pi1inv - Metamath Proof Explorer

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Theorem pi1inv 24959

Description: An inverse in the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)

**Hypotheses**
Ref	Expression
pi1grp.2	⊢ 𝐺 = (𝐽 π₁ 𝑌)
pi1inv.n	⊢ 𝑁 = (inv_g‘𝐺)
pi1inv.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
pi1inv.y	⊢ (𝜑 → 𝑌 ∈ 𝑋)
pi1inv.f	⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
pi1inv.0	⊢ (𝜑 → (𝐹‘0) = 𝑌)
pi1inv.1	⊢ (𝜑 → (𝐹‘1) = 𝑌)
pi1inv.i	⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))

**Assertion**
Ref	Expression
pi1inv	⊢ (𝜑 → (𝑁‘[𝐹]( ≃_ph‘𝐽)) = [𝐼]( ≃_ph‘𝐽))

Distinct variable groups: 𝑥,𝐹 𝑥,𝐺 𝑥,𝐽 𝜑,𝑥 𝑥,𝑌 Allowed substitution hints: 𝐼(𝑥) 𝑁(𝑥) 𝑋(𝑥)

**Proof of Theorem pi1inv**
Step	Hyp	Ref	Expression
1		pi1grp.2	. . . 4 ⊢ 𝐺 = (𝐽 π₁ 𝑌)
2		eqid 2730	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		pi1inv.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
4		pi1inv.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
5		eqid 2730	. . . 4 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
6		pi1inv.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
7		pi1inv.i	. . . . . . . 8 ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))
8	7	pcorevcl 24932	. . . . . . 7 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
9	6, 8	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
10	9	simp1d 1142	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))
11	9	simp2d 1143	. . . . . 6 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1))
12		pi1inv.1	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) = 𝑌)
13	11, 12	eqtrd 2765	. . . . 5 ⊢ (𝜑 → (𝐼‘0) = 𝑌)
14	9	simp3d 1144	. . . . . 6 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))
15		pi1inv.0	. . . . . 6 ⊢ (𝜑 → (𝐹‘0) = 𝑌)
16	14, 15	eqtrd 2765	. . . . 5 ⊢ (𝜑 → (𝐼‘1) = 𝑌)
17	2	a1i 11	. . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
18	1, 3, 4, 17	pi1eluni 24949	. . . . 5 ⊢ (𝜑 → (𝐼 ∈ ∪ (Base‘𝐺) ↔ (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = 𝑌 ∧ (𝐼‘1) = 𝑌)))
19	10, 13, 16, 18	mpbir3and 1343	. . . 4 ⊢ (𝜑 → 𝐼 ∈ ∪ (Base‘𝐺))
20	1, 3, 4, 17	pi1eluni 24949	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ ∪ (Base‘𝐺) ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
21	6, 15, 12, 20	mpbir3and 1343	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ∪ (Base‘𝐺))
22	1, 2, 3, 4, 5, 19, 21	pi1addval 24955	. . 3 ⊢ (𝜑 → ([𝐼]( ≃_ph‘𝐽)(+_g‘𝐺)[𝐹]( ≃_ph‘𝐽)) = [(𝐼(*_𝑝‘𝐽)𝐹)]( ≃_ph‘𝐽))
23		phtpcer 24901	. . . . 5 ⊢ ( ≃_ph‘𝐽) Er (II Cn 𝐽)
24	23	a1i 11	. . . 4 ⊢ (𝜑 → ( ≃_ph‘𝐽) Er (II Cn 𝐽))
25		eqid 2730	. . . . . . 7 ⊢ ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {(𝐹‘1)})
