Theorem pi1xfrcnvlem 24954

Description: Given a path 𝐹 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

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)))
pi1xfrcnv.h	⊢ 𝐻 = ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)

Assertion

Ref	Expression
pi1xfrcnvlem	⊢ (𝜑 → ◡𝐺 ⊆ 𝐻)

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

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

Proof of Theorem pi1xfrcnvlem

Step	Hyp	Ref	Expression
1		pi1xfr.g	. . . 4 ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)
2		fvex 6835	. . . . 5 ⊢ ( ≃ph‘𝐽) ∈ V
3		ecexg 8629	. . . . 5 ⊢ (( ≃ph‘𝐽) ∈ V → [𝑔]( ≃ph‘𝐽) ∈ V)
4	2, 3	mp1i 13	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [𝑔]( ≃ph‘𝐽) ∈ V)
5		ecexg 8629	. . . . 5 ⊢ (( ≃ph‘𝐽) ∈ V → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) ∈ V)
6	2, 5	mp1i 13	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) ∈ V)
7	1, 4, 6	fliftcnv 7248	. . 3 ⊢ (𝜑 → ◡𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)⟩))
8		pi1xfr.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
9		pi1xfr.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))
10	9	pcorevcl 24923	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
11	8, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
12	11	simp1d 1142	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽))
14		pi1xfr.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (𝐽 π1 (𝐹‘0))
15		pi1xfr.j	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
16		iitopon 24770	. . . . . . . . . . . . . 14 ⊢ II ∈ (TopOn‘(0[,]1))
17		cnf2 23134	. . . . . . . . . . . . . 14 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋)
18	16, 15, 8, 17	mp3an2i 1468	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋)
19		0elunit 13372	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
20		ffvelcdm 7015	. . . . . . . . . . . . 13 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ 𝑋)
21	18, 19, 20	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘0) ∈ 𝑋)
22		pi1xfr.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑃)
23	22	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = (Base‘𝑃))
24	14, 15, 21, 23	pi1eluni 24940	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↔ (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0))))
25	24	biimpa 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0)))
26	25	simp1d 1142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔 ∈ (II Cn 𝐽))
27	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
28	25	simp3d 1144	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘1) = (𝐹‘0))
29	26, 27, 28	pcocn 24915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
30	11	simp3d 1144	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))
31	30	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
32	25	simp2d 1143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘0) = (𝐹‘0))
33	31, 32	eqtr4d 2767	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = (𝑔‘0))
34	26, 27	pco0 24912	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘0) = (𝑔‘0))
35	33, 34	eqtr4d 2767	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘0))
36	13, 29, 35	pcocn 24915	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
37	13, 29	pco0 24912	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
38	11	simp2d 1143	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1))
39	38	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
40	37, 39	eqtrd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
41	13, 29	pco1 24913	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘1))
42	26, 27	pco1 24913	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
43	41, 42	eqtrd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
44		pi1xfr.q	. . . . . . . . 9 ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))
45		1elunit 13373	. . . . . . . . . 10 ⊢ 1 ∈ (0[,]1)
46		ffvelcdm 7015	. . . . . . . . . 10 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ 𝑋)
47	18, 45, 46	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) ∈ 𝑋)
48		eqidd 2730	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑄) = (Base‘𝑄))
49	44, 15, 47, 48	pi1eluni 24940	. . . . . . . 8 ⊢ (𝜑 → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄) ↔ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽) ∧ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1) ∧ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))))
50	49	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄) ↔ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽) ∧ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1) ∧ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))))
51	36, 40, 43, 50	mpbir3and 1343	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
52		eqidd 2730	. . . . . 6 ⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))) = (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))))
53		eqidd 2730	. . . . . 6 ⊢ (𝜑 → (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) = (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩))
54		eceq1 8664	. . . . . . 7 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → [ℎ]( ≃ph‘𝐽) = [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))
55		oveq1 7356	. . . . . . . . 9 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → (ℎ(*𝑝‘𝐽)𝐼) = ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))
56	55	oveq2d 7365	. . . . . . . 8 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → (𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼)) = (𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼)))
57	56	eceq1d 8665	. . . . . . 7 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽) = [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽))
58	54, 57	opeq12d 4832	. . . . . 6 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩ = ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)
59	51, 52, 53, 58	fmptco 7063	. . . . 5 ⊢ (𝜑 → ((ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) ∘ (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) = (𝑔 ∈ ∪ 𝐵 ↦ ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩))
60		phtpcer 24892	. . . . . . . . 9 ⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽)
61	60	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ( ≃ph‘𝐽) Er (II Cn 𝐽))
62	13, 26	pco0 24912	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)𝑔)‘0) = (𝐼‘0))
63	62, 39	eqtr2d 2765	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) = ((𝐼(*𝑝‘𝐽)𝑔)‘0))
64	61, 27	erref 8645	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐹( ≃ph‘𝐽)𝐹)
65	61, 13	erref 8645	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐼( ≃ph‘𝐽)𝐼)
66		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ ((0[,]1) × {(𝐹‘0)}) = ((0[,]1) × {(𝐹‘0)})
67	66	pcopt2 24921	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘1) = (𝐹‘0)) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))( ≃ph‘𝐽)𝑔)
68	26, 28, 67	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))( ≃ph‘𝐽)𝑔)
69	39	eqcomd 2735	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
70		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), if(𝑥 ≤ (1 / 4), (2 · 𝑥), (𝑥 + (1 / 4))), ((𝑥 / 2) + (1 / 2)))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), if(𝑥 ≤ (1 / 4), (2 · 𝑥), (𝑥 + (1 / 4))), ((𝑥 / 2) + (1 / 2))))
71	26, 27, 13, 28, 69, 70	pcoass 24922	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)(𝑔(*𝑝‘𝐽)(𝐹(*𝑝‘𝐽)𝐼)))
72	27, 13	pco0 24912	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘0) = (𝐹‘0))
73	28, 72	eqtr4d 2767	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘1) = ((𝐹(*𝑝‘𝐽)𝐼)‘0))
74	61, 26	erref 8645	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔( ≃ph‘𝐽)𝑔)
75	9, 66	pcorev2 24926	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
76	27, 75	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
77	73, 74, 76	pcohtpy 24918	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)(𝐹(*𝑝‘𝐽)𝐼))( ≃ph‘𝐽)(𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)})))
78	61, 71, 77	ertr2d 8642	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))( ≃ph‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼))
79	61, 68, 78	ertr3d 8643	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔( ≃ph‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼))
80	33, 65, 79	pcohtpy 24918	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(𝐼(*𝑝‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)))
81	42, 39	eqtr4d 2767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘1) = (𝐼‘0))
82	13, 29, 13, 35, 81, 70	pcoass 24922	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)(𝐼(*𝑝‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)))
83	61, 80, 82	ertr4d 8644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))
84	63, 64, 83	pcohtpy 24918	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔))( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼)))
85	27, 13, 26, 69, 33, 70	pcoass 24922	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔)))
86	27, 13	pco1 24913	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝐼‘1))
87	86, 33	eqtrd 2764	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝑔‘0))
88	87, 76, 74	pcohtpy 24918	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔))
89	66	pcopt 24920	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0)) → (((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔)
90	26, 32, 89	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔)
91	61, 88, 90	ertrd 8641	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔)
92	61, 85, 91	ertr3d 8643	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔))( ≃ph‘𝐽)𝑔)
93	61, 84, 92	ertr3d 8643	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))( ≃ph‘𝐽)𝑔)
94	61, 93	erthi 8681	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽) = [𝑔]( ≃ph‘𝐽))
95	94	opeq2d 4831	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩ = ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)⟩)
96	95	mpteq2dva 5185	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↦ ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) = (𝑔 ∈ ∪ 𝐵 ↦ ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)⟩))
97	59, 96	eqtrd 2764	. . . 4 ⊢ (𝜑 → ((ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) ∘ (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) = (𝑔 ∈ ∪ 𝐵 ↦ ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)⟩))
98	97	rneqd 5880	. . 3 ⊢ (𝜑 → ran ((ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) ∘ (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)⟩))
99	7, 98	eqtr4d 2767	. 2 ⊢ (𝜑 → ◡𝐺 = ran ((ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) ∘ (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))))
100		rncoss 5918	. . 3 ⊢ ran ((ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) ∘ (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) ⊆ ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)
101		pi1xfrcnv.h	. . 3 ⊢ 𝐻 = ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)
102	100, 101	sseqtrri 3985	. 2 ⊢ ran ((ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) ∘ (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) ⊆ 𝐻
103	99, 102	eqsstrdi 3980	1 ⊢ (𝜑 → ◡𝐺 ⊆ 𝐻)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ⊆ wss 3903  ifcif 4476  {csn 4577  ⟨cop 4583  ∪ cuni 4858   class class class wbr 5092   ↦ cmpt 5173   × cxp 5617  ^◡ccnv 5618  ran crn 5620   ∘ ccom 5623  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   Er wer 8622  [cec 8623  0cc0 11009  1c1 11010   + caddc 11012   · cmul 11014   ≤ cle 11150   − cmin 11347   / cdiv 11777  2c2 12183  4c4 12185  [,]cicc 13251  Basecbs 17120  TopOnctopon 22795   Cn ccn 23109  IIcii 24766   ≃_phcphtpc 24866  *_𝑝cpco 24898   π₁ cpi1 24901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087  ax-addf 11088

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-ec 8627  df-qs 8631  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-fi 9301  df-sup 9332  df-inf 9333  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-q 12850  df-rp 12894  df-xneg 13014  df-xadd 13015  df-xmul 13016  df-ioo 13252  df-icc 13255  df-fz 13411  df-fzo 13558  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-hom 17185  df-cco 17186  df-rest 17326  df-topn 17327  df-0g 17345  df-gsum 17346  df-topgen 17347  df-pt 17348  df-prds 17351  df-xrs 17406  df-qtop 17411  df-imas 17412  df-qus 17413  df-xps 17414  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-submnd 18658  df-mulg 18947  df-cntz 19196  df-cmn 19661  df-psmet 21253  df-xmet 21254  df-met 21255  df-bl 21256  df-mopn 21257  df-cnfld 21262  df-top 22779  df-topon 22796  df-topsp 22818  df-bases 22831  df-cld 22904  df-cn 23112  df-cnp 23113  df-tx 23447  df-hmeo 23640  df-xms 24206  df-ms 24207  df-tms 24208  df-ii 24768  df-htpy 24867  df-phtpy 24868  df-phtpc 24889  df-pco 24903  df-om1 24904  df-pi1 24906

This theorem is referenced by:  pi1xfrcnv  24955