Step | Hyp | Ref
| Expression |
1 | | pi1xfr.j |
. . . 4
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
2 | | iitopon 22490 |
. . . . . . 7
⊢ II ∈
(TopOn‘(0[,]1)) |
3 | 2 | a1i 11 |
. . . . . 6
⊢ (𝜑 → II ∈
(TopOn‘(0[,]1))) |
4 | | pi1xfr.f |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
5 | | cnf2 20863 |
. . . . . 6
⊢ ((II
∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋) |
6 | 3, 1, 4, 5 | syl3anc 1318 |
. . . . 5
⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋) |
7 | | 0elunit 12161 |
. . . . 5
⊢ 0 ∈
(0[,]1) |
8 | | ffvelrn 6265 |
. . . . 5
⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) →
(𝐹‘0) ∈ 𝑋) |
9 | 6, 7, 8 | sylancl 693 |
. . . 4
⊢ (𝜑 → (𝐹‘0) ∈ 𝑋) |
10 | | pi1xfr.p |
. . . . 5
⊢ 𝑃 = (𝐽 π1 (𝐹‘0)) |
11 | 10 | pi1grp 22658 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘0) ∈ 𝑋) → 𝑃 ∈ Grp) |
12 | 1, 9, 11 | syl2anc 691 |
. . 3
⊢ (𝜑 → 𝑃 ∈ Grp) |
13 | | 1elunit 12162 |
. . . . 5
⊢ 1 ∈
(0[,]1) |
14 | | ffvelrn 6265 |
. . . . 5
⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) →
(𝐹‘1) ∈ 𝑋) |
15 | 6, 13, 14 | sylancl 693 |
. . . 4
⊢ (𝜑 → (𝐹‘1) ∈ 𝑋) |
16 | | pi1xfr.q |
. . . . 5
⊢ 𝑄 = (𝐽 π1 (𝐹‘1)) |
17 | 16 | pi1grp 22658 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘1) ∈ 𝑋) → 𝑄 ∈ Grp) |
18 | 1, 15, 17 | syl2anc 691 |
. . 3
⊢ (𝜑 → 𝑄 ∈ Grp) |
19 | 12, 18 | jca 553 |
. 2
⊢ (𝜑 → (𝑃 ∈ Grp ∧ 𝑄 ∈ Grp)) |
20 | | pi1xfr.b |
. . . 4
⊢ 𝐵 = (Base‘𝑃) |
21 | | pi1xfr.g |
. . . 4
⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ 〈[𝑔](
≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)〉) |
22 | | pi1xfr.i |
. . . . . . 7
⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥))) |
23 | 22 | pcorevcl 22633 |
. . . . . 6
⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0))) |
24 | 4, 23 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0))) |
25 | 24 | simp1d 1066 |
. . . 4
⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽)) |
26 | 24 | simp2d 1067 |
. . . . 5
⊢ (𝜑 → (𝐼‘0) = (𝐹‘1)) |
27 | 26 | eqcomd 2616 |
. . . 4
⊢ (𝜑 → (𝐹‘1) = (𝐼‘0)) |
28 | 24 | simp3d 1068 |
. . . 4
⊢ (𝜑 → (𝐼‘1) = (𝐹‘0)) |
29 | 10, 16, 20, 21, 1, 4, 25, 27, 28 | pi1xfrf 22661 |
. . 3
⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄)) |
30 | 20 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 = (Base‘𝑃)) |
31 | 10, 1, 9, 30 | pi1bas2 22649 |
. . . . . . 7
⊢ (𝜑 → 𝐵 = (∪ 𝐵 / (
≃ph‘𝐽))) |
32 | 31 | eleq2d 2673 |
. . . . . 6
⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (∪ 𝐵 / (
≃ph‘𝐽)))) |
33 | 32 | biimpa 500 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (∪ 𝐵 / (
≃ph‘𝐽))) |
34 | | eqid 2610 |
. . . . . 6
⊢ (∪ 𝐵
/ ( ≃ph‘𝐽)) = (∪ 𝐵 / (
≃ph‘𝐽)) |
35 | | oveq1 6556 |
. . . . . . . . 9
⊢ ([𝑓](
≃ph‘𝐽) = 𝑦 → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧) = (𝑦(+g‘𝑃)𝑧)) |
36 | 35 | fveq2d 6107 |
. . . . . . . 8
⊢ ([𝑓](
≃ph‘𝐽) = 𝑦 → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = (𝐺‘(𝑦(+g‘𝑃)𝑧))) |
37 | | fveq2 6103 |
. . . . . . . . 9
⊢ ([𝑓](
≃ph‘𝐽) = 𝑦 → (𝐺‘[𝑓]( ≃ph‘𝐽)) = (𝐺‘𝑦)) |
38 | 37 | oveq1d 6564 |
. . . . . . . 8
⊢ ([𝑓](
≃ph‘𝐽) = 𝑦 → ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))) |
39 | 36, 38 | eqeq12d 2625 |
. . . . . . 7
⊢ ([𝑓](
≃ph‘𝐽) = 𝑦 → ((𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)) ↔ (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))) |
40 | 39 | ralbidv 2969 |
. . . . . 6
⊢ ([𝑓](
≃ph‘𝐽) = 𝑦 → (∀𝑧 ∈ 𝐵 (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))) |
41 | 31 | eleq2d 2673 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (∪ 𝐵 / (
≃ph‘𝐽)))) |
42 | 41 | biimpa 500 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / (
≃ph‘𝐽))) |
43 | 42 | adantlr 747 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / (
≃ph‘𝐽))) |
44 | | oveq2 6557 |
. . . . . . . . . . 11
⊢ ([ℎ](
≃ph‘𝐽) = 𝑧 → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽)) = ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) |
45 | 44 | fveq2d 6107 |
. . . . . . . . . 10
⊢ ([ℎ](
≃ph‘𝐽) = 𝑧 → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧))) |
46 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ ([ℎ](
≃ph‘𝐽) = 𝑧 → (𝐺‘[ℎ]( ≃ph‘𝐽)) = (𝐺‘𝑧)) |
47 | 46 | oveq2d 6565 |
. . . . . . . . . 10
⊢ ([ℎ](
≃ph‘𝐽) = 𝑧 → ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘[ℎ]( ≃ph‘𝐽))) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧))) |
48 | 45, 47 | eqeq12d 2625 |
. . . . . . . . 9
⊢ ([ℎ](
≃ph‘𝐽) = 𝑧 → ((𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘[ℎ]( ≃ph‘𝐽))) ↔ (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)))) |
49 | | phtpcer 22602 |
. . . . . . . . . . . . . 14
⊢ (
≃ph‘𝐽) Er (II Cn 𝐽) |
