Step | Hyp | Ref
| Expression |
1 | | ssrab2 3650 |
. . 3
⊢ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ II |
2 | | cvmliftlem.g |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
3 | 2 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝐺 ∈ (II Cn 𝐽)) |
4 | | simprl 790 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝑗 ∈ 𝐽) |
5 | | cnima 20879 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ (II Cn 𝐽) ∧ 𝑗 ∈ 𝐽) → (◡𝐺 “ 𝑗) ∈ II) |
6 | 3, 4, 5 | syl2anc 691 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (◡𝐺 “ 𝑗) ∈ II) |
7 | | simplr 788 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝑥 ∈ (0[,]1)) |
8 | | simprrl 800 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝐺‘𝑥) ∈ 𝑗) |
9 | | iiuni 22492 |
. . . . . . . . . . . . . 14
⊢ (0[,]1) =
∪ II |
10 | | cvmliftlem.x |
. . . . . . . . . . . . . 14
⊢ 𝑋 = ∪
𝐽 |
11 | 9, 10 | cnf 20860 |
. . . . . . . . . . . . 13
⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋) |
12 | 2, 11 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺:(0[,]1)⟶𝑋) |
13 | 12 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝐺:(0[,]1)⟶𝑋) |
14 | | ffn 5958 |
. . . . . . . . . . 11
⊢ (𝐺:(0[,]1)⟶𝑋 → 𝐺 Fn (0[,]1)) |
15 | | elpreima 6245 |
. . . . . . . . . . 11
⊢ (𝐺 Fn (0[,]1) → (𝑥 ∈ (◡𝐺 “ 𝑗) ↔ (𝑥 ∈ (0[,]1) ∧ (𝐺‘𝑥) ∈ 𝑗))) |
16 | 13, 14, 15 | 3syl 18 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝑥 ∈ (◡𝐺 “ 𝑗) ↔ (𝑥 ∈ (0[,]1) ∧ (𝐺‘𝑥) ∈ 𝑗))) |
17 | 7, 8, 16 | mpbir2and 959 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝑥 ∈ (◡𝐺 “ 𝑗)) |
18 | | simprrr 801 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝑆‘𝑗) ≠ ∅) |
19 | | ffun 5961 |
. . . . . . . . . . . . 13
⊢ (𝐺:(0[,]1)⟶𝑋 → Fun 𝐺) |
20 | | funimacnv 5884 |
. . . . . . . . . . . . 13
⊢ (Fun
𝐺 → (𝐺 “ (◡𝐺 “ 𝑗)) = (𝑗 ∩ ran 𝐺)) |
21 | 13, 19, 20 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝐺 “ (◡𝐺 “ 𝑗)) = (𝑗 ∩ ran 𝐺)) |
22 | | inss1 3795 |
. . . . . . . . . . . 12
⊢ (𝑗 ∩ ran 𝐺) ⊆ 𝑗 |
23 | 21, 22 | syl6eqss 3618 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝐺 “ (◡𝐺 “ 𝑗)) ⊆ 𝑗) |
24 | 23 | ralrimivw 2950 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → ∀𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (◡𝐺 “ 𝑗)) ⊆ 𝑗) |
25 | | r19.2z 4012 |
. . . . . . . . . 10
⊢ (((𝑆‘𝑗) ≠ ∅ ∧ ∀𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (◡𝐺 “ 𝑗)) ⊆ 𝑗) → ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (◡𝐺 “ 𝑗)) ⊆ 𝑗) |
26 | 18, 24, 25 | syl2anc 691 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (◡𝐺 “ 𝑗)) ⊆ 𝑗) |
27 | | eleq2 2677 |
. . . . . . . . . . 11
⊢ (𝑢 = (◡𝐺 “ 𝑗) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ (◡𝐺 “ 𝑗))) |
28 | | imaeq2 5381 |
. . . . . . . . . . . . 13
⊢ (𝑢 = (◡𝐺 “ 𝑗) → (𝐺 “ 𝑢) = (𝐺 “ (◡𝐺 “ 𝑗))) |
29 | 28 | sseq1d 3595 |
. . . . . . . . . . . 12
⊢ (𝑢 = (◡𝐺 “ 𝑗) → ((𝐺 “ 𝑢) ⊆ 𝑗 ↔ (𝐺 “ (◡𝐺 “ 𝑗)) ⊆ 𝑗)) |
30 | 29 | rexbidv 3034 |
. . . . . . . . . . 11
⊢ (𝑢 = (◡𝐺 “ 𝑗) → (∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗 ↔ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (◡𝐺 “ 𝑗)) ⊆ 𝑗)) |
31 | 27, 30 | anbi12d 743 |
. . . . . . . . . 10
⊢ (𝑢 = (◡𝐺 “ 𝑗) → ((𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ (𝑥 ∈ (◡𝐺 “ 𝑗) ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (◡𝐺 “ 𝑗)) ⊆ 𝑗))) |
32 | 31 | rspcev 3282 |
. . . . . . . . 9
⊢ (((◡𝐺 “ 𝑗) ∈ II ∧ (𝑥 ∈ (◡𝐺 “ 𝑗) ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (◡𝐺 “ 𝑗)) ⊆ 𝑗)) → ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗)) |
33 | 6, 17, 26, 32 | syl12anc 1316 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗)) |
34 | | cvmliftlem.f |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
35 | 34 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
36 | 12 | ffvelrnda 6267 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → (𝐺‘𝑥) ∈ 𝑋) |
37 | | cvmliftlem.1 |
. . . . . . . . . 10
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 =
(◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) |
38 | 37, 10 | cvmcov 30499 |
. . . . . . . . 9
⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺‘𝑥) ∈ 𝑋) → ∃𝑗 ∈ 𝐽 ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅)) |
39 | 35, 36, 38 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ∃𝑗 ∈ 𝐽 ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅)) |
40 | 33, 39 | reximddv 3001 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ∃𝑗 ∈ 𝐽 ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗)) |
41 | | r19.42v 3073 |
. . . . . . . . 9
⊢
(∃𝑗 ∈
𝐽 (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ (𝑥 ∈ 𝑢 ∧ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗)) |
