Theorem pi1xfr 24316

Description: Given a path 𝐹 and its inverse 𝐼 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.)

Hypotheses

Ref	Expression
pi1xfr.p	⊢ 𝑃 = (𝐽 π1 (𝐹‘0))
pi1xfr.q	⊢ 𝑄 = (𝐽 π1 (𝐹‘1))
pi1xfr.b	⊢ 𝐵 = (Base‘𝑃)
pi1xfr.g	⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)
pi1xfr.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
pi1xfr.f	⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
pi1xfr.i	⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))

Assertion

Ref	Expression
pi1xfr	⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))

Distinct variable groups:   𝑥,𝑔,𝐵   𝑔,𝐹,𝑥   𝑔,𝐼,𝑥   𝜑,𝑔,𝑥   𝑔,𝐽,𝑥   𝑃,𝑔,𝑥   𝑄,𝑔,𝑥

Allowed substitution hints:   𝐺(𝑥,𝑔)   𝑋(𝑥,𝑔)

Proof of Theorem pi1xfr

Dummy variables 𝑓 ℎ 𝑢 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pi1xfr.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		iitopon 24140	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
3		pi1xfr.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
4		cnf2 22498	. . . . 5 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋)
5	2, 1, 3, 4	mp3an2i 1465	. . . 4 ⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋)
6		0elunit 13294	. . . 4 ⊢ 0 ∈ (0[,]1)
7		ffvelcdm 7009	. . . 4 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ 𝑋)
8	5, 6, 7	sylancl 586	. . 3 ⊢ (𝜑 → (𝐹‘0) ∈ 𝑋)
9		pi1xfr.p	. . . 4 ⊢ 𝑃 = (𝐽 π1 (𝐹‘0))
10	9	pi1grp 24311	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘0) ∈ 𝑋) → 𝑃 ∈ Grp)
11	1, 8, 10	syl2anc 584	. 2 ⊢ (𝜑 → 𝑃 ∈ Grp)
12		1elunit 13295	. . . 4 ⊢ 1 ∈ (0[,]1)
13		ffvelcdm 7009	. . . 4 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ 𝑋)
14	5, 12, 13	sylancl 586	. . 3 ⊢ (𝜑 → (𝐹‘1) ∈ 𝑋)
15		pi1xfr.q	. . . 4 ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))
16	15	pi1grp 24311	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘1) ∈ 𝑋) → 𝑄 ∈ Grp)
17	1, 14, 16	syl2anc 584	. 2 ⊢ (𝜑 → 𝑄 ∈ Grp)
18		pi1xfr.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
19		pi1xfr.g	. . . 4 ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)
20		pi1xfr.i	. . . . . . 7 ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))
21	20	pcorevcl 24286	. . . . . 6 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
22	3, 21	syl 17	. . . . 5 ⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
23	22	simp1d 1141	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))
24	22	simp2d 1142	. . . . 5 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1))
25	24	eqcomd 2742	. . . 4 ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0))
26	22	simp3d 1143	. . . 4 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))
27	9, 15, 18, 19, 1, 3, 23, 25, 26	pi1xfrf 24314	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄))
28	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑃))
29	9, 1, 8, 28	pi1bas2 24302	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / ( ≃ph‘𝐽)))
30	29	eleq2d 2822	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (∪ 𝐵 / ( ≃ph‘𝐽))))
31	30	biimpa 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (∪ 𝐵 / ( ≃ph‘𝐽)))
32		eqid 2736	. . . . . 6 ⊢ (∪ 𝐵 / ( ≃ph‘𝐽)) = (∪ 𝐵 / ( ≃ph‘𝐽))
33		fvoveq1 7352	. . . . . . . 8 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = (𝐺‘(𝑦(+g‘𝑃)𝑧)))
34		fveq2 6819	. . . . . . . . 9 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → (𝐺‘[𝑓]( ≃ph‘𝐽)) = (𝐺‘𝑦))
35	34	oveq1d 7344	. . . . . . . 8 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
36	33, 35	eqeq12d 2752	. . . . . . 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))))
37	36	ralbidv 3170	. . . . . 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))))
38	29	eleq2d 2822	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (∪ 𝐵 / ( ≃ph‘𝐽))))
39	38	biimpa 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / ( ≃ph‘𝐽)))
40	39	adantlr 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / ( ≃ph‘𝐽)))
41		oveq2 7337	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽)) = ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧))
42	41	fveq2d 6823	. . . . . . . . . 10 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)))
43		fveq2 6819	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → (𝐺‘[ℎ]( ≃ph‘𝐽)) = (𝐺‘𝑧))
44	43	oveq2d 7345	. . . . . . . . . 10 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘[ℎ]( ≃ph‘𝐽))) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)))
45	42, 44	eqeq12d 2752	. . . . . . . . 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))))
46		phtpcer 24256	. . . . . . . . . . . . . 14 ⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽)
47	46	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ( ≃ph‘𝐽) Er (II Cn 𝐽))
48	9, 1, 8, 28	pi1eluni 24303	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑓 ∈ ∪ 𝐵 ↔ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0))))
49	48	biimpa 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0)))
50	49	simp1d 1141	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽))
51	50	3adant3 1131	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽))
52	9, 1, 8, 28	pi1eluni 24303	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (ℎ ∈ ∪ 𝐵 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0))))
53	52	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (ℎ ∈ ∪ 𝐵 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0))))
54	53	biimp3a 1468	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0)))
55	54	simp1d 1141	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ (II Cn 𝐽))
56	51, 55	pco0 24275	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘0) = (𝑓‘0))
