼 ∈ (0..^𝑀) ∧ (𝑄‘𝐼) < 𝐶)) |
73 | 72 | simprd 479 |
. . 3
⊢ (𝜑 → (𝑄‘𝐼) < 𝐶) |
74 | 54, 60 | sseldi 3601 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈ ℝ) |
75 | | ltp1 10861 |
. . . . . . . . . 10
⊢ (𝐼 ∈ ℝ → 𝐼 < (𝐼 + 1)) |
76 | | id 22 |
. . . . . . . . . . 11
⊢ (𝐼 ∈ ℝ → 𝐼 ∈
ℝ) |
77 | | peano2re 10209 |
. . . . . . . . . . 11
⊢ (𝐼 ∈ ℝ → (𝐼 + 1) ∈
ℝ) |
78 | 76, 77 | ltnled 10184 |
. . . . . . . . . 10
⊢ (𝐼 ∈ ℝ → (𝐼 < (𝐼 + 1) ↔ ¬ (𝐼 + 1) ≤ 𝐼)) |
79 | 75, 78 | mpbid 222 |
. . . . . . . . 9
⊢ (𝐼 ∈ ℝ → ¬
(𝐼 + 1) ≤ 𝐼) |
80 | 74, 79 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ¬ (𝐼 + 1) ≤ 𝐼) |
81 | 50, 51 | sstri 3612 |
. . . . . . . . . . . 12
⊢
(0..^𝑀) ⊆
ℤ |
82 | 2, 81 | sstri 3612 |
. . . . . . . . . . 11
⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ ℤ |
83 | 82 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ ℤ) |
84 | | elrabi 3359 |
. . . . . . . . . . . . . . 15
⊢ (ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} → ℎ ∈ (0..^𝑀)) |
85 | | elfzo0le 12511 |
. . . . . . . . . . . . . . 15
⊢ (ℎ ∈ (0..^𝑀) → ℎ ≤ 𝑀) |
86 | 84, 85 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} → ℎ ≤ 𝑀) |
87 | 86 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}) → ℎ ≤ 𝑀) |
88 | 87 | ralrimiva 2966 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑀) |
89 | | breq2 4657 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑀 → (ℎ ≤ 𝑚 ↔ ℎ ≤ 𝑀)) |
90 | 89 | ralbidv 2986 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑀 → (∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑚 ↔ ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑀)) |
91 | 90 | rspcev 3309 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℤ ∧
∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑀) → ∃𝑚 ∈ ℤ ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑚) |
92 | 11, 88, 91 | syl2anc 693 |
. . . . . . . . . . 11
⊢ (𝜑 → ∃𝑚 ∈ ℤ ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑚) |
93 | 92 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → ∃𝑚 ∈ ℤ ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑚) |
94 | | elfzuz 12338 |
. . . . . . . . . . . . . 14
⊢ ((𝐼 + 1) ∈ (0...𝑀) → (𝐼 + 1) ∈
(ℤ≥‘0)) |
95 | 65, 94 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐼 + 1) ∈
(ℤ≥‘0)) |
96 | 95 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ∈
(ℤ≥‘0)) |
97 | 11 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → 𝑀 ∈ ℤ) |
98 | 53, 65 | sseldi 3601 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐼 + 1) ∈ ℝ) |
99 | 98 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ∈ ℝ) |
100 | 97 | zred 11482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → 𝑀 ∈ ℝ) |
101 | | elfzle2 12345 |
. . . . . . . . . . . . . . 15
⊢ ((𝐼 + 1) ∈ (0...𝑀) → (𝐼 + 1) ≤ 𝑀) |