42 | 41 | rexbii 3023 |
. . . . . . . 8
⊢
(∃𝑢 ∈ II
∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗)) |
43 | | rexcom 3080 |
. . . . . . . 8
⊢
(∃𝑗 ∈
𝐽 ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ ∃𝑢 ∈ II ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗)) |
44 | | elunirab 4384 |
. . . . . . . 8
⊢ (𝑥 ∈ ∪ {𝑢
∈ II ∣ ∃𝑗
∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ↔ ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗)) |
45 | 42, 43, 44 | 3bitr4i 291 |
. . . . . . 7
⊢
(∃𝑗 ∈
𝐽 ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ 𝑥 ∈ ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗}) |
46 | 40, 45 | sylib 207 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → 𝑥 ∈ ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗}) |
47 | 46 | ex 449 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (0[,]1) → 𝑥 ∈ ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗})) |
48 | 47 | ssrdv 3574 |
. . . 4
⊢ (𝜑 → (0[,]1) ⊆ ∪ {𝑢
∈ II ∣ ∃𝑗
∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗}) |
49 | | uniss 4394 |
. . . . . 6
⊢ ({𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ II → ∪ {𝑢
∈ II ∣ ∃𝑗
∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ ∪
II) |
50 | 1, 49 | mp1i 13 |
. . . . 5
⊢ (𝜑 → ∪ {𝑢
∈ II ∣ ∃𝑗
∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ ∪
II) |
51 | 50, 9 | syl6sseqr 3615 |
. . . 4
⊢ (𝜑 → ∪ {𝑢
∈ II ∣ ∃𝑗
∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ (0[,]1)) |
52 | 48, 51 | eqssd 3585 |
. . 3
⊢ (𝜑 → (0[,]1) = ∪ {𝑢
∈ II ∣ ∃𝑗
∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗}) |
53 | | lebnumii 22573 |
. . 3
⊢ (({𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ II ∧ (0[,]1) = ∪ {𝑢
∈ II ∣ ∃𝑗
∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗}) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣) |
54 | 1, 52, 53 | sylancr 694 |
. 2
⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣) |
55 | | fzfi 12633 |
. . . . 5
⊢
(1...𝑛) ∈
Fin |
56 | | imaeq2 5381 |
. . . . . . . . . 10
⊢ (𝑢 = 𝑣 → (𝐺 “ 𝑢) = (𝐺 “ 𝑣)) |
57 | 56 | sseq1d 3595 |
. . . . . . . . 9
⊢ (𝑢 = 𝑣 → ((𝐺 “ 𝑢) ⊆ 𝑗 ↔ (𝐺 “ 𝑣) ⊆ 𝑗)) |
58 | 57 | 2rexbidv 3039 |
. . . . . . . 8
⊢ (𝑢 = 𝑣 → (∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗 ↔ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗)) |
59 | 58 | rexrab 3337 |
. . . . . . 7
⊢
(∃𝑣 ∈
{𝑢 ∈ II ∣
∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 ↔ ∃𝑣 ∈ II (∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗 ∧ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣)) |
60 | | vex 3176 |
. . . . . . . . . . . . 13
⊢ 𝑗 ∈ V |
61 | | vex 3176 |
. . . . . . . . . . . . 13
⊢ 𝑠 ∈ V |
62 | 60, 61 | op1std 7069 |
. . . . . . . . . . . 12
⊢ (𝑢 = 〈𝑗, 𝑠〉 → (1st ‘𝑢) = 𝑗) |
63 | 62 | sseq2d 3596 |
. . . . . . . . . . 11
⊢ (𝑢 = 〈𝑗, 𝑠〉 → ((𝐺 “ 𝑣) ⊆ (1st ‘𝑢) ↔ (𝐺 “ 𝑣) ⊆ 𝑗)) |
64 | 63 | rexiunxp 5184 |
. . . . . . . . . 10
⊢
(∃𝑢 ∈
∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ 𝑣) ⊆ (1st ‘𝑢) ↔ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗) |
65 | | imass2 5420 |
. . . . . . . . . . . 12
⊢ ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (𝐺 “ 𝑣)) |
66 | | sstr2 3575 |
. . . . . . . . . . . 12
⊢ ((𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (𝐺 “ 𝑣) → ((𝐺 “ 𝑣) ⊆ (1st ‘𝑢) → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘𝑢))) |
67 | 65, 66 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ((𝐺 “ 𝑣) ⊆ (1st ‘𝑢) → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘𝑢))) |
68 | 67 | reximdv 2999 |
. . . . . . . . . 10
⊢ ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (∃𝑢 ∈ ∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ 𝑣) ⊆ (1st ‘𝑢) → ∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘𝑢))) |
69 | 64, 68 | syl5bir 232 |
. . . . . . . . 9
⊢ ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗 → ∃𝑢 ∈ ∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘𝑢))) |
70 | 69 | impcom 445 |
. . . . . . . 8
⊢
((∃𝑗 ∈
𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗 ∧ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣) → ∃𝑢 ∈ ∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘𝑢)) |
71 | 70 | rexlimivw 3011 |
. . . . . . 7
⊢
(∃𝑣 ∈ II
(∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗 ∧ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣) → ∃𝑢 ∈ ∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘𝑢)) |
72 | 59, 71 | sylbi 206 |
. . . . . 6
⊢
(∃𝑣 ∈
{𝑢 ∈ II ∣
∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∃𝑢 ∈ ∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘𝑢)) |
73 | 72 | ralimi 2936 |
