Proof of Theorem pi1xfrcnvlem
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 | | fvex 6113 |
. . . . 5
⊢ (
≃ph‘𝐽) ∈ V |
3 | | ecexg 7633 |
. . . . 5
⊢ ((
≃ph‘𝐽) ∈ V → [𝑔]( ≃ph‘𝐽) ∈ V) |
4 | 2, 3 | mp1i 13 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [𝑔]( ≃ph‘𝐽) ∈ V) |
5 | | ecexg 7633 |
. . . . 5
⊢ ((
≃ph‘𝐽) ∈ V → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) ∈ V) |
6 | 2, 5 | mp1i 13 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) ∈ V) |
7 | 1, 4, 6 | fliftcnv 6461 |
. . 3
⊢ (𝜑 → ◡𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ 〈[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)〉)) |
8 | | pi1xfr.f |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
9 | | pi1xfr.i |
. . . . . . . . . . . 12
⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥))) |
10 | 9 | pcorevcl 22633 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0))) |
11 | 8, 10 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0))) |
12 | 11 | simp1d 1066 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽)) |
13 | 12 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽)) |
14 | | pi1xfr.p |
. . . . . . . . . . . 12
⊢ 𝑃 = (𝐽 π1 (𝐹‘0)) |
15 | | pi1xfr.j |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
16 | | iitopon 22490 |
. . . . . . . . . . . . . . 15
⊢ II ∈
(TopOn‘(0[,]1)) |
17 | 16 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → II ∈
(TopOn‘(0[,]1))) |
18 | | cnf2 20863 |
. . . . . . . . . . . . . 14
⊢ ((II
∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋) |
19 | 17, 15, 8, 18 | syl3anc 1318 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋) |
20 | | 0elunit 12161 |
. . . . . . . . . . . . 13
⊢ 0 ∈
(0[,]1) |
21 | | ffvelrn 6265 |
. . . . . . . . . . . . 13
⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) →
(𝐹‘0) ∈ 𝑋) |
22 | 19, 20, 21 | sylancl 693 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐹‘0) ∈ 𝑋) |
23 | | pi1xfr.b |
. . . . . . . . . . . . 13
⊢ 𝐵 = (Base‘𝑃) |
24 | 23 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 = (Base‘𝑃)) |
25 | 14, 15, 22, 24 | pi1eluni 22650 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↔ (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0)))) |
26 | 25 | biimpa 500 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0))) |
27 | 26 | simp1d 1066 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔 ∈ (II Cn 𝐽)) |
28 | 8 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽)) |
29 | 26 | simp3d 1068 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘1) = (𝐹‘0)) |
30 | 27, 28, 29 | pcocn 22625 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽)) |
31 | 11 | simp3d 1068 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐼‘1) = (𝐹‘0)) |
32 | 31 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0)) |
33 | 26 | simp2d 1067 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘0) = (𝐹‘0)) |
34 | 32, 33 | eqtr4d 2647 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = (𝑔‘0)) |
35 | 27, 28 | pco0 22622 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘0) = (𝑔‘0)) |
36 | 34, 35 | eqtr4d 2647 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘0)) |
37 | 13, 30, 36 | pcocn 22625 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽)) |
38 | 13, 30 | pco0 22622 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0)) |
39 | 11 | simp2d 1067 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼‘0) = (𝐹‘1)) |
40 | 39 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1)) |
41 | 38, 40 | eqtrd 2644 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1)) |
42 | 13, 30 | pco1 22623 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘1)) |
43 | 27, 28 | pco1 22623 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1)) |
44 | 42, 43 | eqtrd 2644 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)) |
45 | | pi1xfr.q |
. . . . . . . . 9
⊢ 𝑄 = (𝐽 π1 (𝐹‘1)) |
46 | | 1elunit 12162 |
. . . . . . . . . 10
⊢ 1 ∈
(0[,]1) |
47 | | ffvelrn 6265 |
. . . . . . . . . 10
⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) →
(𝐹‘1) ∈ 𝑋) |