102 | 65, 101 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐼 + 1) ≤ 𝑀) |
103 | 102 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ≤ 𝑀) |
104 | | simpr 477 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝑄‘(𝐼 + 1)) < 𝐶) |
105 | 66 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝑄‘(𝐼 + 1)) ∈ ℝ) |
106 | 27 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → 𝐶 ∈ ℝ) |
107 | 105, 106 | ltnled 10184 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → ((𝑄‘(𝐼 + 1)) < 𝐶 ↔ ¬ 𝐶 ≤ (𝑄‘(𝐼 + 1)))) |
108 | 104, 107 | mpbid 222 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → ¬ 𝐶 ≤ (𝑄‘(𝐼 + 1))) |
109 | | iccleub 12229 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑄‘0) ∈
ℝ* ∧ (𝑄‘𝑀) ∈ ℝ* ∧ 𝐶 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → 𝐶 ≤ (𝑄‘𝑀)) |
110 | 28, 29, 26, 109 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐶 ≤ (𝑄‘𝑀)) |
111 | 110 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑀 = (𝐼 + 1)) → 𝐶 ≤ (𝑄‘𝑀)) |
112 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 = (𝐼 + 1) → (𝑄‘𝑀) = (𝑄‘(𝐼 + 1))) |
113 | 112 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑀 = (𝐼 + 1)) → (𝑄‘𝑀) = (𝑄‘(𝐼 + 1))) |
114 | 111, 113 | breqtrd 4679 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑀 = (𝐼 + 1)) → 𝐶 ≤ (𝑄‘(𝐼 + 1))) |
115 | 114 | adantlr 751 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) ∧ 𝑀 = (𝐼 + 1)) → 𝐶 ≤ (𝑄‘(𝐼 + 1))) |
116 | 108, 115 | mtand 691 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → ¬ 𝑀 = (𝐼 + 1)) |
117 | 116 | neqned 2801 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → 𝑀 ≠ (𝐼 + 1)) |
118 | 99, 100, 103, 117 | leneltd 10191 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) < 𝑀) |
119 | | elfzo2 12473 |
. . . . . . . . . . . 12
⊢ ((𝐼 + 1) ∈ (0..^𝑀) ↔ ((𝐼 + 1) ∈
(ℤ≥‘0) ∧ 𝑀 ∈ ℤ ∧ (𝐼 + 1) < 𝑀)) |
120 | 96, 97, 118, 119 | syl3anbrc 1246 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ∈ (0..^𝑀)) |
121 | | fveq2 6191 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (𝐼 + 1) → (𝑄‘𝑘) = (𝑄‘(𝐼 + 1))) |
122 | 121 | breq1d 4663 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝐼 + 1) → ((𝑄‘𝑘) < 𝐶 ↔ (𝑄‘(𝐼 + 1)) < 𝐶)) |
123 | 122 | elrab 3363 |
. . . . . . . . . . 11
⊢ ((𝐼 + 1) ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ↔ ((𝐼 + 1) ∈ (0..^𝑀) ∧ (𝑄‘(𝐼 + 1)) < 𝐶)) |
124 | 120, 104,
123 | sylanbrc 698 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}) |
125 | | suprzub 11779 |
. . . . . . . . . 10
⊢ (({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ ℤ ∧ ∃𝑚 ∈ ℤ ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑚 ∧ (𝐼 + 1) ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}) → (𝐼 + 1) ≤ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < )) |
