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 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋)) |
6 | | iitopon 22490 |
. . . . . . . . 9
⊢ II ∈
(TopOn‘(0[,]1)) |
7 | 6 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → II ∈
(TopOn‘(0[,]1))) |
8 | | pi1xfr.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
9 | | cnf2 20863 |
. . . . . . . 8
⊢ ((II
∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋) |
10 | 7, 4, 8, 9 | syl3anc 1318 |
. . . . . . 7
⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋) |
11 | | 0elunit 12161 |
. . . . . . 7
⊢ 0 ∈
(0[,]1) |
12 | | ffvelrn 6265 |
. . . . . . 7
⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) →
(𝐹‘0) ∈ 𝑋) |
13 | 10, 11, 12 | sylancl 693 |
. . . . . 6
⊢ (𝜑 → (𝐹‘0) ∈ 𝑋) |
14 | 13 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘0) ∈ 𝑋) |
15 | 3 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 = (Base‘𝑃)) |
16 | 2, 4, 13, 15 | pi1eluni 22650 |
. . . . . . 7
⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↔ (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0)))) |
17 | 16 | biimpa 500 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0))) |
18 | 17 | simp1d 1066 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔 ∈ (II Cn 𝐽)) |
19 | 17 | simp2d 1067 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘0) = (𝐹‘0)) |
20 | 17 | simp3d 1068 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘1) = (𝐹‘0)) |
21 | 2, 3, 5, 14, 18, 19, 20 | elpi1i 22654 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [𝑔]( ≃ph‘𝐽) ∈ 𝐵) |
22 | | pi1xfr.q |
. . . . 5
⊢ 𝑄 = (𝐽 π1 (𝐹‘1)) |
23 | | eqid 2610 |
. . . . 5
⊢
(Base‘𝑄) =
(Base‘𝑄) |
24 | | 1elunit 12162 |
. . . . . . 7
⊢ 1 ∈
(0[,]1) |
25 | | ffvelrn 6265 |
. . . . . . 7
⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) →
(𝐹‘1) ∈ 𝑋) |
26 | 10, 24, 25 | sylancl 693 |
. . . . . 6
⊢ (𝜑 → (𝐹‘1) ∈ 𝑋) |
27 | 26 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) ∈ 𝑋) |
28 | | pi1xfrval.i |
. . . . . . 7
⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽)) |
29 | 28 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽)) |
30 | 8 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽)) |
31 | 18, 30, 20 | pcocn 22625 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽)) |
32 | 18, 30 | pco0 22622 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘0) = (𝑔‘0)) |
33 | | pi1xfrval.2 |
. . . . . . . 8
⊢ (𝜑 → (𝐼‘1) = (𝐹‘0)) |
34 | 33 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0)) |
35 | 19, 32, 34 | 3eqtr4rd 2655 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘0)) |
36 | 29, 31, 35 | pcocn 22625 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽)) |
37 | 29, 31 | pco0 22622 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0)) |
38 | | pi1xfrval.1 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘1) = (𝐼‘0)) |
39 | 38 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0)) |
40 | 37, 39 | eqtr4d 2647 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1)) |
41 | 29, 31 | pco1 22623 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘1)) |
42 | 18, 30 | pco1 22623 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1)) |
43 | 41, 42 | eqtrd 2644 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)) |
44 | 22, 23, 5, 27, 36, 40, 43 | elpi1i 22654 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) ∈ (Base‘𝑄)) |
