Step | Hyp | Ref
| Expression |
1 | | pi1xfr.j |
. . . 4
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
2 | | iitopon 23094 |
. . . . . . 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 21465 |
. . . . . 6
⊢ ((II
∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋) |
6 | 3, 1, 4, 5 | syl3anc 1439 |
. . . . 5
⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋) |
7 | | 0elunit 12609 |
. . . . 5
⊢ 0 ∈
(0[,]1) |
8 | | ffvelrn 6623 |
. . . . 5
⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) →
(𝐹‘0) ∈ 𝑋) |
9 | 6, 7, 8 | sylancl 580 |
. . . 4
⊢ (𝜑 → (𝐹‘0) ∈ 𝑋) |
10 | | pi1xfr.p |
. . . . 5
⊢ 𝑃 = (𝐽 π1 (𝐹‘0)) |
11 | 10 | pi1grp 23261 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘0) ∈ 𝑋) → 𝑃 ∈ Grp) |
12 | 1, 9, 11 | syl2anc 579 |
. . 3
⊢ (𝜑 → 𝑃 ∈ Grp) |
13 | | 1elunit 12610 |
. . . . 5
⊢ 1 ∈
(0[,]1) |
14 | | ffvelrn 6623 |
. . . . 5
⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) →
(𝐹‘1) ∈ 𝑋) |
15 | 6, 13, 14 | sylancl 580 |
. . . 4
⊢ (𝜑 → (𝐹‘1) ∈ 𝑋) |
16 | | pi1xfr.q |
. . . . 5
⊢ 𝑄 = (𝐽 π1 (𝐹‘1)) |
17 | 16 | pi1grp 23261 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘1) ∈ 𝑋) → 𝑄 ∈ Grp) |
18 | 1, 15, 17 | syl2anc 579 |
. . 3
⊢ (𝜑 → 𝑄 ∈ Grp) |
19 | 12, 18 | jca 507 |
. 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 23236 |
. . . . . 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 1133 |
. . . 4
⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽)) |
26 | 24 | simp2d 1134 |
. . . . 5
⊢ (𝜑 → (𝐼‘0) = (𝐹‘1)) |
27 | 26 | eqcomd 2784 |
. . . 4
⊢ (𝜑 → (𝐹‘1) = (𝐼‘0)) |
28 | 24 | simp3d 1135 |
. . . 4
⊢ (𝜑 → (𝐼‘1) = (𝐹‘0)) |
29 | 10, 16, 20, 21, 1, 4, 25, 27, 28 | pi1xfrf 23264 |
. . 3
⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄)) |
30 | 20 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 = (Base‘𝑃)) |
31 | 10, 1, 9, 30 | pi1bas2 23252 |
. . . . . . 7
⊢ (𝜑 → 𝐵 = (∪ 𝐵 / (
≃ph‘𝐽))) |
32 | 31 | eleq2d 2845 |
. . . . . 6
⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (∪ 𝐵 / (
≃ph‘𝐽)))) |
33 | 32 | biimpa 470 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (∪ 𝐵 / (
≃ph‘𝐽))) |
34 | | eqid 2778 |
. . . . . 6
⊢ (∪ 𝐵
/ ( ≃ph‘𝐽)) = (∪ 𝐵 / (
≃ph‘𝐽)) |
35 | | fvoveq1 6947 |
. . . . . . . 8
⊢ ([𝑓](
≃ph‘𝐽) = 𝑦 → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = (𝐺‘(𝑦(+g‘𝑃)𝑧))) |
36 | | fveq2 6448 |
. . . . . . . . 9
⊢ ([𝑓](
≃ph‘𝐽) = 𝑦 → (𝐺‘[𝑓]( ≃ph‘𝐽)) = (𝐺‘𝑦)) |
37 | 36 | oveq1d 6939 |
. . . . . . . 8
⊢ ([𝑓](
≃ph‘𝐽) = 𝑦 → ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))) |
38 | 35, 37 | eqeq12d 2793 |
. . . . . . 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)))) |
39 | 38 | ralbidv 3168 |
. . . . . 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)))) |
40 | 31 | eleq2d 2845 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (∪ 𝐵 / (
≃ph‘𝐽)))) |
41 | 40 | biimpa 470 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / (
≃ph‘𝐽))) |
42 | 41 | adantlr 705 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / (
≃ph‘𝐽))) |
43 | | oveq2 6932 |
. . . . . . . . . . 11
⊢ ([ℎ](
≃ph‘𝐽) = 𝑧 → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽)) = ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) |
44 | 43 | fveq2d 6452 |
. . . . . . . . . 10
⊢ ([ℎ](
≃ph‘𝐽) = 𝑧 → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧))) |
45 | | fveq2 6448 |
. . . . . . . . . . 11
⊢ ([ℎ](
≃ph‘𝐽) = 𝑧 → (𝐺‘[ℎ]( ≃ph‘𝐽)) = (𝐺‘𝑧)) |
46 | 45 | oveq2d 6940 |
. . . . . . . . . 10
⊢ ([ℎ](
≃ph‘𝐽) = 𝑧 → ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘[ℎ]( ≃ph‘𝐽))) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧))) |
47 | 44, 46 | eqeq12d 2793 |
. . . . . . . . 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)))) |
48 | | phtpcer 23206 |
. . . . . . . . . . . . . 14
⊢ (
≃ph‘𝐽) Er (II Cn 𝐽) |
49 | 48 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (
≃ph‘𝐽) Er (II Cn 𝐽)) |
50 | 10, 1, 9, 30 | pi1eluni 23253 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑓 ∈ ∪ 𝐵 ↔ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0)))) |
51 | 50 | biimpa 470 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0))) |