. . . . 5
⊢
(∀𝑘 ∈
(1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∀𝑘 ∈ (1...𝑛)∃𝑢 ∈ ∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘𝑢)) |
74 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑢 = (𝑔‘𝑘) → (1st ‘𝑢) = (1st
‘(𝑔‘𝑘))) |
75 | 74 | sseq2d 3596 |
. . . . . 6
⊢ (𝑢 = (𝑔‘𝑘) → ((𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘𝑢) ↔ (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘)))) |
76 | 75 | ac6sfi 8089 |
. . . . 5
⊢
(((1...𝑛) ∈ Fin
∧ ∀𝑘 ∈
(1...𝑛)∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘𝑢)) → ∃𝑔(𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘)))) |
77 | 55, 73, 76 | sylancr 694 |
. . . 4
⊢
(∀𝑘 ∈
(1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∃𝑔(𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘)))) |
78 | | cvmliftlem.b |
. . . . . . 7
⊢ 𝐵 = ∪
𝐶 |
79 | 34 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘)))) → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
80 | 2 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘)))) → 𝐺 ∈ (II Cn 𝐽)) |
81 | | cvmliftlem.p |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ 𝐵) |
82 | 81 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘)))) → 𝑃 ∈ 𝐵) |
83 | | cvmliftlem.e |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) |
84 | 83 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘)))) → (𝐹‘𝑃) = (𝐺‘0)) |
85 | | simplr 788 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘)))) → 𝑛 ∈ ℕ) |
86 | | simprl 790 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘)))) → 𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) |
87 | | sneq 4135 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑎 → {𝑗} = {𝑎}) |
88 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑎 → (𝑆‘𝑗) = (𝑆‘𝑎)) |
89 | 87, 88 | xpeq12d 5064 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑎 → ({𝑗} × (𝑆‘𝑗)) = ({𝑎} × (𝑆‘𝑎))) |
90 | 89 | cbviunv 4495 |
. . . . . . . . 9
⊢ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) = ∪
𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎)) |
91 | | feq3 5941 |
. . . . . . . . 9
⊢ (∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) = ∪
𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎)) → (𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ↔ 𝑔:(1...𝑛)⟶∪
𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎)))) |
92 | 90, 91 | ax-mp 5 |
. . . . . . . 8
⊢ (𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ↔ 𝑔:(1...𝑛)⟶∪
𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎))) |
93 | 86, 92 | sylib 207 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘)))) → 𝑔:(1...𝑛)⟶∪
𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎))) |
94 | | simprr 792 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘)))) → ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘))) |
95 | | eqid 2610 |
. . . . . . 7
⊢
(topGen‘ran (,)) = (topGen‘ran (,)) |
96 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑧 → (𝐺‘𝑡) = (𝐺‘𝑧)) |
97 | 96 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑧 → (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)) = (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑧))) |
98 | 97 | cbvmptv 4678 |
. . . . . . . . . 10
⊢ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡))) = (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑧))) |
99 | | eleq2 2677 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 = 𝑏 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐 ↔ (𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) |
100 | 99 | cbvriotav 6522 |
. . . . . . . . . . . . . . 15
⊢
(℩𝑐
∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) |
101 | | fveq1 6102 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = 𝑥 → (𝑦‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑤 − 1) / 𝑛))) |
102 | 101 | eleq1d 2672 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑥 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) |
103 | 102 | riotabidv 6513 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑥 → (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) = (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) |
104 | 100, 103 | syl5eq 2656 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑥 → (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) |
105 | 104 | reseq2d 5317 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑥 → (𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐)) = (𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))) |