50 | 49 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (
≃ph‘𝐽) Er (II Cn 𝐽)) |
51 | 10, 1, 9, 30 | pi1eluni 22650 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑓 ∈ ∪ 𝐵 ↔ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0)))) |
52 | 51 | biimpa 500 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0))) |
53 | 52 | simp1d 1066 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽)) |
54 | 53 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽)) |
55 | 10, 1, 9, 30 | pi1eluni 22650 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (ℎ ∈ ∪ 𝐵 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0)))) |
56 | 55 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (ℎ ∈ ∪ 𝐵 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0)))) |
57 | 56 | biimp3a 1424 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0))) |
58 | 57 | simp1d 1066 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ (II Cn 𝐽)) |
59 | 54, 58 | pco0 22622 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘0) = (𝑓‘0)) |
60 | 52 | simp2d 1067 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0)) |
61 | 60 | 3adant3 1074 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0)) |
62 | 59, 61 | eqtrd 2644 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘0) = (𝐹‘0)) |
63 | 52 | simp3d 1068 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0)) |
64 | 63 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0)) |
65 | 57 | simp2d 1067 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘0) = (𝐹‘0)) |
66 | 64, 65 | eqtr4d 2647 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (ℎ‘0)) |
67 | 54, 58, 66 | pcocn 22625 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽)) |
68 | 4 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽)) |
69 | 67, 68 | pco0 22622 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹)‘0) = ((𝑓(*𝑝‘𝐽)ℎ)‘0)) |
70 | 28 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0)) |
71 | 62, 69, 70 | 3eqtr4rd 2655 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹)‘0)) |
72 | 25 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽)) |
73 | 50, 72 | erref 7649 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼( ≃ph‘𝐽)𝐼) |
74 | 57 | simp3d 1068 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘1) = (𝐹‘0)) |
75 | | eqid 2610 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑢 ∈ (0[,]1) ↦ if(𝑢 ≤ (1 / 2), if(𝑢 ≤ (1 / 4), (2 · 𝑢), (𝑢 + (1 / 4))), ((𝑢 / 2) + (1 / 2)))) = (𝑢 ∈ (0[,]1) ↦ if(𝑢 ≤ (1 / 2), if(𝑢 ≤ (1 / 4), (2 · 𝑢), (𝑢 + (1 / 4))), ((𝑢 / 2) + (1 / 2)))) |
76 | 54, 58, 68, 66, 74, 75 | pcoass 22632 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)(𝑓(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))) |
77 | 58, 68 | pco0 22622 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*𝑝‘𝐽)𝐹)‘0) = (ℎ‘0)) |
78 | 66, 77 | eqtr4d 2647 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘0)) |
79 | 50, 54 | erref 7649 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓( ≃ph‘𝐽)𝑓) |
80 | 68, 72 | pco1 22623 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝐼‘1)) |
81 | 65, 77, 70 | 3eqtr4rd 2655 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘0)) |
82 | 80, 81 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘0)) |
83 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((0[,]1)
× {(𝐹‘0)}) =
((0[,]1) × {(𝐹‘0)}) |
84 | 22, 83 | pcorev2 22636 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})) |
85 | 68, 84 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})) |
86 | 58, 68, 74 | pcocn 22625 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽)) |
87 | 50, 86 | erref 7649 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) |
88 | 82, 85, 87 | pcohtpy 22628 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)(((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))) |
89 | 77, 65 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*𝑝‘𝐽)𝐹)‘0) = (𝐹‘0)) |
90 | 83 | pcopt 22630 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((ℎ(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽) ∧ ((ℎ(*𝑝‘𝐽)𝐹)‘0) = (𝐹‘0)) → (((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) |
91 | 86, 89, 90 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (((0[,]1) ×
{(𝐹‘0)})(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) |
92 | 50, 88, 91 | ertrd 7645 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) |
93 | 26 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1)) |
94 | 93 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0)) |
95 | 68, 72, 86, 94, 81, 75 | pcoass 22632 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)))) |