52 | 51 | simp1d 1133 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽)) |
53 | 52 | 3adant3 1123 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽)) |
54 | 10, 1, 9, 30 | pi1eluni 23253 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (ℎ ∈ ∪ 𝐵 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0)))) |
55 | 54 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (ℎ ∈ ∪ 𝐵 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0)))) |
56 | 55 | biimp3a 1542 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0))) |
57 | 56 | simp1d 1133 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ (II Cn 𝐽)) |
58 | 53, 57 | pco0 23225 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘0) = (𝑓‘0)) |
59 | 51 | simp2d 1134 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0)) |
60 | 59 | 3adant3 1123 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0)) |
61 | 58, 60 | eqtrd 2814 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘0) = (𝐹‘0)) |
62 | 51 | simp3d 1135 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0)) |
63 | 62 | 3adant3 1123 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0)) |
64 | 56 | simp2d 1134 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘0) = (𝐹‘0)) |
65 | 63, 64 | eqtr4d 2817 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (ℎ‘0)) |
66 | 53, 57, 65 | pcocn 23228 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽)) |
67 | 4 | 3ad2ant1 1124 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽)) |
68 | 66, 67 | pco0 23225 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹)‘0) = ((𝑓(*𝑝‘𝐽)ℎ)‘0)) |
69 | 28 | 3ad2ant1 1124 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0)) |
70 | 61, 68, 69 | 3eqtr4rd 2825 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹)‘0)) |
71 | 25 | 3ad2ant1 1124 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽)) |
72 | 49, 71 | erref 8048 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼( ≃ph‘𝐽)𝐼) |
73 | 56 | simp3d 1135 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘1) = (𝐹‘0)) |
74 | | eqid 2778 |
. . . . . . . . . . . . . . . . 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)))) |
75 | 53, 57, 67, 65, 73, 74 | pcoass 23235 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)(𝑓(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))) |
76 | 57, 67 | pco0 23225 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*𝑝‘𝐽)𝐹)‘0) = (ℎ‘0)) |
77 | 65, 76 | eqtr4d 2817 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘0)) |
78 | 49, 53 | erref 8048 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓( ≃ph‘𝐽)𝑓) |
79 | 67, 71 | pco1 23226 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝐼‘1)) |
80 | 64, 76, 69 | 3eqtr4rd 2825 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘0)) |
81 | 79, 80 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘0)) |
82 | | eqid 2778 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((0[,]1)
× {(𝐹‘0)}) =
((0[,]1) × {(𝐹‘0)}) |
83 | 22, 82 | pcorev2 23239 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})) |
84 | 67, 83 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})) |
85 | 57, 67, 73 | pcocn 23228 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽)) |
86 | 49, 85 | erref 8048 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) |
87 | 81, 84, 86 | pcohtpy 23231 |
. . . . . . . . . . . . . . . . . . . 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))) |
88 | 76, 64 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*𝑝‘𝐽)𝐹)‘0) = (𝐹‘0)) |
89 | 82 | pcopt 23233 |
. . . . . . . . . . . . . . . . . . . . 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)) |
90 | 85, 88, 89 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (((0[,]1) ×
{(𝐹‘0)})(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) |
91 | 49, 87, 90 | ertrd 8044 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) |
92 | 26 | 3ad2ant1 1124 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1)) |
93 | 92 | eqcomd 2784 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0)) |
