pi1inv - Metamath Proof Explorer

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Theorem pi1inv 25099

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 2735	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		pi1inv.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
4		pi1inv.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
5		eqid 2735	. . . 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 25072	. . . . . . 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 1141	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))
11	9	simp2d 1142	. . . . . 6 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1))
12		pi1inv.1	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) = 𝑌)
13	11, 12	eqtrd 2775	. . . . 5 ⊢ (𝜑 → (𝐼‘0) = 𝑌)
14	9	simp3d 1143	. . . . . 6 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))
15		pi1inv.0	. . . . . 6 ⊢ (𝜑 → (𝐹‘0) = 𝑌)
16	14, 15	eqtrd 2775	. . . . 5 ⊢ (𝜑 → (𝐼‘1) = 𝑌)
17	2	a1i 11	. . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
18	1, 3, 4, 17	pi1eluni 25089	. . . . 5 ⊢ (𝜑 → (𝐼 ∈ ∪ (Base‘𝐺) ↔ (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = 𝑌 ∧ (𝐼‘1) = 𝑌)))
19	10, 13, 16, 18	mpbir3and 1341	. . . 4 ⊢ (𝜑 → 𝐼 ∈ ∪ (Base‘𝐺))
20	1, 3, 4, 17	pi1eluni 25089	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ ∪ (Base‘𝐺) ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
21	6, 15, 12, 20	mpbir3and 1341	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ∪ (Base‘𝐺))
22	1, 2, 3, 4, 5, 19, 21	pi1addval 25095	. . 3 ⊢ (𝜑 → ([𝐼]( ≃_ph‘𝐽)(+_g‘𝐺)[𝐹]( ≃_ph‘𝐽)) = [(𝐼(*_𝑝‘𝐽)𝐹)]( ≃_ph‘𝐽))
23		phtpcer 25041	. . . . 5 ⊢ ( ≃_ph‘𝐽) Er (II Cn 𝐽)
24	23	a1i 11	. . . 4 ⊢ (𝜑 → ( ≃_ph‘𝐽) Er (II Cn 𝐽))
25		eqid 2735	. . . . . . 7 ⊢ ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {(𝐹‘1)})
26	7, 25	pcorev 25074	. . . . . 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 4643	. . . . . 6 ⊢ (𝜑 → {(𝐹‘1)} = {𝑌})
29	28	xpeq2d 5719	. . . . 5 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {𝑌}))
30	27, 29	breqtrd 5174	. . . 4 ⊢ (𝜑 → (𝐼(*_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)((0[,]1) × {𝑌}))
31	24, 30	erthi 8797	. . 3 ⊢ (𝜑 → [(𝐼(*_𝑝‘𝐽)𝐹)]( ≃_ph‘𝐽) = [((0[,]1) × {𝑌})]( ≃_ph‘𝐽))
32		eqid 2735	. . . . 5 ⊢ ((0[,]1) × {𝑌}) = ((0[,]1) × {𝑌})
33	1, 2, 3, 4, 32	pi1grplem 25096	. . . 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 2779	. 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 25093	. . 3 ⊢ (𝜑 → [𝐹]( ≃_ph‘𝐽) ∈ (Base‘𝐺))
38	1, 2, 3, 4, 10, 13, 16	elpi1i 25093	. . 3 ⊢ (𝜑 → [𝐼]( ≃_ph‘𝐽) ∈ (Base‘𝐺))
39		eqid 2735	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
40		pi1inv.n	. . . 4 ⊢ 𝑁 = (inv_g‘𝐺)
41	2, 5, 39, 40	grpinvid2 19023	. . 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 1370	. 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 1537 ∈ wcel 2106 {csn 4631 ∪ cuni 4912 class class class wbr 5148 ↦ cmpt 5231 × cxp 5687 ‘cfv 6563 (class class class)co 7431 Er wer 8741 [cec 8742 0cc0 11153 1c1 11154 − cmin 11490 [,]cicc 13387 Basecbs 17245 +_gcplusg 17298 0_gc0g 17486 Grpcgrp 18964 inv_gcminusg 18965 TopOnctopon 22932 Cn ccn 23248 IIcii 24915 ≃_phcphtpc 25015 *_𝑝cpco 25047 π₁ cpi1 25050

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-qus 17556 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-cn 23251 df-cnp 23252 df-tx 23586 df-hmeo 23779 df-xms 24346 df-ms 24347 df-tms 24348 df-ii 24917 df-htpy 25016 df-phtpy 25017 df-phtpc 25038 df-pco 25052 df-om1 25053 df-pi1 25055

This theorem is referenced by: (None)

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