26	7, 25	pcorev 24934	. . . . . 6 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐼(*_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)((0[,]1) × {(𝐹‘1)}))
27	6, 26	syl 17	. . . . 5 ⊢ (𝜑 → (𝐼(*_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)((0[,]1) × {(𝐹‘1)}))
28	12	sneqd 4604	. . . . . 6 ⊢ (𝜑 → {(𝐹‘1)} = {𝑌})
29	28	xpeq2d 5671	. . . . 5 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {𝑌}))
30	27, 29	breqtrd 5136	. . . 4 ⊢ (𝜑 → (𝐼(*_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)((0[,]1) × {𝑌}))
31	24, 30	erthi 8730	. . 3 ⊢ (𝜑 → [(𝐼(*_𝑝‘𝐽)𝐹)]( ≃_ph‘𝐽) = [((0[,]1) × {𝑌})]( ≃_ph‘𝐽))
32		eqid 2730	. . . . 5 ⊢ ((0[,]1) × {𝑌}) = ((0[,]1) × {𝑌})
33	1, 2, 3, 4, 32	pi1grplem 24956	. . . 4 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ [((0[,]1) × {𝑌})]( ≃_ph‘𝐽) = (0_g‘𝐺)))
34	33	simprd 495	. . 3 ⊢ (𝜑 → [((0[,]1) × {𝑌})]( ≃_ph‘𝐽) = (0_g‘𝐺))
35	22, 31, 34	3eqtrd 2769	. 2 ⊢ (𝜑 → ([𝐼]( ≃_ph‘𝐽)(+_g‘𝐺)[𝐹]( ≃_ph‘𝐽)) = (0_g‘𝐺))
36	33	simpld 494	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
37	1, 2, 3, 4, 6, 15, 12	elpi1i 24953	. . 3 ⊢ (𝜑 → [𝐹]( ≃_ph‘𝐽) ∈ (Base‘𝐺))
38	1, 2, 3, 4, 10, 13, 16	elpi1i 24953	. . 3 ⊢ (𝜑 → [𝐼]( ≃_ph‘𝐽) ∈ (Base‘𝐺))
39		eqid 2730	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
40		pi1inv.n	. . . 4 ⊢ 𝑁 = (inv_g‘𝐺)
41	2, 5, 39, 40	grpinvid2 18931	. . 3 ⊢ ((𝐺 ∈ Grp ∧ [𝐹]( ≃_ph‘𝐽) ∈ (Base‘𝐺) ∧ [𝐼]( ≃_ph‘𝐽) ∈ (Base‘𝐺)) → ((𝑁‘[𝐹]( ≃_ph‘𝐽)) = [𝐼]( ≃_ph‘𝐽) ↔ ([𝐼]( ≃_ph‘𝐽)(+_g‘𝐺)[𝐹]( ≃_ph‘𝐽)) = (0_g‘𝐺)))
42	36, 37, 38, 41	syl3anc 1373	. 2 ⊢ (𝜑 → ((𝑁‘[𝐹]( ≃_ph‘𝐽)) = [𝐼]( ≃_ph‘𝐽) ↔ ([𝐼]( ≃_ph‘𝐽)(+_g‘𝐺)[𝐹]( ≃_ph‘𝐽)) = (0_g‘𝐺)))
43	35, 42	mpbird 257	1 ⊢ (𝜑 → (𝑁‘[𝐹]( ≃_ph‘𝐽)) = [𝐼]( ≃_ph‘𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {csn 4592 ∪ cuni 4874 class class class wbr 5110 ↦ cmpt 5191 × cxp 5639 ‘cfv 6514 (class class class)co 7390 Er wer 8671 [cec 8672 0cc0 11075 1c1 11076 − cmin 11412 [,]cicc 13316 Basecbs 17186 +_gcplusg 17227 0_gc0g 17409 Grpcgrp 18872 inv_gcminusg 18873 TopOnctopon 22804 Cn ccn 23118 IIcii 24775 ≃_phcphtpc 24875 *_𝑝cpco 24907 π₁ cpi1 24910

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-ec 8676 df-qs 8680 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-icc 13320 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-qus 17479 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-cn 23121 df-cnp 23122 df-tx 23456 df-hmeo 23649 df-xms 24215 df-ms 24216 df-tms 24217 df-ii 24777 df-htpy 24876 df-phtpy 24877 df-phtpc 24898 df-pco 24912 df-om1 24913 df-pi1 24915

This theorem is referenced by: (None)

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