94 | 67, 71, 85, 93, 80, 74 | pcoass 23235 |
. . . . . . . . . . . . . . . . . . 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)))) |
95 | 49, 91, 94 | ertr3d 8046 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)))) |
96 | 77, 78, 95 | pcohtpy 23231 |
. . . . . . . . . . . . . . . . 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))))) |
97 | 71, 85, 80 | pcocn 23228 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽)) |
98 | 71, 85 | pco0 23225 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0)) |
99 | 98, 92 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1)) |
100 | 99 | eqcomd 2784 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0)) |
101 | 53, 67, 97, 63, 100, 74 | pcoass 23235 |
. . . . . . . . . . . . . . . . 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))))) |
102 | 49, 96, 101 | ertr4d 8047 |
. . . . . . . . . . . . . . . 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)))) |
103 | 49, 75, 102 | ertrd 8044 |
. . . . . . . . . . . . . . 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)))) |
104 | 70, 72, 103 | pcohtpy 23231 |
. . . . . . . . . . . . . 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))))) |
105 | 4 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽)) |
106 | 52, 105, 62 | pcocn 23228 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽)) |
107 | 106 | 3adant3 1123 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽)) |
108 | 52, 105 | pco0 23225 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘0) = (𝑓‘0)) |
109 | 28 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0)) |
110 | 59, 108, 109 | 3eqtr4rd 2825 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*𝑝‘𝐽)𝐹)‘0)) |
111 | 110 | 3adant3 1123 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*𝑝‘𝐽)𝐹)‘0)) |
112 | 53, 67 | pco1 23226 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1)) |
113 | 112, 99 | eqtr4d 2817 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘1) = ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0)) |
114 | 71, 107, 97, 111, 113, 74 | pcoass 23235 |
. . . . . . . . . . . . . 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))))) |
115 | 49, 104, 114 | ertr4d 8047 |
. . . . . . . . . . . . 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)))) |
116 | 49, 115 | erthi 8077 |
. . . . . . . . . . . 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)) |
117 | 1 | 3ad2ant1 1124 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋)) |
118 | 53, 57 | pco1 23226 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘1) = (ℎ‘1)) |
119 | 118, 73 | eqtrd 2814 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘1) = (𝐹‘0)) |
120 | 10, 1, 9, 30 | pi1eluni 23253 |
. . . . . . . . . . . . . . 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)))) |
121 | 120 | 3ad2ant1 1124 |
. . . . . . . . . . . . . 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)))) |
122 | 66, 61, 119, 121 | mpbir3and 1399 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)ℎ) ∈ ∪ 𝐵) |
123 | 10, 16, 20, 21, 117, 67, 71, 93, 69, 122 | pi1xfrval 23265 |
. . . . . . . . . . . 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)) |
124 | | eqid 2778 |
. . . . . . . . . . . . 13
⊢
(Base‘𝑄) =
(Base‘𝑄) |
125 | 15 | 3ad2ant1 1124 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) ∈ 𝑋) |
126 | | eqid 2778 |
. . . . . . . . . . . . 13
⊢
(+g‘𝑄) = (+g‘𝑄) |
127 | 25 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽)) |
128 | 127, 106,
110 | pcocn 23228 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽)) |
129 | 128 | 3adant3 1123 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽)) |
130 | 127, 106 | pco0 23225 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0)) |
131 | 26 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1)) |
132 | 130, 131 | eqtrd 2814 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1)) |
133 | 132 | 3adant3 1123 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1)) |
134 | 127, 106 | pco1 23226 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = ((𝑓(*𝑝‘𝐽)𝐹)‘1)) |
