pi1xfr - Metamath Proof Explorer

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Theorem pi1xfr 24562

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 24386	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
3		pi1xfr.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
4		cnf2 22744	. . . . 5 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋)
5	2, 1, 3, 4	mp3an2i 1466	. . . 4 ⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋)
6		0elunit 13442	. . . 4 ⊢ 0 ∈ (0[,]1)
7		ffvelcdm 7080	. . . 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 24557	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘0) ∈ 𝑋) → 𝑃 ∈ Grp)
11	1, 8, 10	syl2anc 584	. 2 ⊢ (𝜑 → 𝑃 ∈ Grp)
12		1elunit 13443	. . . 4 ⊢ 1 ∈ (0[,]1)
13		ffvelcdm 7080	. . . 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 24557	. . 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 24532	. . . . . 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 1142	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))
24	22	simp2d 1143	. . . . 5 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1))
25	24	eqcomd 2738	. . . 4 ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0))
26	22	simp3d 1144	. . . 4 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))
27	9, 15, 18, 19, 1, 3, 23, 25, 26	pi1xfrf 24560	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄))
28	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑃))
29	9, 1, 8, 28	pi1bas2 24548	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / ( ≃_ph‘𝐽)))
30	29	eleq2d 2819	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽))))
31	30	biimpa 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽)))
32		eqid 2732	. . . . . 6 ⊢ (∪ 𝐵 / ( ≃_ph‘𝐽)) = (∪ 𝐵 / ( ≃_ph‘𝐽))
33		fvoveq1 7428	. . . . . . . 8 ⊢ ([𝑓]( ≃_ph‘𝐽) = 𝑦 → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)) = (𝐺‘(𝑦(+_g‘𝑃)𝑧)))
34		fveq2 6888	. . . . . . . . 9 ⊢ ([𝑓]( ≃_ph‘𝐽) = 𝑦 → (𝐺‘[𝑓]( ≃_ph‘𝐽)) = (𝐺‘𝑦))
35	34	oveq1d 7420	. . . . . . . 8 ⊢ ([𝑓]( ≃_ph‘𝐽) = 𝑦 → ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧)))
36	33, 35	eqeq12d 2748	. . . . . . 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 3177	. . . . . 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 2819	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽))))
39	38	biimpa 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽)))
40	39	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽)))
41		oveq2 7413	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃_ph‘𝐽) = 𝑧 → ([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽)) = ([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧))
42	41	fveq2d 6892	. . . . . . . . . 10 ⊢ ([ℎ]( ≃_ph‘𝐽) = 𝑧 → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽))) = (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)))
43		fveq2 6888	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃_ph‘𝐽) = 𝑧 → (𝐺‘[ℎ]( ≃_ph‘𝐽)) = (𝐺‘𝑧))
44	43	oveq2d 7421	. . . . . . . . . 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 2748	. . . . . . . . 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 24502	. . . . . . . . . . . . . 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 24549	. . . . . . . . . . . . . . . . . . . . 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 1142	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽))
51	50	3adant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽))
52	9, 1, 8, 28	pi1eluni 24549	. . . . . . . . . . . . . . . . . . . . 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 1469	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0)))
55	54	simp1d 1142	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ (II Cn 𝐽))
56	51, 55	pco0 24521	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)ℎ)‘0) = (𝑓‘0))
57	49	simp2d 1143	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0))
58	57	3adant3 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0))
59	56, 58	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)ℎ)‘0) = (𝐹‘0))
60	49	simp3d 1144	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0))
61	60	3adant3 1132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0))
62	54	simp2d 1143	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘0) = (𝐹‘0))
63	61, 62	eqtr4d 2775	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (ℎ‘0))
64	51, 55, 63	pcocn 24524	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*_𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽))
65	3	3ad2ant1 1133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
66	64, 65	pco0 24521	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹)‘0) = ((𝑓(*_𝑝‘𝐽)ℎ)‘0))
67	26	3ad2ant1 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
68	59, 66, 67	3eqtr4rd 2783	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹)‘0))
69	23	3ad2ant1 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽))
70	47, 69	erref 8719	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼( ≃_ph‘𝐽)𝐼)
71	54	simp3d 1144	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘1) = (𝐹‘0))
72		eqid 2732	. . . . . . . . . . . . . . . . 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 24531	. . . . . . . . . . . . . . . 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 24521	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*_𝑝‘𝐽)𝐹)‘0) = (ℎ‘0))
75	63, 74	eqtr4d 2775	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = ((ℎ(*_𝑝‘𝐽)𝐹)‘0))
76	47, 51	erref 8719	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓( ≃_ph‘𝐽)𝑓)
77	65, 69	pco1 24522	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*_𝑝‘𝐽)𝐼)‘1) = (𝐼‘1))
78	62, 74, 67	3eqtr4rd 2783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((ℎ(*_𝑝‘𝐽)𝐹)‘0))
79	77, 78	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(_𝑝‘𝐽)𝐼)‘1) = ((ℎ(_𝑝‘𝐽)𝐹)‘0))
80		eqid 2732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0[,]1) × {(𝐹‘0)}) = ((0[,]1) × {(𝐹‘0)})
81	20, 80	pcorev2 24535	. . . . . . . . . . . . . . . . . . . . . 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 24524	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*_𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
84	47, 83	erref 8719	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))
85	79, 82, 84	pcohtpy 24527	. . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*_𝑝‘𝐽)𝐹)‘0) = (𝐹‘0))
87	80	pcopt 24529	. . . . . . . . . . . . . . . . . . . . 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 8715	. . . . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
91	90	eqcomd 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