96 | 50, 92, 95 | ertr3d 7647 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)))) |
97 | 78, 79, 96 | pcohtpy 22628 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)(𝑓(*𝑝‘𝐽)(𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))))) |
98 | 72, 86, 81 | pcocn 22625 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽)) |
99 | 72, 86 | pco0 22622 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0)) |
100 | 99, 93 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1)) |
101 | 100 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0)) |
102 | 54, 68, 98, 64, 101, 75 | pcoass 22632 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)))( ≃ph‘𝐽)(𝑓(*𝑝‘𝐽)(𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))))) |
103 | 50, 97, 102 | ertr4d 7648 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)((𝑓(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)))) |
104 | 50, 76, 103 | ertrd 7645 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)((𝑓(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)))) |
105 | 71, 73, 104 | pcohtpy 22628 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)(𝐼(*𝑝‘𝐽)((𝑓(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))))) |
106 | 4 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽)) |
107 | 53, 106, 63 | pcocn 22625 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽)) |
108 | 107 | 3adant3 1074 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽)) |
109 | 53, 106 | pco0 22622 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘0) = (𝑓‘0)) |
110 | 28 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0)) |
111 | 60, 109, 110 | 3eqtr4rd 2655 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*𝑝‘𝐽)𝐹)‘0)) |
112 | 111 | 3adant3 1074 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*𝑝‘𝐽)𝐹)‘0)) |
113 | 54, 68 | pco1 22623 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1)) |
114 | 113, 100 | eqtr4d 2647 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘1) = ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0)) |
115 | 72, 108, 98, 112, 114, 75 | pcoass 22632 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)))( ≃ph‘𝐽)(𝐼(*𝑝‘𝐽)((𝑓(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))))) |
116 | 50, 105, 115 | ertr4d 7648 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)))) |
117 | 50, 116 | erthi 7680 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → [(𝐼(*𝑝‘𝐽)((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) = [((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)))](
≃ph‘𝐽)) |
118 | 1 | 3ad2ant1 1075 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋)) |
119 | 54, 58 | pco1 22623 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘1) = (ℎ‘1)) |
120 | 119, 74 | eqtrd 2644 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘1) = (𝐹‘0)) |
121 | 10, 1, 9, 30 | pi1eluni 22650 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑓(*𝑝‘𝐽)ℎ) ∈ ∪ 𝐵 ↔ ((𝑓(*𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽) ∧ ((𝑓(*𝑝‘𝐽)ℎ)‘0) = (𝐹‘0) ∧ ((𝑓(*𝑝‘𝐽)ℎ)‘1) = (𝐹‘0)))) |
122 | 121 | 3ad2ant1 1075 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ) ∈ ∪ 𝐵 ↔ ((𝑓(*𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽) ∧ ((𝑓(*𝑝‘𝐽)ℎ)‘0) = (𝐹‘0) ∧ ((𝑓(*𝑝‘𝐽)ℎ)‘1) = (𝐹‘0)))) |
123 | 67, 62, 120, 122 | mpbir3and 1238 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)ℎ) ∈ ∪ 𝐵) |
124 | 10, 16, 20, 21, 118, 68, 72, 94, 70, 123 | pi1xfrval 22662 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
125 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(Base‘𝑄) =
(Base‘𝑄) |
126 | 15 | 3ad2ant1 1075 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) ∈ 𝑋) |
127 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(+g‘𝑄) = (+g‘𝑄) |
128 | 25 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽)) |
129 | 128, 107,
111 | pcocn 22625 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽)) |
130 | 129 | 3adant3 1074 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽)) |
131 | 128, 107 | pco0 22622 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0)) |
132 | 26 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1)) |
133 | 131, 132 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1)) |
134 | 133 | 3adant3 1074 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1)) |
135 | 128, 107 | pco1 22623 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = ((𝑓(*𝑝‘𝐽)𝐹)‘1)) |
136 | 53, 106 | pco1 22623 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1)) |
137 | 135, 136 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)) |
138 | 137 | 3adant3 1074 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)) |
