Theorem pi1xfrcnvlem 22856

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 6201	. . . . 5 ⊢ ( ≃ph‘𝐽) ∈ V
3		ecexg 7746	. . . . 5 ⊢ (( ≃ph‘𝐽) ∈ V → [𝑔]( ≃ph‘𝐽) ∈ V)
4	2, 3	mp1i 13	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [𝑔]( ≃ph‘𝐽) ∈ V)
5		ecexg 7746	. . . . 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 6561	. . 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 22825	. . . . . . . . . . 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 1073	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))
13	12	adantr 481	. . . . . . . 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 22682	. . . . . . . . . . . . . . 15 ⊢ II ∈ (TopOn‘(0[,]1))
17	16	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
18		cnf2 21053	. . . . . . . . . . . . . 14 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋)
19	17, 15, 8, 18	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋)
20		0elunit 12290	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
21		ffvelrn 6357	. . . . . . . . . . . . 13 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ 𝑋)
22	19, 20, 21	sylancl 694	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘0) ∈ 𝑋)
23		pi1xfr.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑃)
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = (Base‘𝑃))
25	14, 15, 22, 24	pi1eluni 22842	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↔ (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0))))
26	25	biimpa 501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0)))
27	26	simp1d 1073	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔 ∈ (II Cn 𝐽))
28	8	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
29	26	simp3d 1075	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘1) = (𝐹‘0))
30	27, 28, 29	pcocn 22817	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
31	11	simp3d 1075	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))
32	31	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
33	26	simp2d 1074	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘0) = (𝐹‘0))
34	32, 33	eqtr4d 2659	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = (𝑔‘0))
35	27, 28	pco0 22814	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘0) = (𝑔‘0))
36	34, 35	eqtr4d 2659	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘0))
37	13, 30, 36	pcocn 22817	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
38	13, 30	pco0 22814	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
39	11	simp2d 1074	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1))
40	39	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
41	38, 40	eqtrd 2656	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
42	13, 30	pco1 22815	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘1))
43	27, 28	pco1 22815	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
44	42, 43	eqtrd 2656	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
45		pi1xfr.q	. . . . . . . . 9 ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))
46		1elunit 12291	. . . . . . . . . 10 ⊢ 1 ∈ (0[,]1)
47		ffvelrn 6357	. . . . . . . . . 10 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ 𝑋)
48	19, 46, 47	sylancl 694	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) ∈ 𝑋)
49		eqidd 2623	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑄) = (Base‘𝑄))
50	45, 15, 48, 49	pi1eluni 22842	. . . . . . . 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))))
51	50	adantr 481	. . . . . . 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))))
52	37, 41, 44, 51	mpbir3and 1245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
53		eqidd 2623	. . . . . 6 ⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))) = (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))))
54		eqidd 2623	. . . . . 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)⟩))
55		eceq1 7782	. . . . . . 7 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → [ℎ]( ≃ph‘𝐽) = [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))
56		oveq1 6657	. . . . . . . . 9 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → (ℎ(*𝑝‘𝐽)𝐼) = ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))
57	56	oveq2d 6666	. . . . . . . 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)))
58	57	eceq1d 7783	. . . . . . 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))
59	55, 58	opeq12d 4410	. . . . . 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)⟩)
60	52, 53, 54, 59	fmptco 6396	. . . . 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)⟩))
61		phtpcer 22794	. . . . . . . . 9 ⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽)
62	61	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ( ≃ph‘𝐽) Er (II Cn 𝐽))
63	13, 27	pco0 22814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)𝑔)‘0) = (𝐼‘0))
64	63, 40	eqtr2d 2657	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) = ((𝐼(*𝑝‘𝐽)𝑔)‘0))
65	62, 28	erref 7762	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐹( ≃ph‘𝐽)𝐹)
66	62, 13	erref 7762	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐼( ≃ph‘𝐽)𝐼)
67		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ ((0[,]1) × {(𝐹‘0)}) = ((0[,]1) × {(𝐹‘0)})
68	67	pcopt2 22823	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘1) = (𝐹‘0)) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))( ≃ph‘𝐽)𝑔)
69	27, 29, 68	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))( ≃ph‘𝐽)𝑔)
70	40	eqcomd 2628	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
71		eqid 2622	. . . . . . . . . . . . . . 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))))
72	27, 28, 13, 29, 70, 71	pcoass 22824	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)(𝑔(*𝑝‘𝐽)(𝐹(*𝑝‘𝐽)𝐼)))
73	28, 13	pco0 22814	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘0) = (𝐹‘0))
74	29, 73	eqtr4d 2659	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘1) = ((𝐹(*𝑝‘𝐽)𝐼)‘0))
75	62, 27	erref 7762	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔( ≃ph‘𝐽)𝑔)
76	9, 67	pcorev2 22828	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
77	28, 76	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
78	74, 75, 77	pcohtpy 22820	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)(𝐹(*𝑝‘𝐽)𝐼))( ≃ph‘𝐽)(𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)})))
79	62, 72, 78	ertr2d 7759	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))( ≃ph‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼))
80	62, 69, 79	ertr3d 7760	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔( ≃ph‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼))
81	34, 66, 80	pcohtpy 22820	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(𝐼(*𝑝‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)))
82	43, 40	eqtr4d 2659	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘1) = (𝐼‘0))
83	13, 30, 13, 36, 82, 71	pcoass 22824	. . . . . . . . . . 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)))
84	62, 81, 83	ertr4d 7761	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))
85	64, 65, 84	pcohtpy 22820	. . . . . . . . 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)))
86	28, 13, 27, 70, 34, 71	pcoass 22824	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔)))
87	28, 13	pco1 22815	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝐼‘1))
88	87, 34	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝑔‘0))
89	88, 77, 75	pcohtpy 22820	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔))
90	67	pcopt 22822	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0)) → (((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔)
91	27, 33, 90	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔)
92	62, 89, 91	ertrd 7758	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔)
93	62, 86, 92	ertr3d 7760	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔))( ≃ph‘𝐽)𝑔)
94	62, 85, 93	ertr3d 7760	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))( ≃ph‘𝐽)𝑔)
95	62, 94	erthi 7793	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽) = [𝑔]( ≃ph‘𝐽))
96	95	opeq2d 4409	. . . . . 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)⟩)
97	96	mpteq2dva 4744	. . . . 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)⟩))
98	60, 97	eqtrd 2656	. . . 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)⟩))
99	98	rneqd 5353	. . 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)⟩))
100	7, 99	eqtr4d 2659	. 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)))))
101		rncoss 5386	. . 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)⟩)
102		pi1xfrcnv.h	. . 3 ⊢ 𝐻 = ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)
103	101, 102	sseqtr4i 3638	. 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
104	100, 103	syl6eqss 3655	1 ⊢ (𝜑 → ◡𝐺 ⊆ 𝐻)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  ifcif 4086  {csn 4177  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  ran crn 5115   ∘ ccom 5118  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   Er wer 7739  [cec 7740  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   ≤ cle 10075   − cmin 10266   / cdiv 10684  2c2 11070  4c4 11072  [,]cicc 12178  Basecbs 15857  TopOnctopon 20715   Cn ccn 21028  IIcii 22678   ≃_phcphtpc 22768  *_𝑝cpco 22800   π₁ cpi1 22803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-qus 16169  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-ii 22680  df-htpy 22769  df-phtpy 22770  df-phtpc 22791  df-pco 22805  df-om1 22806  df-pi1 22808

This theorem is referenced by:  pi1xfrcnv  22857