92	65, 69, 83, 91, 78, 72	pcoass 24531	. . . . . . . . . . . . . . . . . . 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 8717	. . . . . . . . . . . . . . . . . 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 24527	. . . . . . . . . . . . . . . . 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 24524	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
96	69, 83	pco0 24521	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
97	96, 90	eqtrd 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
98	97	eqcomd 2738	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘0))
99	51, 65, 95, 61, 98, 72	pcoass 24531	. . . . . . . . . . . . . . . . 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 8718	. . . . . . . . . . . . . . . 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 8715	. . . . . . . . . . . . . . 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 24527	. . . . . . . . . . . . . 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 24524	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓(*_𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
105	104	3adant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*_𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
106	50, 103	pco0 24521	. . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*_𝑝‘𝐽)𝐹)‘0))
109	108	3adant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*_𝑝‘𝐽)𝐹)‘0))
110	51, 65	pco1 24522	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
111	110, 97	eqtr4d 2775	. . . . . . . . . . . . . . 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 24531	. . . . . . . . . . . . . 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 8718	. . . . . . . . . . . . 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 8750	. . . . . . . . . . . 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 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋))
116	51, 55	pco1 24522	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)ℎ)‘1) = (ℎ‘1))
117	116, 71	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)ℎ)‘1) = (𝐹‘0))
118	9, 1, 8, 28	pi1eluni 24549	. . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . 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 1342	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*_𝑝‘𝐽)ℎ) ∈ ∪ 𝐵)
121	9, 15, 18, 19, 115, 65, 69, 91, 67, 120	pi1xfrval 24561	. . . . . . . . . . . 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 2732	. . . . . . . . . . . . 13 ⊢ (Base‘𝑄) = (Base‘𝑄)
123	14	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) ∈ 𝑋)
124		eqid 2732	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑄) = (+_g‘𝑄)
125	23	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽))
126	125, 104, 108	pcocn 24524	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
127	126	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
128	125, 104	pco0 24521	. . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
131	130	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
132	125, 104	pco1 24522	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘1) = ((𝑓(*_𝑝‘𝐽)𝐹)‘1))
133	50, 103	pco1 24522	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
134	132, 133	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
135	134	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
136		eqidd 2733	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (Base‘𝑄) = (Base‘𝑄))
137	15, 115, 123, 136	pi1eluni 24549	. . . . . . . . . . . . . 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 1342	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
139	69, 83	pco1 24522	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘1) = ((ℎ(*_𝑝‘𝐽)𝐹)‘1))
140	55, 65	pco1 24522	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*_𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
141	139, 140	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
142	15, 115, 123, 136	pi1eluni 24549	. . . . . . . . . . . . . 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 1342	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
144	15, 122, 115, 123, 124, 138, 143	pi1addval 24555	. . . . . . . . . . . 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 2782	. . . . . . . . . . 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 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘0) ∈ 𝑋)
147		eqid 2732	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
148		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵)
149		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ ∪ 𝐵)
150	9, 18, 115, 146, 147, 148, 149	pi1addval 24555	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽)) = [(𝑓(*_𝑝‘𝐽)ℎ)]( ≃_ph‘𝐽))
151	150	fveq2d 6892	. . . . . . . . . . 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 24561	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐺‘[𝑓]( ≃_ph‘𝐽)) = [(𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽))
156	155	3adant3 1132	. . . . . . . . . . . 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 24561	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[ℎ]( ≃_ph‘𝐽)) = [(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽))
158	156, 157	oveq12d 7423	. . . . . . . . . . 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 2782	. . . . . . . . . 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 1118	. . . . . . . . 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 8774	. . . . . . . 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 3146	. . . . . 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 8774	. . . . 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 3146	. . 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 19086	. 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 1344	1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ifcif 4527 {csn 4627 ⟨cop 4633 ∪ cuni 4907 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ran crn 5676 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 Er wer 8696 [cec 8697 / cqs 8698 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 ≤ cle 11245 − cmin 11440 / cdiv 11867 2c2 12263 4c4 12265 [,]cicc 13323 Basecbs 17140 +_gcplusg 17193 Grpcgrp 18815 GrpHom cghm 19083 TopOnctopon 22403 Cn ccn 22719 IIcii 24382 ≃_phcphtpc 24476 *_𝑝cpco 24507 π₁ cpi1 24510

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-qus 17451 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-mulg 18945 df-ghm 19084 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-cn 22722 df-cnp 22723 df-tx 23057 df-hmeo 23250 df-xms 23817 df-ms 23818 df-tms 23819 df-ii 24384 df-htpy 24477 df-phtpy 24478 df-phtpc 24499 df-pco 24512 df-om1 24513 df-pi1 24515

This theorem is referenced by: pi1xfrcnv 24564 pi1xfrgim 24565

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