106 | 105 | cnveqd 5220 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑥 → ◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐)) = ◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))) |
107 | 106 | fveq1d 6105 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑥 → (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑧)) = (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧))) |
108 | 107 | mpteq2dv 4673 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑥 → (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑧))) = (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)))) |
109 | 98, 108 | syl5eq 2656 |
. . . . . . . . 9
⊢ (𝑦 = 𝑥 → (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡))) = (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)))) |
110 | | oveq1 6556 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑚 → (𝑤 − 1) = (𝑚 − 1)) |
111 | 110 | oveq1d 6564 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑚 → ((𝑤 − 1) / 𝑛) = ((𝑚 − 1) / 𝑛)) |
112 | | oveq1 6556 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑚 → (𝑤 / 𝑛) = (𝑚 / 𝑛)) |
113 | 111, 112 | oveq12d 6567 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑚 → (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) = (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛))) |
114 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑚 → (𝑔‘𝑤) = (𝑔‘𝑚)) |
115 | 114 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑚 → (2nd ‘(𝑔‘𝑤)) = (2nd ‘(𝑔‘𝑚))) |
116 | 111 | fveq2d 6107 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑚 → (𝑥‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑚 − 1) / 𝑛))) |
117 | 116 | eleq1d 2672 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑚 → ((𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏)) |
118 | 115, 117 | riotaeqbidv 6514 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑚 → (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) = (℩𝑏 ∈ (2nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏)) |
119 | 118 | reseq2d 5317 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑚 → (𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) = (𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))) |
120 | 119 | cnveqd 5220 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑚 → ◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) = ◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))) |
121 | 120 | fveq1d 6105 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑚 → (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)) = (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧))) |
122 | 113, 121 | mpteq12dv 4663 |
. . . . . . . . 9
⊢ (𝑤 = 𝑚 → (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧))) = (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)))) |
123 | 109, 122 | cbvmpt2v 6633 |
. . . . . . . 8
⊢ (𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)))) = (𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)))) |
124 | | seqeq2 12667 |
. . . . . . . 8
⊢ ((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)))) = (𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)))) → seq0((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)))), (( I ↾ ℕ) ∪ {〈0,
{〈0, 𝑃〉}〉}))
= seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {〈0,
{〈0, 𝑃〉}〉}))) |
125 | 123, 124 | ax-mp 5 |
. . . . . . 7
⊢
seq0((𝑦 ∈ V,
𝑤 ∈ ℕ ↦
(𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)))), (( I ↾ ℕ) ∪ {〈0,
{〈0, 𝑃〉}〉}))
= seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {〈0,
{〈0, 𝑃〉}〉})) |
126 | | eqid 2610 |
. . . . . . 7
⊢ ∪ 𝑘 ∈ (1...𝑛)(seq0((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)))), (( I ↾ ℕ) ∪ {〈0,
{〈0, 𝑃〉}〉}))‘𝑘) = ∪ 𝑘 ∈ (1...𝑛)(seq0((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (◡(𝐹 ↾ (℩𝑐 ∈ (2nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)))), (( I ↾ ℕ) ∪ {〈0,
{〈0, 𝑃〉}〉}))‘𝑘) |
127 | 37, 78, 10, 79, 80, 82, 84, 85, 93, 94, 95, 125, 126 | cvmliftlem14 30533 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘)))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) |
128 | 127 | ex 449 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))) |
129 | 128 | exlimdv 1848 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∃𝑔(𝑔:(1...𝑛)⟶∪
𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔‘𝑘))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))) |
130 | 77, 129 | syl5 33 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))) |
131 | 130 | rexlimdva 3013 |
. 2
⊢ (𝜑 → (∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))) |
132 | 54, 131 | mpd 15 |
1
⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) |