48 | 19, 46, 47 | sylancl 693 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘1) ∈ 𝑋) |
49 | | eqidd 2611 |
. . . . . . . . 9
⊢ (𝜑 → (Base‘𝑄) = (Base‘𝑄)) |
50 | 45, 15, 48, 49 | pi1eluni 22650 |
. . . . . . . 8
⊢ (𝜑 → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ ∪
(Base‘𝑄) ↔
((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽) ∧ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1) ∧ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)))) |
51 | 50 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ ∪
(Base‘𝑄) ↔
((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽) ∧ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1) ∧ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)))) |
52 | 37, 41, 44, 51 | mpbir3and 1238 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ ∪
(Base‘𝑄)) |
53 | | eqidd 2611 |
. . . . . 6
⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))) = (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) |
54 | | eqidd 2611 |
. . . . . 6
⊢ (𝜑 → (ℎ ∈ ∪
(Base‘𝑄) ↦
〈[ℎ](
≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) = (ℎ ∈ ∪
(Base‘𝑄) ↦
〈[ℎ](
≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉)) |
55 | | eceq1 7669 |
. . . . . . 7
⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → [ℎ]( ≃ph‘𝐽) = [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
56 | | oveq1 6556 |
. . . . . . . . 9
⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → (ℎ(*𝑝‘𝐽)𝐼) = ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼)) |
57 | 56 | oveq2d 6565 |
. . . . . . . 8
⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → (𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼)) = (𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))) |
58 | 57 | eceq1d 7670 |
. . . . . . 7
⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽) = [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)) |
59 | 55, 58 | opeq12d 4348 |
. . . . . 6
⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → 〈[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉 = 〈[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) |
60 | 52, 53, 54, 59 | fmptco 6303 |
. . . . 5
⊢ (𝜑 → ((ℎ ∈ ∪
(Base‘𝑄) ↦
〈[ℎ](
≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) ∘ (𝑔 ∈ ∪ 𝐵
↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) = (𝑔 ∈ ∪ 𝐵 ↦ 〈[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉)) |
61 | | phtpcer 22602 |
. . . . . . . . 9
⊢ (
≃ph‘𝐽) Er (II Cn 𝐽) |
62 | 61 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (
≃ph‘𝐽) Er (II Cn 𝐽)) |
63 | 13, 27 | pco0 22622 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)𝑔)‘0) = (𝐼‘0)) |
64 | 63, 40 | eqtr2d 2645 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) = ((𝐼(*𝑝‘𝐽)𝑔)‘0)) |
65 | 62, 28 | erref 7649 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐹( ≃ph‘𝐽)𝐹) |
66 | 62, 13 | erref 7649 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐼( ≃ph‘𝐽)𝐼) |
67 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢ ((0[,]1)
× {(𝐹‘0)}) =
((0[,]1) × {(𝐹‘0)}) |
68 | 67 | pcopt2 22631 |
. . . . . . . . . . . . . 14
⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘1) = (𝐹‘0)) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))(
≃ph‘𝐽)𝑔) |
69 | 27, 29, 68 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))(
≃ph‘𝐽)𝑔) |
70 | 40 | eqcomd 2616 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0)) |
71 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), if(𝑥 ≤ (1 / 4), (2 · 𝑥), (𝑥 + (1 / 4))), ((𝑥 / 2) + (1 / 2)))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), if(𝑥 ≤ (1 / 4), (2 · 𝑥), (𝑥 + (1 / 4))), ((𝑥 / 2) + (1 / 2)))) |
72 | 27, 28, 13, 29, 70, 71 | pcoass 22632 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)(𝑔(*𝑝‘𝐽)(𝐹(*𝑝‘𝐽)𝐼))) |
73 | 28, 13 | pco0 22622 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘0) = (𝐹‘0)) |
74 | 29, 73 | eqtr4d 2647 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘1) = ((𝐹(*𝑝‘𝐽)𝐼)‘0)) |
75 | 62, 27 | erref 7649 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔( ≃ph‘𝐽)𝑔) |
76 | 9, 67 | pcorev2 22636 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})) |
