Theorem pi1xfr 25403

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 25227	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
3		pi1xfr.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
4		cnf2 23594	. . . . 5 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋)
5	2, 1, 3, 4	mp3an2i 1496	. . . 4 ⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋)
6		0elunit 13624	. . . 4 ⊢ 0 ∈ (0[,]1)
7		ffvelcdm 7102	. . . 4 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ 𝑋)
8	5, 6, 7	sylancl 598	. . 3 ⊢ (𝜑 → (𝐹‘0) ∈ 𝑋)
9		pi1xfr.p	. . . 4 ⊢ 𝑃 = (𝐽 π1 (𝐹‘0))
10	9	pi1grp 25398	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘0) ∈ 𝑋) → 𝑃 ∈ Grp)
11	1, 8, 10	syl2anc 596	. 2 ⊢ (𝜑 → 𝑃 ∈ Grp)
12		1elunit 13625	. . . 4 ⊢ 1 ∈ (0[,]1)
13		ffvelcdm 7102	. . . 4 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ 𝑋)
14	5, 12, 13	sylancl 598	. . 3 ⊢ (𝜑 → (𝐹‘1) ∈ 𝑋)
15		pi1xfr.q	. . . 4 ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))
16	15	pi1grp 25398	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘1) ∈ 𝑋) → 𝑄 ∈ Grp)
17	1, 14, 16	syl2anc 596	. 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 25373	. . . . . 6 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
22	3, 21	syl 18	. . . . 5 ⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
23	22	simp1d 1160	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))
24	22	simp2d 1161	. . . . 5 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1))
25	24	eqcomd 2769	. . . 4 ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0))
26	22	simp3d 1162	. . . 4 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))
27	9, 15, 18, 19, 1, 3, 23, 25, 26	pi1xfrf 25401	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄))
28	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑃))
29	9, 1, 8, 28	pi1bas2 25389	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / ( ≃ph‘𝐽)))
30	29	eleq2d 2850	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (∪ 𝐵 / ( ≃ph‘𝐽))))
31	30	biimpa 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (∪ 𝐵 / ( ≃ph‘𝐽)))
32		eqid 2763	. . . . . 6 ⊢ (∪ 𝐵 / ( ≃ph‘𝐽)) = (∪ 𝐵 / ( ≃ph‘𝐽))
33		fvoveq1 7463	. . . . . . . 8 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = (𝐺‘(𝑦(+g‘𝑃)𝑧)))
34		fveq2 6905	. . . . . . . . 9 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → (𝐺‘[𝑓]( ≃ph‘𝐽)) = (𝐺‘𝑦))
35	34	oveq1d 7455	. . . . . . . 8 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
36	33, 35	eqeq12d 2779	. . . . . . 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 3189	. . . . . 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 2850	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (∪ 𝐵 / ( ≃ph‘𝐽))))
39	38	biimpa 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / ( ≃ph‘𝐽)))
40	39	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / ( ≃ph‘𝐽)))
41		oveq2 7448	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽)) = ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧))
42	41	fveq2d 6909	. . . . . . . . . 10 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)))
43		fveq2 6905	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → (𝐺‘[ℎ]( ≃ph‘𝐽)) = (𝐺‘𝑧))
44	43	oveq2d 7456	. . . . . . . . . 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 2779	. . . . . . . . 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 25343	. . . . . . . . . . . . . 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 25390	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑓 ∈ ∪ 𝐵 ↔ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0))))
49	48	biimpa 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0)))
50	49	simp1d 1160	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽))
51	50	3adant3 1150	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽))
52	9, 1, 8, 28	pi1eluni 25390	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (ℎ ∈ ∪ 𝐵 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0))))
53	52	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (ℎ ∈ ∪ 𝐵 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0))))
54	53	biimp3a 1499	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0)))
55	54	simp1d 1160	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ (II Cn 𝐽))
56	51, 55	pco0 25362	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘0) = (𝑓‘0))
57	49	simp2d 1161	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0))
58	57	3adant3 1150	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0))
59	56, 58	eqtrd 2798	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘0) = (𝐹‘0))
60	49	simp3d 1162	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0))
61	60	3adant3 1150	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0))
62	54	simp2d 1161	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘0) = (𝐹‘0))
63	61, 62	eqtr4d 2801	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (ℎ‘0))
64	51, 55, 63	pcocn 25365	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽))
65	3	3ad2ant1 1151	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
66	64, 65	pco0 25362	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹)‘0) = ((𝑓(*𝑝‘𝐽)ℎ)‘0))
67	26	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