139 | | eqidd 2611 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (Base‘𝑄) = (Base‘𝑄)) |
140 | 16, 118, 126, 139 | pi1eluni 22650 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ ∪
(Base‘𝑄) ↔
((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽) ∧ ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1) ∧ ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)))) |
141 | 130, 134,
138, 140 | mpbir3and 1238 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ ∪
(Base‘𝑄)) |
142 | 72, 86 | pco1 22623 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘1)) |
143 | 58, 68 | pco1 22623 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1)) |
144 | 142, 143 | eqtrd 2644 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)) |
145 | 16, 118, 126, 139 | pi1eluni 22650 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) ∈ ∪
(Base‘𝑄) ↔
((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽) ∧ ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1) ∧ ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)))) |
146 | 98, 100, 144, 145 | mpbir3and 1238 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) ∈ ∪
(Base‘𝑄)) |
147 | 16, 125, 118, 126, 127, 141, 146 | pi1addval 22656 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ([(𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)(+g‘𝑄)[(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) = [((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)))](
≃ph‘𝐽)) |
148 | 117, 124,
147 | 3eqtr4d 2654 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽)) = ([(𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)(+g‘𝑄)[(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))) |
149 | 9 | 3ad2ant1 1075 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘0) ∈ 𝑋) |
150 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(+g‘𝑃) = (+g‘𝑃) |
151 | | simp2 1055 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵) |
152 | | simp3 1056 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ ∪ 𝐵) |
153 | 10, 20, 118, 149, 150, 151, 152 | pi1addval 22656 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽)) = [(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽)) |
154 | 153 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = (𝐺‘[(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽))) |
155 | 1 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋)) |
156 | 27 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0)) |
157 | | simpr 476 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵) |
158 | 10, 16, 20, 21, 155, 106, 128, 156, 110, 157 | pi1xfrval 22662 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐺‘[𝑓]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
159 | 158 | 3adant3 1074 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[𝑓]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
160 | 10, 16, 20, 21, 118, 68, 72, 94, 70, 152 | pi1xfrval 22662 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[ℎ]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
161 | 159, 160 | oveq12d 6567 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘[ℎ]( ≃ph‘𝐽))) = ([(𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)(+g‘𝑄)[(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))) |
162 | 148, 154,
161 | 3eqtr4d 2654 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘[ℎ]( ≃ph‘𝐽)))) |
163 | 162 | 3expa 1257 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘[ℎ]( ≃ph‘𝐽)))) |
164 | 34, 48, 163 | ectocld 7701 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ (∪ 𝐵 / (
≃ph‘𝐽))) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧))) |
165 | 43, 164 | syldan 486 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧))) |
166 | 165 | ralrimiva 2949 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ∀𝑧 ∈ 𝐵 (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧))) |
167 | 34, 40, 166 | ectocld 7701 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐵 / (
≃ph‘𝐽))) → ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))) |
168 | 33, 167 | syldan 486 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))) |
169 | 168 | ralrimiva 2949 |
. . 3
⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))) |
170 | 29, 169 | jca 553 |
. 2
⊢ (𝜑 → (𝐺:𝐵⟶(Base‘𝑄) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))) |
171 | 20, 125, 150, 127 | isghm 17483 |
. 2
⊢ (𝐺 ∈ (𝑃 GrpHom 𝑄) ↔ ((𝑃 ∈ Grp ∧ 𝑄 ∈ Grp) ∧ (𝐺:𝐵⟶(Base‘𝑄) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))))) |
172 | 19, 170, 171 | sylanbrc 695 |
1
⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄)) |