pi1inv - Metamath Proof Explorer

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Theorem pi1inv 22852

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 2622	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		pi1inv.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
4		pi1inv.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
5		eqid 2622	. . . 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 22825	. . . . . . 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 1073	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))
11	9	simp2d 1074	. . . . . 6 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1))
12		pi1inv.1	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) = 𝑌)
13	11, 12	eqtrd 2656	. . . . 5 ⊢ (𝜑 → (𝐼‘0) = 𝑌)
14	9	simp3d 1075	. . . . . 6 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))
15		pi1inv.0	. . . . . 6 ⊢ (𝜑 → (𝐹‘0) = 𝑌)
16	14, 15	eqtrd 2656	. . . . 5 ⊢ (𝜑 → (𝐼‘1) = 𝑌)
17	2	a1i 11	. . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
18	1, 3, 4, 17	pi1eluni 22842	. . . . 5 ⊢ (𝜑 → (𝐼 ∈ ∪ (Base‘𝐺) ↔ (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = 𝑌 ∧ (𝐼‘1) = 𝑌)))
19	10, 13, 16, 18	mpbir3and 1245	. . . 4 ⊢ (𝜑 → 𝐼 ∈ ∪ (Base‘𝐺))
20	1, 3, 4, 17	pi1eluni 22842	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ ∪ (Base‘𝐺) ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
21	6, 15, 12, 20	mpbir3and 1245	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ∪ (Base‘𝐺))
22	1, 2, 3, 4, 5, 19, 21	pi1addval 22848	. . 3 ⊢ (𝜑 → ([𝐼]( ≃_ph‘𝐽)(+_g‘𝐺)[𝐹]( ≃_ph‘𝐽)) = [(𝐼(*_𝑝‘𝐽)𝐹)]( ≃_ph‘𝐽))
23		phtpcer 22794	. . . . 5 ⊢ ( ≃_ph‘𝐽) Er (II Cn 𝐽)
24	23	a1i 11	. . . 4 ⊢ (𝜑 → ( ≃_ph‘𝐽) Er (II Cn 𝐽))
25		eqid 2622	. . . . . . 7 ⊢ ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {(𝐹‘1)})
26	7, 25	pcorev 22827	. . . . . 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 4189	. . . . . 6 ⊢ (𝜑 → {(𝐹‘1)} = {𝑌})
29	28	xpeq2d 5139	. . . . 5 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {𝑌}))
30	27, 29	breqtrd 4679	. . . 4 ⊢ (𝜑 → (𝐼(*_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)((0[,]1) × {𝑌}))
31	24, 30	erthi 7793	. . 3 ⊢ (𝜑 → [(𝐼(*_𝑝‘𝐽)𝐹)]( ≃_ph‘𝐽) = [((0[,]1) × {𝑌})]( ≃_ph‘𝐽))
32		eqid 2622	. . . . 5 ⊢ ((0[,]1) × {𝑌}) = ((0[,]1) × {𝑌})
33	1, 2, 3, 4, 32	pi1grplem 22849	. . . 4 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ [((0[,]1) × {𝑌})]( ≃_ph‘𝐽) = (0_g‘𝐺)))
34	33	simprd 479	. . 3 ⊢ (𝜑 → [((0[,]1) × {𝑌})]( ≃_ph‘𝐽) = (0_g‘𝐺))
35	22, 31, 34	3eqtrd 2660	. 2 ⊢ (𝜑 → ([𝐼]( ≃_ph‘𝐽)(+_g‘𝐺)[𝐹]( ≃_ph‘𝐽)) = (0_g‘𝐺))
36	33	simpld 475	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
37	1, 2, 3, 4, 6, 15, 12	elpi1i 22846	. . 3 ⊢ (𝜑 → [𝐹]( ≃_ph‘𝐽) ∈ (Base‘𝐺))
38	1, 2, 3, 4, 10, 13, 16	elpi1i 22846	. . 3 ⊢ (𝜑 → [𝐼]( ≃_ph‘𝐽) ∈ (Base‘𝐺))
39		eqid 2622	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
40		pi1inv.n	. . . 4 ⊢ 𝑁 = (inv_g‘𝐺)
41	2, 5, 39, 40	grpinvid2 17471	. . 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 1326	. 2 ⊢ (𝜑 → ((𝑁‘[𝐹]( ≃_ph‘𝐽)) = [𝐼]( ≃_ph‘𝐽) ↔ ([𝐼]( ≃_ph‘𝐽)(+_g‘𝐺)[𝐹]( ≃_ph‘𝐽)) = (0_g‘𝐺)))
43	35, 42	mpbird 247	1 ⊢ (𝜑 → (𝑁‘[𝐹]( ≃_ph‘𝐽)) = [𝐼]( ≃_ph‘𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {csn 4177 ∪ cuni 4436 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ‘cfv 5888 (class class class)co 6650 Er wer 7739 [cec 7740 0cc0 9936 1c1 9937 − cmin 10266 [,]cicc 12178 Basecbs 15857 +_gcplusg 15941 0_gc0g 16100 Grpcgrp 17422 inv_gcminusg 17423 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-grp 17425 df-minusg 17426 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: (None)

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