68	59, 66, 67	3eqtr4rd 2809	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (((𝑓(*𝑝‘𝐽)ℎ)(*𝑝‘𝐽)𝐹)‘0))
69	23	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽))
70	47, 69	erref 8763	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼( ≃ph‘𝐽)𝐼)
71	54	simp3d 1162	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘1) = (𝐹‘0))
72		eqid 2763	. . . . . . . . . . . . . . . . 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 25372	. . . . . . . . . . . . . . . 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 25362	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*𝑝‘𝐽)𝐹)‘0) = (ℎ‘0))
75	63, 74	eqtr4d 2801	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘0))
76	47, 51	erref 8763	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓( ≃ph‘𝐽)𝑓)
77	65, 69	pco1 25363	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝐼‘1))
78	62, 74, 67	3eqtr4rd 2809	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘0))
79	77, 78	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘0))
80		eqid 2763	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0[,]1) × {(𝐹‘0)}) = ((0[,]1) × {(𝐹‘0)})
81	20, 80	pcorev2 25376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
82	65, 81	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
83	55, 65, 71	pcocn 25365	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
84	47, 83	erref 8763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))
85	79, 82, 84	pcohtpy 25368	. . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*𝑝‘𝐽)𝐹)‘0) = (𝐹‘0))
87	80	pcopt 25370	. . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . 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 8759	. . . . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
91	90	eqcomd 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
92	65, 69, 83, 91, 78, 72	pcoass 25372	. . . . . . . . . . . . . . . . . . 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 8761	. . . . . . . . . . . . . . . . . 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 25368	. . . . . . . . . . . . . . . . 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 25365	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
96	69, 83	pco0 25362	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
97	96, 90	eqtrd 2798	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
98	97	eqcomd 2769	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘0))
99	51, 65, 95, 61, 98, 72	pcoass 25372	. . . . . . . . . . . . . . . . 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 8762	. . . . . . . . . . . . . . . 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 8759	. . . . . . . . . . . . . . 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 25368	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
104	50, 103, 60	pcocn 25365	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
105	104	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
106	50, 103	pco0 25362	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘0) = (𝑓‘0))
107	26	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
108	57, 106, 107	3eqtr4rd 2809	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*𝑝‘𝐽)𝐹)‘0))
109	108	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*𝑝‘𝐽)𝐹)‘0))
110	51, 65	pco1 25363	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
111	110, 97	eqtr4d 2801	. . . . . . . . . . . . . . 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 25372	. . . . . . . . . . . . . 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 8762	. . . . . . . . . . . . 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 8799	. . . . . . . . . . . 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 1151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋))
116	51, 55	pco1 25363	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘1) = (ℎ‘1))
117	116, 71	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)ℎ)‘1) = (𝐹‘0))
118	9, 1, 8, 28	pi1eluni 25390	. . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . 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 1361	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*𝑝‘𝐽)ℎ) ∈ ∪ 𝐵)
121	9, 15, 18, 19, 115, 65, 69, 91, 67, 120	pi1xfrval 25402	. . . . . . . . . . . 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 2763	. . . . . . . . . . . . 13 ⊢ (Base‘𝑄) = (Base‘𝑄)
123	14	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) ∈ 𝑋)
124		eqid 2763	. . . . . . . . . . . . 13 ⊢ (+g‘𝑄) = (+g‘𝑄)
125	23	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽))
126	125, 104, 108	pcocn 25365	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
127	126	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
128	125, 104	pco0 25362	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
129	24	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
130	128, 129	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
131	130	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
132	125, 104	pco1 25363	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = ((𝑓(*𝑝‘𝐽)𝐹)‘1))
133	50, 103	pco1 25363	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
134	132, 133	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
135	134	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