135 | 52, 105 | pco1 23226 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1)) |
136 | 134, 135 | eqtrd 2814 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)) |
137 | 136 | 3adant3 1123 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)) |
138 | | eqidd 2779 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (Base‘𝑄) = (Base‘𝑄)) |
139 | 16, 117, 125, 138 | pi1eluni 23253 |
. . . . . . . . . . . . . 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)))) |
140 | 129, 133,
137, 139 | mpbir3and 1399 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ ∪
(Base‘𝑄)) |
141 | 71, 85 | pco1 23226 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘1)) |
142 | 57, 67 | pco1 23226 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1)) |
143 | 141, 142 | eqtrd 2814 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1)) |
144 | 16, 117, 125, 138 | pi1eluni 23253 |
. . . . . . . . . . . . . 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)))) |
145 | 97, 99, 143, 144 | mpbir3and 1399 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) ∈ ∪
(Base‘𝑄)) |
146 | 16, 124, 117, 125, 126, 140, 145 | pi1addval 23259 |
. . . . . . . . . . . 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)) |
147 | 116, 123,
146 | 3eqtr4d 2824 |
. . . . . . . . . . 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))) |
148 | 9 | 3ad2ant1 1124 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘0) ∈ 𝑋) |
149 | | eqid 2778 |
. . . . . . . . . . . . 13
⊢
(+g‘𝑃) = (+g‘𝑃) |
150 | | simp2 1128 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵) |
151 | | simp3 1129 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ ∪ 𝐵) |
152 | 10, 20, 117, 148, 149, 150, 151 | pi1addval 23259 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽)) = [(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽)) |
153 | 152 | fveq2d 6452 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = (𝐺‘[(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽))) |
154 | 1 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋)) |
155 | 27 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0)) |
156 | | simpr 479 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵) |
157 | 10, 16, 20, 21, 154, 105, 127, 155, 109, 156 | pi1xfrval 23265 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐺‘[𝑓]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
158 | 157 | 3adant3 1123 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[𝑓]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
159 | 10, 16, 20, 21, 117, 67, 71, 93, 69, 151 | pi1xfrval 23265 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[ℎ]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)) |
160 | 158, 159 | oveq12d 6942 |
. . . . . . . . . . 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))) |
161 | 147, 153,
160 | 3eqtr4d 2824 |
. . . . . . . . . 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)))) |
162 | 161 | 3expa 1108 |
. . . . . . . . 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)))) |
163 | 34, 47, 162 | ectocld 8099 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ (∪ 𝐵 / (
≃ph‘𝐽))) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧))) |
164 | 42, 163 | syldan 585 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧))) |
165 | 164 | ralrimiva 3148 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ∀𝑧 ∈ 𝐵 (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧))) |
166 | 34, 39, 165 | ectocld 8099 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐵 / (
≃ph‘𝐽))) → ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))) |
167 | 33, 166 | syldan 585 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))) |
168 | 167 | ralrimiva 3148 |
. . 3
⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))) |
169 | 29, 168 | jca 507 |
. 2
⊢ (𝜑 → (𝐺:𝐵⟶(Base‘𝑄) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))) |
170 | 20, 124, 149, 126 | isghm 18048 |
. 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))))) |
171 | 19, 169, 170 | sylanbrc 578 |
1
⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄)) |