126 | 83, 93, 124, 125 | syl3anc 1326 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ≤ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < )) |
127 | 126, 1 | syl6breqr 4695 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ≤ 𝐼) |
128 | 80, 127 | mtand 691 |
. . . . . . 7
⊢ (𝜑 → ¬ (𝑄‘(𝐼 + 1)) < 𝐶) |
129 | | eqcom 2629 |
. . . . . . . . . . 11
⊢ ((𝑄‘(𝐼 + 1)) = 𝐶 ↔ 𝐶 = (𝑄‘(𝐼 + 1))) |
130 | 129 | biimpi 206 |
. . . . . . . . . 10
⊢ ((𝑄‘(𝐼 + 1)) = 𝐶 → 𝐶 = (𝑄‘(𝐼 + 1))) |
131 | 130 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) = 𝐶) → 𝐶 = (𝑄‘(𝐼 + 1))) |
132 | 35 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) = 𝐶) → 𝑄 Fn (0...𝑀)) |
133 | 65 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) = 𝐶) → (𝐼 + 1) ∈ (0...𝑀)) |
134 | | fnfvelrn 6356 |
. . . . . . . . . 10
⊢ ((𝑄 Fn (0...𝑀) ∧ (𝐼 + 1) ∈ (0...𝑀)) → (𝑄‘(𝐼 + 1)) ∈ ran 𝑄) |
135 | 132, 133,
134 | syl2anc 693 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) = 𝐶) → (𝑄‘(𝐼 + 1)) ∈ ran 𝑄) |
136 | 131, 135 | eqeltrd 2701 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) = 𝐶) → 𝐶 ∈ ran 𝑄) |
137 | 32, 136 | mtand 691 |
. . . . . . 7
⊢ (𝜑 → ¬ (𝑄‘(𝐼 + 1)) = 𝐶) |
138 | 128, 137 | jca 554 |
. . . . . 6
⊢ (𝜑 → (¬ (𝑄‘(𝐼 + 1)) < 𝐶 ∧ ¬ (𝑄‘(𝐼 + 1)) = 𝐶)) |
139 | | pm4.56 516 |
. . . . . 6
⊢ ((¬
(𝑄‘(𝐼 + 1)) < 𝐶 ∧ ¬ (𝑄‘(𝐼 + 1)) = 𝐶) ↔ ¬ ((𝑄‘(𝐼 + 1)) < 𝐶 ∨ (𝑄‘(𝐼 + 1)) = 𝐶)) |
140 | 138, 139 | sylib 208 |
. . . . 5
⊢ (𝜑 → ¬ ((𝑄‘(𝐼 + 1)) < 𝐶 ∨ (𝑄‘(𝐼 + 1)) = 𝐶)) |
141 | 66, 27 | leloed 10180 |
. . . . 5
⊢ (𝜑 → ((𝑄‘(𝐼 + 1)) ≤ 𝐶 ↔ ((𝑄‘(𝐼 + 1)) < 𝐶 ∨ (𝑄‘(𝐼 + 1)) = 𝐶))) |
142 | 140, 141 | mtbird 315 |
. . . 4
⊢ (𝜑 → ¬ (𝑄‘(𝐼 + 1)) ≤ 𝐶) |
143 | 27, 66 | ltnled 10184 |
. . . 4
⊢ (𝜑 → (𝐶 < (𝑄‘(𝐼 + 1)) ↔ ¬ (𝑄‘(𝐼 + 1)) ≤ 𝐶)) |
144 | 142, 143 | mpbird 247 |
. . 3
⊢ (𝜑 → 𝐶 < (𝑄‘(𝐼 + 1))) |
145 | 63, 67, 27, 73, 144 | eliood 39720 |
. 2
⊢ (𝜑 → 𝐶 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) |
146 | | fveq2 6191 |
. . . . 5
⊢ (𝑗 = 𝐼 → (𝑄‘𝑗) = (𝑄‘𝐼)) |
147 | | oveq1 6657 |
. . . . . 6
⊢ (𝑗 = 𝐼 → (𝑗 + 1) = (𝐼 + 1)) |
148 | 147 | fveq2d 6195 |
. . . . 5
⊢ (𝑗 = 𝐼 → (𝑄‘(𝑗 + 1)) = (𝑄‘(𝐼 + 1))) |
149 | 146, 148 | oveq12d 6668 |
. . . 4
⊢ (𝑗 = 𝐼 → ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1))) = ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) |
150 | 149 | eleq2d 2687 |
. . 3
⊢ (𝑗 = 𝐼 → (𝐶 ∈ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1))) ↔ 𝐶 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))) |
151 | 150 | rspcev 3309 |
. 2
⊢ ((𝐼 ∈ (0..^𝑀) ∧ 𝐶 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → ∃𝑗 ∈ (0..^𝑀)𝐶 ∈ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) |
152 | 60, 145, 151 | syl2anc 693 |
1
⊢ (𝜑 → ∃𝑗 ∈ (0..^𝑀)𝐶 ∈ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) |