pi1inv - Metamath Proof Explorer

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Theorem pi1inv 24985

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 2729	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		pi1inv.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
4		pi1inv.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
5		eqid 2729	. . . 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 24958	. . . . . . 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 2764	. . . . 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 2764	. . . . 5 ⊢ (𝜑 → (𝐼‘1) = 𝑌)
17	2	a1i 11	. . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
18	1, 3, 4, 17	pi1eluni 24975	. . . . 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 24975	. . . . 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 24981	. . 3 ⊢ (𝜑 → ([𝐼]( ≃_ph‘𝐽)(+_g‘𝐺)[𝐹]( ≃_ph‘𝐽)) = [(𝐼(*_𝑝‘𝐽)𝐹)]( ≃_ph‘𝐽))
23		phtpcer 24927	. . . . 5 ⊢ ( ≃_ph‘𝐽) Er (II Cn 𝐽)
24	23	a1i 11	. . . 4 ⊢ (𝜑 → ( ≃_ph‘𝐽) Er (II Cn 𝐽))
25		eqid 2729	. . . . . . 7 ⊢ ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {(𝐹‘1)})
26	7, 25	pcorev 24960	. . . . . 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 4597	. . . . . 6 ⊢ (𝜑 → {(𝐹‘1)} = {𝑌})
29	28	xpeq2d 5661	. . . . 5 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {𝑌}))
30	27, 29	breqtrd 5128	. . . 4 ⊢ (𝜑 → (𝐼(*_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)((0[,]1) × {𝑌}))
31	24, 30	erthi 8704	. . 3 ⊢ (𝜑 → [(𝐼(*_𝑝‘𝐽)𝐹)]( ≃_ph‘𝐽) = [((0[,]1) × {𝑌})]( ≃_ph‘𝐽))
32		eqid 2729	. . . . 5 ⊢ ((0[,]1) × {𝑌}) = ((0[,]1) × {𝑌})
33	1, 2, 3, 4, 32	pi1grplem 24982	. . . 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 2768	. 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 24979	. . 3 ⊢ (𝜑 → [𝐹]( ≃_ph‘𝐽) ∈ (Base‘𝐺))
38	1, 2, 3, 4, 10, 13, 16	elpi1i 24979	. . 3 ⊢ (𝜑 → [𝐼]( ≃_ph‘𝐽) ∈ (Base‘𝐺))
39		eqid 2729	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
40		pi1inv.n	. . . 4 ⊢ 𝑁 = (inv_g‘𝐺)
41	2, 5, 39, 40	grpinvid2 18906	. . 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 4585 ∪ cuni 4867 class class class wbr 5102 ↦ cmpt 5183 × cxp 5629 ‘cfv 6499 (class class class)co 7369 Er wer 8645 [cec 8646 0cc0 11044 1c1 11045 − cmin 11381 [,]cicc 13285 Basecbs 17155 +_gcplusg 17196 0_gc0g 17378 Grpcgrp 18847 inv_gcminusg 18848 TopOnctopon 22830 Cn ccn 23144 IIcii 24801 ≃_phcphtpc 24901 *_𝑝cpco 24933 π₁ cpi1 24936

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-ec 8650 df-qs 8654 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-qus 17448 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-grp 18850 df-minusg 18851 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-cn 23147 df-cnp 23148 df-tx 23482 df-hmeo 23675 df-xms 24241 df-ms 24242 df-tms 24243 df-ii 24803 df-htpy 24902 df-phtpy 24903 df-phtpc 24924 df-pco 24938 df-om1 24939 df-pi1 24941

This theorem is referenced by: (None)

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