77 | 28, 76 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})) |
78 | 74, 75, 77 | pcohtpy 22628 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)(𝐹(*𝑝‘𝐽)𝐼))( ≃ph‘𝐽)(𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))) |
79 | 62, 72, 78 | ertr2d 7646 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))(
≃ph‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)) |
80 | 62, 69, 79 | ertr3d 7647 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔( ≃ph‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)) |
81 | 34, 66, 80 | pcohtpy 22628 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(𝐼(*𝑝‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼))) |
82 | 43, 40 | eqtr4d 2647 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘1) = (𝐼‘0)) |
83 | 13, 30, 13, 36, 82, 71 | pcoass 22632 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)(𝐼(*𝑝‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼))) |
84 | 62, 81, 83 | ertr4d 7648 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼)) |
85 | 64, 65, 84 | pcohtpy 22628 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔))( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))) |
86 | 28, 13, 27, 70, 34, 71 | pcoass 22632 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔))) |
87 | 28, 13 | pco1 22623 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝐼‘1)) |
88 | 87, 34 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝑔‘0)) |
89 | 88, 77, 75 | pcohtpy 22628 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔)) |
90 | 67 | pcopt 22630 |
. . . . . . . . . . . 12
⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0)) → (((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔) |
91 | 27, 33, 90 | syl2anc 691 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (((0[,]1) ×
{(𝐹‘0)})(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔) |
92 | 62, 89, 91 | ertrd 7645 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔) |
93 | 62, 86, 92 | ertr3d 7647 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔))( ≃ph‘𝐽)𝑔) |
94 | 62, 85, 93 | ertr3d 7647 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))( ≃ph‘𝐽)𝑔) |
95 | 62, 94 | erthi 7680 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽) = [𝑔]( ≃ph‘𝐽)) |
96 | 95 | opeq2d 4347 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 〈[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉 = 〈[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)〉) |
97 | 96 | mpteq2dva 4672 |
. . . . 5
⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↦ 〈[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) = (𝑔 ∈ ∪ 𝐵 ↦ 〈[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)〉)) |
98 | 60, 97 | eqtrd 2644 |
. . . 4
⊢ (𝜑 → ((ℎ ∈ ∪
(Base‘𝑄) ↦
〈[ℎ](
≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) ∘ (𝑔 ∈ ∪ 𝐵
↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) = (𝑔 ∈ ∪ 𝐵 ↦ 〈[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)〉)) |
99 | 98 | rneqd 5274 |
. . 3
⊢ (𝜑 → ran ((ℎ ∈ ∪
(Base‘𝑄) ↦
〈[ℎ](
≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) ∘ (𝑔 ∈ ∪ 𝐵
↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) = ran (𝑔 ∈ ∪ 𝐵 ↦ 〈[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)〉)) |
100 | 7, 99 | eqtr4d 2647 |
. 2
⊢ (𝜑 → ◡𝐺 = ran ((ℎ ∈ ∪
(Base‘𝑄) ↦
〈[ℎ](
≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) ∘ (𝑔 ∈ ∪ 𝐵
↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))))) |
101 | | rncoss 5307 |
. . 3
⊢ ran
((ℎ ∈ ∪ (Base‘𝑄) ↦ 〈[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) ∘ (𝑔 ∈ ∪ 𝐵
↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) ⊆ ran (ℎ ∈ ∪
(Base‘𝑄) ↦
〈[ℎ](
≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) |
102 | | pi1xfrcnv.h |
. . 3
⊢ 𝐻 = ran (ℎ ∈ ∪
(Base‘𝑄) ↦
〈[ℎ](
≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) |
103 | 101, 102 | sseqtr4i 3601 |
. 2
⊢ ran
((ℎ ∈ ∪ (Base‘𝑄) ↦ 〈[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)〉) ∘ (𝑔 ∈ ∪ 𝐵
↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) ⊆ 𝐻 |
104 | 100, 103 | syl6eqss 3618 |
1
⊢ (𝜑 → ◡𝐺 ⊆ 𝐻) |