Step | Hyp | Ref
| Expression |
1 | | pi1xfr.g |
. . . 4
⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ 〈[𝑔](
≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)〉) |
2 | | pi1xfr.p |
. . . . 5
⊢ 𝑃 = (𝐽 π1 (𝐹‘0)) |
3 | | pi1xfr.b |
. . . . 5
⊢ 𝐵 = (Base‘𝑃) |
4 | | pi1xfr.j |
. . . . . 6
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
5 | 4 | adantr 483 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋)) |
6 | | iitopon 23489 |
. . . . . . . 8
⊢ II ∈
(TopOn‘(0[,]1)) |
7 | | pi1xfr.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
8 | | cnf2 21859 |
. . . . . . . 8
⊢ ((II
∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋) |
9 | 6, 4, 7, 8 | mp3an2i 1462 |
. . . . . . 7
⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋) |
10 | | 0elunit 12858 |
. . . . . . 7
⊢ 0 ∈
(0[,]1) |
11 | | ffvelrn 6851 |
. . . . . . 7
⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) →
(𝐹‘0) ∈ 𝑋) |
12 | 9, 10, 11 | sylancl 588 |
. . . . . 6
⊢ (𝜑 → (𝐹‘0) ∈ 𝑋) |
13 | 12 | adantr 483 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘0) ∈ 𝑋) |
14 | 3 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 = (Base‘𝑃)) |
15 | 2, 4, 12, 14 | pi1eluni 23648 |
. . . . . . 7
⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↔ (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0)))) |
16 | 15 | biimpa 479 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0))) |
17 | 16 | simp1d 1138 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔 ∈ (II Cn 𝐽)) |
18 | 16 | simp2d 1139 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘0) = (𝐹‘0)) |
19 | 16 | simp3d 1140 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘1) = (𝐹‘0)) |
20 | 2, 3, 5, 13, 17, 18, 19 | elpi1i 23652 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [𝑔]( ≃ph‘𝐽) ∈ 𝐵) |
21 | | pi1xfr.q |
. . . . 5
⊢ 𝑄 = (𝐽 π1 (𝐹‘1)) |
22 | | eqid 2823 |
. . . . 5
⊢
(Base‘𝑄) =
(Base‘𝑄) |
23 | | 1elunit 12859 |
. . . . . . 7
⊢ 1 ∈
(0[,]1) |
24 | | ffvelrn 6851 |
. . . . . . 7
⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) →
(𝐹‘1) ∈ 𝑋) |
25 | 9, 23, 24 | sylancl 588 |
. . . . . 6
⊢ (𝜑 → (𝐹‘1) ∈ 𝑋) |
26 | 25 | adantr 483 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) ∈ 𝑋) |
27 | | pi1xfrval.i |
. . . . . . 7
⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽)) |
28 | 27 | adantr 483 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽)) |
29 | 7 | adantr 483 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽)) |
30 | 17, 29, 19 | pcocn 23623 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽)) |
31 | 17, 29 | pco0 23620 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘0) = (𝑔‘0)) |
32 | | pi1xfrval.2 |
. . . . . . . 8
⊢ (𝜑 → (𝐼‘1) = (𝐹‘0)) |
33 | 32 | adantr 483 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0)) |
34 | 18, 31, 33 | 3eqtr4rd 2869 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘0)) |
35 | 28, 30, 34 | pcocn 23623 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽)) |
36 | 28, 30 | pco0 23620 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0)) |
37 | | pi1xfrval.1 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘1) = (𝐼‘0)) |
38 | 37 | adantr 483 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0)) |
39 | 36, 38 | eqtr4d 2861 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1)) |
40 | 28, 30 | pco1 23621 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘1)) |
41 | 17, 29 | pco1 23621 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1)) |