57	49	simp2d 1142	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0))
58	57	3adant3 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0))
59	56, 58	eqtrd 2776	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘0) = (𝐹‘0))
60	49	simp3d 1143	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0))
61	60	3adant3 1131	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0))
62	54	simp2d 1142	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘0) = (𝐹‘0))
63	61, 62	eqtr4d 2779	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (ℎ‘0))
64	51, 55, 63	pcocn 24278	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽))
65	3	3ad2ant1 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
66	64, 65	pco0 24275	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹)‘0) = ((𝑓(*𝑝‘𝐽)ℎ)‘0))
67	26	3ad2ant1 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
68	59, 66, 67	3eqtr4rd 2787	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹)‘0))
69	23	3ad2ant1 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽))
70	47, 69	erref 8581	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼( ≃ph‘𝐽)𝐼)
71	54	simp3d 1143	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘1) = (𝐹‘0))
72		eqid 2736	. . . . . . . . . . . . . . . . 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))))
73	51, 55, 65, 63, 71, 72	pcoass 24285	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)(𝑓(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)))
74	55, 65	pco0 24275	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*𝑝‘𝐽)𝐹)‘0) = (ℎ‘0))
75	63, 74	eqtr4d 2779	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘0))
76	47, 51	erref 8581	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓( ≃ph‘𝐽)𝑓)
77	65, 69	pco1 24276	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝐼‘1))
78	62, 74, 67	3eqtr4rd 2787	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘0))
79	77, 78	eqtrd 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘0))
80		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0[,]1) × {(𝐹‘0)}) = ((0[,]1) × {(𝐹‘0)})
81	20, 80	pcorev2 24289	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
82	65, 81	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
83	55, 65, 71	pcocn 24278	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
84	47, 83	erref 8581	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))
85	79, 82, 84	pcohtpy 24281	. . . . . . . . . . . . . . . . . . . 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)))
86	74, 62	eqtrd 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*𝑝‘𝐽)𝐹)‘0) = (𝐹‘0))
87	80	pcopt 24283	. . . . . . . . . . . . . . . . . . . . 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))
88	83, 86, 87	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))
89	47, 85, 88	ertrd 8577	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))
90	24	3ad2ant1 1132	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
91	90	eqcomd 2742	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
92	65, 69, 83, 91, 78, 72	pcoass 24285	. . . . . . . . . . . . . . . . . . 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))))
93	47, 89, 92	ertr3d 8579	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))))
94	75, 76, 93	pcohtpy 24281	. . . . . . . . . . . . . . . . 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)))))
95	69, 83, 78	pcocn 24278	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
96	69, 83	pco0 24275	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
97	96, 90	eqtrd 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
98	97	eqcomd 2742	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0))
99	51, 65, 95, 61, 98, 72	pcoass 24285	. . . . . . . . . . . . . . . . 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)))))
100	47, 94, 99	ertr4d 8580	. . . . . . . . . . . . . . . 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))))
101	47, 73, 100	ertrd 8577	. . . . . . . . . . . . . . 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))))
102	68, 70, 101	pcohtpy 24281	. . . . . . . . . . . . . 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)))))
103	3	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
104	50, 103, 60	pcocn 24278	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
105	104	3adant3 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
106	50, 103	pco0 24275	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘0) = (𝑓‘0))
107	26	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
108	57, 106, 107	3eqtr4rd 2787	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*𝑝‘𝐽)𝐹)‘0))
109	108	3adant3 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*𝑝‘𝐽)𝐹)‘0))
110	51, 65	pco1 24276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
111	110, 97	eqtr4d 2779	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘1) = ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0))
112	69, 105, 95, 109, 111, 72	pcoass 24285	. . . . . . . . . . . . . 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)))))
113	47, 102, 112	ertr4d 8580	. . . . . . . . . . . . 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))))
114	47, 113	erthi 8612	. . . . . . . . . . . 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))
115	1	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋))
116	51, 55	pco1 24276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘1) = (ℎ‘1))
117	116, 71	eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘1) = (𝐹‘0))
118	9, 1, 8, 28	pi1eluni 24303	. . . . . . . . . . . . . . 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))))
