|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > pi1inv | Structured version Visualization version GIF version | ||
| Description: An inverse in the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| pi1grp.2 | ⊢ 𝐺 = (𝐽 π1 𝑌) | 
| pi1inv.n | ⊢ 𝑁 = (invg‘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))) | 
| Ref | Expression | 
|---|---|
| pi1inv | ⊢ (𝜑 → (𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pi1grp.2 | . . . 4 ⊢ 𝐺 = (𝐽 π1 𝑌) | |
| 2 | eqid 2737 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 3 | pi1inv.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
| 4 | pi1inv.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
| 5 | eqid 2737 | . . . 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 25058 | . . . . . . 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 1143 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽)) | 
| 11 | 9 | simp2d 1144 | . . . . . 6 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1)) | 
| 12 | pi1inv.1 | . . . . . 6 ⊢ (𝜑 → (𝐹‘1) = 𝑌) | |
| 13 | 11, 12 | eqtrd 2777 | . . . . 5 ⊢ (𝜑 → (𝐼‘0) = 𝑌) | 
| 14 | 9 | simp3d 1145 | . . . . . 6 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0)) | 
| 15 | pi1inv.0 | . . . . . 6 ⊢ (𝜑 → (𝐹‘0) = 𝑌) | |
| 16 | 14, 15 | eqtrd 2777 | . . . . 5 ⊢ (𝜑 → (𝐼‘1) = 𝑌) | 
| 17 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) | 
| 18 | 1, 3, 4, 17 | pi1eluni 25075 | . . . . 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 25075 | . . . . 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 25081 | . . 3 ⊢ (𝜑 → ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)𝐹)]( ≃ph‘𝐽)) | 
| 23 | phtpcer 25027 | . . . . 5 ⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽) | |
| 24 | 23 | a1i 11 | . . . 4 ⊢ (𝜑 → ( ≃ph‘𝐽) Er (II Cn 𝐽)) | 
| 25 | eqid 2737 | . . . . . . 7 ⊢ ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {(𝐹‘1)}) | |
| 26 | 7, 25 | pcorev 25060 | . . . . . 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 4638 | . . . . . 6 ⊢ (𝜑 → {(𝐹‘1)} = {𝑌}) | 
| 29 | 28 | xpeq2d 5715 | . . . . 5 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {𝑌})) | 
| 30 | 27, 29 | breqtrd 5169 | . . . 4 ⊢ (𝜑 → (𝐼(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)((0[,]1) × {𝑌})) | 
| 31 | 24, 30 | erthi 8798 | . . 3 ⊢ (𝜑 → [(𝐼(*𝑝‘𝐽)𝐹)]( ≃ph‘𝐽) = [((0[,]1) × {𝑌})]( ≃ph‘𝐽)) | 
| 32 | eqid 2737 | . . . . 5 ⊢ ((0[,]1) × {𝑌}) = ((0[,]1) × {𝑌}) | |
| 33 | 1, 2, 3, 4, 32 | pi1grplem 25082 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ [((0[,]1) × {𝑌})]( ≃ph‘𝐽) = (0g‘𝐺))) | 
| 34 | 33 | simprd 495 | . . 3 ⊢ (𝜑 → [((0[,]1) × {𝑌})]( ≃ph‘𝐽) = (0g‘𝐺)) | 
| 35 | 22, 31, 34 | 3eqtrd 2781 | . 2 ⊢ (𝜑 → ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = (0g‘𝐺)) | 
| 36 | 33 | simpld 494 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) | 
| 37 | 1, 2, 3, 4, 6, 15, 12 | elpi1i 25079 | . . 3 ⊢ (𝜑 → [𝐹]( ≃ph‘𝐽) ∈ (Base‘𝐺)) | 
| 38 | 1, 2, 3, 4, 10, 13, 16 | elpi1i 25079 | . . 3 ⊢ (𝜑 → [𝐼]( ≃ph‘𝐽) ∈ (Base‘𝐺)) | 
| 39 | eqid 2737 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 40 | pi1inv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
| 41 | 2, 5, 39, 40 | grpinvid2 19010 | . . 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)) = (0g‘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)) = (0g‘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 1087 = wceq 1540 ∈ wcel 2108 {csn 4626 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ‘cfv 6561 (class class class)co 7431 Er wer 8742 [cec 8743 0cc0 11155 1c1 11156 − cmin 11492 [,]cicc 13390 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Grpcgrp 18951 invgcminusg 18952 TopOnctopon 22916 Cn ccn 23232 IIcii 24901 ≃phcphtpc 25001 *𝑝cpco 25033 π1 cpi1 25036 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-qus 17554 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-cn 23235 df-cnp 23236 df-tx 23570 df-hmeo 23763 df-xms 24330 df-ms 24331 df-tms 24332 df-ii 24903 df-htpy 25002 df-phtpy 25003 df-phtpc 25024 df-pco 25038 df-om1 25039 df-pi1 25041 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |