Theorem pi1xfrcnvlem 24803

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 6903	. . . . 5 ⊢ ( ≃ph‘𝐽) ∈ V
3		ecexg 8709	. . . . 5 ⊢ (( ≃ph‘𝐽) ∈ V → [𝑔]( ≃ph‘𝐽) ∈ V)
4	2, 3	mp1i 13	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [𝑔]( ≃ph‘𝐽) ∈ V)
5		ecexg 8709	. . . . 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 7310	. . 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 24772	. . . . . . . . . . 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 1140	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))
13	12	adantr 479	. . . . . . . 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 24619	. . . . . . . . . . . . . 14 ⊢ II ∈ (TopOn‘(0[,]1))
17		cnf2 22973	. . . . . . . . . . . . . 14 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋)
18	16, 15, 8, 17	mp3an2i 1464	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋)
19		0elunit 13450	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
20		ffvelcdm 7082	. . . . . . . . . . . . 13 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ 𝑋)
21	18, 19, 20	sylancl 584	. . . . . . . . . . . 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 24789	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↔ (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0))))
25	24	biimpa 475	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0)))
26	25	simp1d 1140	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔 ∈ (II Cn 𝐽))
27	8	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
28	25	simp3d 1142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘1) = (𝐹‘0))
29	26, 27, 28	pcocn 24764	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
30	11	simp3d 1142	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))
31	30	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
32	25	simp2d 1141	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘0) = (𝐹‘0))
33	31, 32	eqtr4d 2773	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = (𝑔‘0))
34	26, 27	pco0 24761	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘0) = (𝑔‘0))
35	33, 34	eqtr4d 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘0))
36	13, 29, 35	pcocn 24764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
37	13, 29	pco0 24761	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
38	11	simp2d 1141	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1))
39	38	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
40	37, 39	eqtrd 2770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
41	13, 29	pco1 24762	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘1))
42	26, 27	pco1 24762	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
43	41, 42	eqtrd 2770	. . . . . . 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 13451	. . . . . . . . . 10 ⊢ 1 ∈ (0[,]1)
46		ffvelcdm 7082	. . . . . . . . . 10 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ 𝑋)
47	18, 45, 46	sylancl 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) ∈ 𝑋)
48		eqidd 2731	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑄) = (Base‘𝑄))
49	44, 15, 47, 48	pi1eluni 24789	. . . . . . . 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 479	. . . . . . 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 1340	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
52		eqidd 2731	. . . . . 6 ⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))) = (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))))
53		eqidd 2731	. . . . . 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 8743	. . . . . . 7 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → [ℎ]( ≃ph‘𝐽) = [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))
55		oveq1 7418	. . . . . . . . 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 7427	. . . . . . . 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 8744	. . . . . . 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 4880	. . . . . 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 7128	. . . . 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 24741	. . . . . . . . 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 24761	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)𝑔)‘0) = (𝐼‘0))
63	62, 39	eqtr2d 2771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) = ((𝐼(*𝑝‘𝐽)𝑔)‘0))
64	61, 27	erref 8725	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐹( ≃ph‘𝐽)𝐹)
65	61, 13	erref 8725	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐼( ≃ph‘𝐽)𝐼)
66		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ ((0[,]1) × {(𝐹‘0)}) = ((0[,]1) × {(𝐹‘0)})
67	66	pcopt2 24770	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘1) = (𝐹‘0)) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))( ≃ph‘𝐽)𝑔)
68	26, 28, 67	syl2anc 582	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))( ≃ph‘𝐽)𝑔)
69	39	eqcomd 2736	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
70		eqid 2730	. . . . . . . . . . . . . . 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 24771	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)(𝑔(*𝑝‘𝐽)(𝐹(*𝑝‘𝐽)𝐼)))
72	27, 13	pco0 24761	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘0) = (𝐹‘0))
73	28, 72	eqtr4d 2773	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘1) = ((𝐹(*𝑝‘𝐽)𝐼)‘0))
74	61, 26	erref 8725	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔( ≃ph‘𝐽)𝑔)
75	9, 66	pcorev2 24775	. . . . . . . . . . . . . . . 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 24767	. . . . . . . . . . . . . 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 8722	. . . . . . . . . . . . 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 8723	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔( ≃ph‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼))
80	33, 65, 79	pcohtpy 24767	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(𝐼(*𝑝‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)))
81	42, 39	eqtr4d 2773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘1) = (𝐼‘0))
82	13, 29, 13, 35, 81, 70	pcoass 24771	. . . . . . . . . . 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 8724	. . . . . . . . . 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 24767	. . . . . . . . 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 24771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔)))
86	27, 13	pco1 24762	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝐼‘1))
87	86, 33	eqtrd 2770	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝑔‘0))
88	87, 76, 74	pcohtpy 24767	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔))
89	66	pcopt 24769	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0)) → (((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔)
90	26, 32, 89	syl2anc 582	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔)
91	61, 88, 90	ertrd 8721	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔)
92	61, 85, 91	ertr3d 8723	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔))( ≃ph‘𝐽)𝑔)
93	61, 84, 92	ertr3d 8723	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))( ≃ph‘𝐽)𝑔)
94	61, 93	erthi 8756	. . . . . . 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 4879	. . . . . 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 5247	. . . . 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 2770	. . . 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 5936	. . 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 2773	. 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 5970	. . 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 4018	. 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 4035	1 ⊢ (𝜑 → ◡𝐺 ⊆ 𝐻)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ⊆ wss 3947  ifcif 4527  {csn 4627  ⟨cop 4633  ∪ cuni 4907   class class class wbr 5147   ↦ cmpt 5230   × cxp 5673  ^◡ccnv 5674  ran crn 5676   ∘ ccom 5679  ⟶wf 6538  ‘cfv 6542  (class class class)co 7411   Er wer 8702  [cec 8703  0cc0 11112  1c1 11113   + caddc 11115   · cmul 11117   ≤ cle 11253   − cmin 11448   / cdiv 11875  2c2 12271  4c4 12273  [,]cicc 13331  Basecbs 17148  TopOnctopon 22632   Cn ccn 22948  IIcii 24615   ≃_phcphtpc 24715  *_𝑝cpco 24747   π₁ cpi1 24750

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-ec 8707  df-qs 8711  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-ioo 13332  df-icc 13335  df-fz 13489  df-fzo 13632  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-hom 17225  df-cco 17226  df-rest 17372  df-topn 17373  df-0g 17391  df-gsum 17392  df-topgen 17393  df-pt 17394  df-prds 17397  df-xrs 17452  df-qtop 17457  df-imas 17458  df-qus 17459  df-xps 17460  df-mre 17534  df-mrc 17535  df-acs 17537  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-mulg 18987  df-cntz 19222  df-cmn 19691  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-cnfld 21145  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-cn 22951  df-cnp 22952  df-tx 23286  df-hmeo 23479  df-xms 24046  df-ms 24047  df-tms 24048  df-ii 24617  df-htpy 24716  df-phtpy 24717  df-phtpc 24738  df-pco 24752  df-om1 24753  df-pi1 24755

This theorem is referenced by:  pi1xfrcnv  24804