119	118	3ad2ant1 1132	. . . . . . . . . . . . . 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))))
120	64, 59, 117, 119	mpbir3and 1341	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)ℎ) ∈ ∪ 𝐵)
121	9, 15, 18, 19, 115, 65, 69, 91, 67, 120	pi1xfrval 24315	. . . . . . . . . . . 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))
122		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘𝑄) = (Base‘𝑄)
123	14	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) ∈ 𝑋)
124		eqid 2736	. . . . . . . . . . . . 13 ⊢ (+g‘𝑄) = (+g‘𝑄)
125	23	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽))
126	125, 104, 108	pcocn 24278	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
127	126	3adant3 1131	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
128	125, 104	pco0 24275	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
129	24	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
130	128, 129	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
131	130	3adant3 1131	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
132	125, 104	pco1 24276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = ((𝑓(*𝑝‘𝐽)𝐹)‘1))
133	50, 103	pco1 24276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
134	132, 133	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
135	134	3adant3 1131	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
136		eqidd 2737	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (Base‘𝑄) = (Base‘𝑄))
137	15, 115, 123, 136	pi1eluni 24303	. . . . . . . . . . . . . 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))))
138	127, 131, 135, 137	mpbir3and 1341	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
139	69, 83	pco1 24276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘1))
140	55, 65	pco1 24276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
141	139, 140	eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
142	15, 115, 123, 136	pi1eluni 24303	. . . . . . . . . . . . . 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))))
143	95, 97, 141, 142	mpbir3and 1341	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
144	15, 122, 115, 123, 124, 138, 143	pi1addval 24309	. . . . . . . . . . . 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))
145	114, 121, 144	3eqtr4d 2786	. . . . . . . . . . 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)))
146	8	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘0) ∈ 𝑋)
147		eqid 2736	. . . . . . . . . . . . 13 ⊢ (+g‘𝑃) = (+g‘𝑃)
148		simp2 1136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵)
149		simp3 1137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ ∪ 𝐵)
150	9, 18, 115, 146, 147, 148, 149	pi1addval 24309	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽)) = [(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽))
151	150	fveq2d 6823	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = (𝐺‘[(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽)))
152	1	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋))
153	25	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
154		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵)
155	9, 15, 18, 19, 152, 103, 125, 153, 107, 154	pi1xfrval 24315	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐺‘[𝑓]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))
156	155	3adant3 1131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[𝑓]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))
157	9, 15, 18, 19, 115, 65, 69, 91, 67, 149	pi1xfrval 24315	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[ℎ]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))
158	156, 157	oveq12d 7347	. . . . . . . . . . 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)))
159	145, 151, 158	3eqtr4d 2786	. . . . . . . . . 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))))
160	159	3expa 1117	. . . . . . . . 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))))
161	32, 45, 160	ectocld 8636	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ (∪ 𝐵 / ( ≃ph‘𝐽))) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)))
162	40, 161	syldan 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)))
163	162	ralrimiva 3139	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ∀𝑧 ∈ 𝐵 (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)))
164	32, 37, 163	ectocld 8636	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐵 / ( ≃ph‘𝐽))) → ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
165	31, 164	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
166	165	ralrimiva 3139	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
167	27, 166	jca 512	. 2 ⊢ (𝜑 → (𝐺:𝐵⟶(Base‘𝑄) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))))
168	18, 122, 147, 124	isghm 18922	. 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)))))
169	11, 17, 167, 168	syl21anbrc 1343	1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3061  ifcif 4472  {csn 4572  ⟨cop 4578  ∪ cuni 4851   class class class wbr 5089   ↦ cmpt 5172   × cxp 5612  ran crn 5615  ⟶wf 6469  ‘cfv 6473  (class class class)co 7329   Er wer 8558  [cec 8559   / cqs 8560  0cc0 10964  1c1 10965   + caddc 10967   · cmul 10969   ≤ cle 11103   − cmin 11298   / cdiv 11725  2c2 12121  4c4 12123  [,]cicc 13175  Basecbs 17001  +_gcplusg 17051  Grpcgrp 18665   GrpHom cghm 18919  TopOnctopon 22157   Cn ccn 22473  IIcii 24136   ≃_phcphtpc 24230  *_𝑝cpco 24261   π₁ cpi1 24264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041  ax-pre-sup 11042  ax-addf 11043  ax-mulf 11044