136		eqidd 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (Base‘𝑄) = (Base‘𝑄))
137	15, 115, 123, 136	pi1eluni 25390	. . . . . . . . . . . . . 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 1361	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
139	69, 83	pco1 25363	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘1) = ((ℎ(*𝑝‘𝐽)𝐹)‘1))
140	55, 65	pco1 25363	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
141	139, 140	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
142	15, 115, 123, 136	pi1eluni 25390	. . . . . . . . . . . . . 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 1361	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
144	15, 122, 115, 123, 124, 138, 143	pi1addval 25396	. . . . . . . . . . . 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 2808	. . . . . . . . . . 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 1151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘0) ∈ 𝑋)
147		eqid 2763	. . . . . . . . . . . . 13 ⊢ (+g‘𝑃) = (+g‘𝑃)
148		simp2 1155	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵)
149		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ ∪ 𝐵)
150	9, 18, 115, 146, 147, 148, 149	pi1addval 25396	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽)) = [(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽))
151	150	fveq2d 6909	. . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋))
153	25	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
154		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵)
155	9, 15, 18, 19, 152, 103, 125, 153, 107, 154	pi1xfrval 25402	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐺‘[𝑓]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝑓(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))
156	155	3adant3 1150	. . . . . . . . . . . 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 25402	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[ℎ]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))
158	156, 157	oveq12d 7458	. . . . . . . . . . 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 2808	. . . . . . . . . 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 1136	. . . . . . . . 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 8828	. . . . . . . 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 603	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)))
163	162	ralrimiva 3158	. . . . . 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 8828	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐵 / ( ≃ph‘𝐽))) → ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
165	31, 164	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
166	165	ralrimiva 3158	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
167	27, 166	jca 521	. 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 19456	. 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 1363	1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ifcif 4485  {csn 4588  ⟨cop 4838  ∪ cuni 4870   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653  ran crn 5656  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   Er wer 8739  [cec 8740   / cqs 8741  0cc0 11224  1c1 11225   + caddc 11227   · cmul 11229   ≤ cle 11368   − cmin 11565   / cdiv 11997  2c2 12421  4c4 12423  [,]cicc 13503  Basecbs 17413  +_gcplusg 17454  Grpcgrp 19170   GrpHom cghm 19453  TopOnctopon 23255   Cn ccn 23569  IIcii 25223   ≃_phcphtpc 25317  *_𝑝cpco 25348   π₁ cpi1 25351

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302  ax-addf 11303

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-om 7899  df-1st 8022  df-2nd 8023  df-supp 8202  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-er 8742  df-ec 8744  df-qs 8748  df-map 8874  df-ixp 8951  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-fsupp 9379  df-fi 9427  df-sup 9458  df-inf 9459  df-oi 9528  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-q 13100  df-rp 13145  df-xneg 13265  df-xadd 13266  df-xmul 13267  df-ioo 13504  df-icc 13507  df-fz 13664  df-fzo 13813  df-seq 14169  df-exp 14229  df-hash 14499  df-cj 15292  df-re 15293  df-im 15294  df-sqrt 15428  df-abs 15429  df-struct 17351  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-starv 17469  df-sca 17470  df-vsca 17471  df-ip 17472  df-tset 17473  df-ple 17474  df-ds 17476  df-unif 17477  df-hom 17478  df-cco 17479  df-rest 17619  df-topn 17620  df-0g 17638  df-gsum 17639  df-topgen 17640  df-pt 17641  df-prds 17644  df-xrs 17700  df-qtop 17705  df-imas 17706  df-qus 17707  df-xps 17708  df-mre 17782  df-mrc 17783  df-acs 17785  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-submnd 19005  df-grp 19173  df-mulg 19304  df-ghm 19454  df-cntz 19558  df-cmn 20023  df-psmet 21697  df-xmet 21698  df-met 21699  df-bl 21700  df-mopn 21701  df-cnfld 21706  df-top 23239  df-topon 23256  df-topsp 23278  df-bases 23291  df-cld 23364  df-cn 23572  df-cnp 23573  df-tx 23908  df-hmeo 24101  df-xms 24666  df-ms 24667  df-tms 24668  df-ii 25225  df-htpy 25318  df-phtpy 25319  df-phtpc 25340  df-pco 25353  df-om1 25354  df-pi1 25356

This theorem is used by:  pi1xfrcnv  25405  pi1xfrgim  25406