42 | 40, 41 | eqtrd 2858 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)) |
43 | 21, 22, 5, 26, 35, 39, 42 | elpi1i 23652 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) ∈ (Base‘𝑄)) |
44 | | eceq1 8329 |
. . . 4
⊢ (𝑔 = ℎ → [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽)) |
45 | | oveq1 7165 |
. . . . . 6
⊢ (𝑔 = ℎ → (𝑔(*𝑝‘𝐽)𝐹) = (ℎ(*𝑝‘𝐽)𝐹)) |
46 | 45 | oveq2d 7174 |
. . . . 5
⊢ (𝑔 = ℎ → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) = (𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))) |
47 | 46 | eceq1d 8330 |
. . . 4
⊢ (𝑔 = ℎ → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) = [(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
48 | | phtpcer 23601 |
. . . . . 6
⊢ (
≃ph‘𝐽) Er (II Cn 𝐽) |
49 | 48 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (
≃ph‘𝐽) Er (II Cn 𝐽)) |
50 | 18 | 3ad2antr1 1184 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝑔‘0) = (𝐹‘0)) |
51 | 17 | 3ad2antr1 1184 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → 𝑔 ∈ (II Cn 𝐽)) |
52 | 7 | adantr 483 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → 𝐹 ∈ (II Cn 𝐽)) |
53 | 51, 52 | pco0 23620 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → ((𝑔(*𝑝‘𝐽)𝐹)‘0) = (𝑔‘0)) |
54 | 32 | adantr 483 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝐼‘1) = (𝐹‘0)) |
55 | 50, 53, 54 | 3eqtr4rd 2869 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝐼‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘0)) |
56 | 27 | adantr 483 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → 𝐼 ∈ (II Cn 𝐽)) |
57 | 49, 56 | erref 8311 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → 𝐼( ≃ph‘𝐽)𝐼) |
58 | 19 | 3ad2antr1 1184 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝑔‘1) = (𝐹‘0)) |
59 | | simpr3 1192 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽)) |
60 | 49, 51 | erth 8340 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝑔( ≃ph‘𝐽)ℎ ↔ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) |
61 | 59, 60 | mpbird 259 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → 𝑔( ≃ph‘𝐽)ℎ) |
62 | 49, 52 | erref 8311 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → 𝐹( ≃ph‘𝐽)𝐹) |
63 | 58, 61, 62 | pcohtpy 23626 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝑔(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) |
64 | 55, 57, 63 | pcohtpy 23626 |
. . . . 5
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))) |
65 | 49, 64 | erthi 8342 |
. . . 4
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) = [(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
66 | 1, 20, 43, 44, 47, 65 | fliftfund 7068 |
. . 3
⊢ (𝜑 → Fun 𝐺) |
67 | 1, 20, 43 | fliftf 7070 |
. . 3
⊢ (𝜑 → (Fun 𝐺 ↔ 𝐺:ran (𝑔 ∈ ∪ 𝐵 ↦ [𝑔]( ≃ph‘𝐽))⟶(Base‘𝑄))) |
68 | 66, 67 | mpbid 234 |
. 2
⊢ (𝜑 → 𝐺:ran (𝑔 ∈ ∪ 𝐵 ↦ [𝑔]( ≃ph‘𝐽))⟶(Base‘𝑄)) |
69 | 2, 4, 12, 14 | pi1bas2 23647 |
. . . 4
⊢ (𝜑 → 𝐵 = (∪ 𝐵 / (
≃ph‘𝐽))) |
70 | | df-qs 8297 |
. . . . 5
⊢ (∪ 𝐵
/ ( ≃ph‘𝐽)) = {𝑠 ∣ ∃𝑔 ∈ ∪ 𝐵𝑠 = [𝑔]( ≃ph‘𝐽)} |
71 | | eqid 2823 |
. . . . . 6
⊢ (𝑔 ∈ ∪ 𝐵
↦ [𝑔](
≃ph‘𝐽)) = (𝑔 ∈ ∪ 𝐵 ↦ [𝑔]( ≃ph‘𝐽)) |
72 | 71 | rnmpt 5829 |
. . . . 5
⊢ ran
(𝑔 ∈ ∪ 𝐵
↦ [𝑔](
≃ph‘𝐽)) = {𝑠 ∣ ∃𝑔 ∈ ∪ 𝐵𝑠 = [𝑔]( ≃ph‘𝐽)} |
73 | 70, 72 | eqtr4i 2849 |
. . . 4
⊢ (∪ 𝐵
/ ( ≃ph‘𝐽)) = ran (𝑔 ∈ ∪ 𝐵 ↦ [𝑔]( ≃ph‘𝐽)) |
74 | 69, 73 | syl6eq 2874 |
. . 3
⊢ (𝜑 → 𝐵 = ran (𝑔 ∈ ∪ 𝐵 ↦ [𝑔]( ≃ph‘𝐽))) |
75 | 74 | feq2d 6502 |
. 2
⊢ (𝜑 → (𝐺:𝐵⟶(Base‘𝑄) ↔ 𝐺:ran (𝑔 ∈ ∪ 𝐵 ↦ [𝑔]( ≃ph‘𝐽))⟶(Base‘𝑄))) |
76 | 68, 75 | mpbird 259 |
1
⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄)) |