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-iin 4941  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-se 5570  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-isom 6482  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-of 7587  df-om 7773  df-1st 7891  df-2nd 7892  df-supp 8040  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-1o 8359  df-2o 8360  df-er 8561  df-ec 8563  df-qs 8567  df-map 8680  df-ixp 8749  df-en 8797  df-dom 8798  df-sdom 8799  df-fin 8800  df-fsupp 9219  df-fi 9260  df-sup 9291  df-inf 9292  df-oi 9359  df-card 9788  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-div 11726  df-nn 12067  df-2 12129  df-3 12130  df-4 12131  df-5 12132  df-6 12133  df-7 12134  df-8 12135  df-9 12136  df-n0 12327  df-z 12413  df-dec 12531  df-uz 12676  df-q 12782  df-rp 12824  df-xneg 12941  df-xadd 12942  df-xmul 12943  df-ioo 13176  df-icc 13179  df-fz 13333  df-fzo 13476  df-seq 13815  df-exp 13876  df-hash 14138  df-cj 14901  df-re 14902  df-im 14903  df-sqrt 15037  df-abs 15038  df-struct 16937  df-sets 16954  df-slot 16972  df-ndx 16984  df-base 17002  df-ress 17031  df-plusg 17064  df-mulr 17065  df-starv 17066  df-sca 17067  df-vsca 17068  df-ip 17069  df-tset 17070  df-ple 17071  df-ds 17073  df-unif 17074  df-hom 17075  df-cco 17076  df-rest 17222  df-topn 17223  df-0g 17241  df-gsum 17242  df-topgen 17243  df-pt 17244  df-prds 17247  df-xrs 17302  df-qtop 17307  df-imas 17308  df-qus 17309  df-xps 17310  df-mre 17384  df-mrc 17385  df-acs 17387  df-mgm 18415  df-sgrp 18464  df-mnd 18475  df-submnd 18520  df-grp 18668  df-mulg 18789  df-ghm 18920  df-cntz 19011  df-cmn 19475  df-psmet 20687  df-xmet 20688  df-met 20689  df-bl 20690  df-mopn 20691  df-cnfld 20696  df-top 22141  df-topon 22158  df-topsp 22180  df-bases 22194  df-cld 22268  df-cn 22476  df-cnp 22477  df-tx 22811  df-hmeo 23004  df-xms 23571  df-ms 23572  df-tms 23573  df-ii 24138  df-htpy 24231  df-phtpy 24232  df-phtpc 24253  df-pco 24266  df-om1 24267  df-pi1 24269

This theorem is referenced by:  pi1xfrcnv  24318  pi1xfrgim  24319