45 | | eceq1 7669 |
. . . 4
⊢ (𝑔 = ℎ → [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽)) |
46 | | oveq1 6556 |
. . . . . 6
⊢ (𝑔 = ℎ → (𝑔(*𝑝‘𝐽)𝐹) = (ℎ(*𝑝‘𝐽)𝐹)) |
47 | 46 | oveq2d 6565 |
. . . . 5
⊢ (𝑔 = ℎ → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) = (𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))) |
48 | 47 | eceq1d 7670 |
. . . 4
⊢ (𝑔 = ℎ → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) = [(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
49 | | phtpcer 22602 |
. . . . . 6
⊢ (
≃ph‘𝐽) Er (II Cn 𝐽) |
50 | 49 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (
≃ph‘𝐽) Er (II Cn 𝐽)) |
51 | 19 | 3ad2antr1 1219 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝑔‘0) = (𝐹‘0)) |
52 | 18 | 3ad2antr1 1219 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → 𝑔 ∈ (II Cn 𝐽)) |
53 | 8 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → 𝐹 ∈ (II Cn 𝐽)) |
54 | 52, 53 | pco0 22622 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → ((𝑔(*𝑝‘𝐽)𝐹)‘0) = (𝑔‘0)) |
55 | 33 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝐼‘1) = (𝐹‘0)) |
56 | 51, 54, 55 | 3eqtr4rd 2655 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝐼‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘0)) |
57 | 28 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → 𝐼 ∈ (II Cn 𝐽)) |
58 | 50, 57 | erref 7649 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → 𝐼( ≃ph‘𝐽)𝐼) |
59 | 20 | 3ad2antr1 1219 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝑔‘1) = (𝐹‘0)) |
60 | | simpr3 1062 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽)) |
61 | 50, 52 | erth 7678 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝑔( ≃ph‘𝐽)ℎ ↔ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) |
62 | 60, 61 | mpbird 246 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → 𝑔( ≃ph‘𝐽)ℎ) |
63 | 50, 53 | erref 7649 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → 𝐹( ≃ph‘𝐽)𝐹) |
64 | 59, 62, 63 | pcohtpy 22628 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑔 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵 ∧ [𝑔]( ≃ph‘𝐽) = [ℎ]( ≃ph‘𝐽))) → (𝑔(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) |
65 | 56, 58, 64 | pcohtpy 22628 |
. . . . 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))) |
66 | 50, 65 | erthi 7680 |
. . . 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)) |
67 | 1, 21, 44, 45, 48, 66 | fliftfund 6463 |
. . 3
⊢ (𝜑 → Fun 𝐺) |
68 | 1, 21, 44 | fliftf 6465 |
. . 3
⊢ (𝜑 → (Fun 𝐺 ↔ 𝐺:ran (𝑔 ∈ ∪ 𝐵 ↦ [𝑔]( ≃ph‘𝐽))⟶(Base‘𝑄))) |
69 | 67, 68 | mpbid 221 |
. 2
⊢ (𝜑 → 𝐺:ran (𝑔 ∈ ∪ 𝐵 ↦ [𝑔]( ≃ph‘𝐽))⟶(Base‘𝑄)) |
70 | 2, 4, 13, 15 | pi1bas2 22649 |
. . . 4
⊢ (𝜑 → 𝐵 = (∪ 𝐵 / (
≃ph‘𝐽))) |
71 | | df-qs 7635 |
. . . . 5
⊢ (∪ 𝐵
/ ( ≃ph‘𝐽)) = {𝑠 ∣ ∃𝑔 ∈ ∪ 𝐵𝑠 = [𝑔]( ≃ph‘𝐽)} |
72 | | eqid 2610 |
. . . . . 6
⊢ (𝑔 ∈ ∪ 𝐵
↦ [𝑔](
≃ph‘𝐽)) = (𝑔 ∈ ∪ 𝐵 ↦ [𝑔]( ≃ph‘𝐽)) |
73 | 72 | rnmpt 5292 |
. . . . 5
⊢ ran
(𝑔 ∈ ∪ 𝐵
↦ [𝑔](
≃ph‘𝐽)) = {𝑠 ∣ ∃𝑔 ∈ ∪ 𝐵𝑠 = [𝑔]( ≃ph‘𝐽)} |
74 | 71, 73 | eqtr4i 2635 |
. . . 4
⊢ (∪ 𝐵
/ ( ≃ph‘𝐽)) = ran (𝑔 ∈ ∪ 𝐵 ↦ [𝑔]( ≃ph‘𝐽)) |
75 | 70, 74 | syl6eq 2660 |
. . 3
⊢ (𝜑 → 𝐵 = ran (𝑔 ∈ ∪ 𝐵 ↦ [𝑔]( ≃ph‘𝐽))) |
76 | 75 | feq2d 5944 |
. 2
⊢ (𝜑 → (𝐺:𝐵⟶(Base‘𝑄) ↔ 𝐺:ran (𝑔 ∈ ∪ 𝐵 ↦ [𝑔]( ≃ph‘𝐽))⟶(Base‘𝑄))) |
77 | 69, 76 | mpbird 246 |
1
⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄)) |