pi1inv - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pi1inv		Structured version Visualization version GIF version

Theorem pi1inv 24950

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 24923	. . . . . . 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 24940	. . . . 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 24940	. . . . 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 24946	. . 3 ⊢ (𝜑 → ([𝐼]( ≃_ph‘𝐽)(+_g‘𝐺)[𝐹]( ≃_ph‘𝐽)) = [(𝐼(*_𝑝‘𝐽)𝐹)]( ≃_ph‘𝐽))
23		phtpcer 24892	. . . . 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 24925	. . . . . 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 4589	. . . . . 6 ⊢ (𝜑 → {(𝐹‘1)} = {𝑌})
29	28	xpeq2d 5649	. . . . 5 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {𝑌}))
30	27, 29	breqtrd 5118	. . . 4 ⊢ (𝜑 → (𝐼(*_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)((0[,]1) × {𝑌}))
31	24, 30	erthi 8681	. . 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 24947	. . . 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 24944	. . 3 ⊢ (𝜑 → [𝐹]( ≃_ph‘𝐽) ∈ (Base‘𝐺))
38	1, 2, 3, 4, 10, 13, 16	elpi1i 24944	. . 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 18871	. . 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 4577 ∪ cuni 4858 class class class wbr 5092 ↦ cmpt 5173 × cxp 5617 ‘cfv 6482 (class class class)co 7349 Er wer 8622 [cec 8623 0cc0 11009 1c1 11010 − cmin 11347 [,]cicc 13251 Basecbs 17120 +_gcplusg 17161 0_gc0g 17343 Grpcgrp 18812 inv_gcminusg 18813 TopOnctopon 22795 Cn ccn 23109 IIcii 24766 ≃_phcphtpc 24866 *_𝑝cpco 24898 π₁ cpi1 24901

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-ec 8627 df-qs 8631 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-icc 13255 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-qus 17413 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-grp 18815 df-minusg 18816 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-cn 23112 df-cnp 23113 df-tx 23447 df-hmeo 23640 df-xms 24206 df-ms 24207 df-tms 24208 df-ii 24768 df-htpy 24867 df-phtpy 24868 df-phtpc 24889 df-pco 24903 df-om1 24904 df-pi1 24906

This theorem is referenced by: (None)

W3C validator