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| Mirrors > Home > MPE Home > Th. List > pi1cpbl | Structured version Visualization version GIF version | ||
| Description: The group operation, loop concatenation, is compatible with homotopy equivalence. (Contributed by Mario Carneiro, 10-Jul-2015.) |
| Ref | Expression |
|---|---|
| pi1val.g | ⊢ 𝐺 = (𝐽 π1 𝑌) |
| pi1val.1 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
| pi1val.2 | ⊢ (𝜑 → 𝑌 ∈ 𝑋) |
| pi1bas2.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
| pi1bas3.r | ⊢ 𝑅 = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)) |
| pi1cpbl.o | ⊢ 𝑂 = (𝐽 Ω1 𝑌) |
| pi1cpbl.a | ⊢ + = (+g‘𝑂) |
| Ref | Expression |
|---|---|
| pi1cpbl | ⊢ (𝜑 → ((𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pi1cpbl.o | . . . . 5 ⊢ 𝑂 = (𝐽 Ω1 𝑌) | |
| 2 | pi1val.1 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
| 3 | 2 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝐽 ∈ (TopOn‘𝑋)) |
| 4 | pi1val.2 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
| 5 | 4 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑌 ∈ 𝑋) |
| 6 | pi1val.g | . . . . . 6 ⊢ 𝐺 = (𝐽 π1 𝑌) | |
| 7 | pi1bas2.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
| 8 | 7 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝐵 = (Base‘𝐺)) |
| 9 | eqidd 2779 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (Base‘𝑂) = (Base‘𝑂)) | |
| 10 | 6, 3, 5, 1, 8, 9 | pi1buni 23247 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → ∪ 𝐵 = (Base‘𝑂)) |
| 11 | simprl 761 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀𝑅𝑁) | |
| 12 | pi1bas3.r | . . . . . . . . 9 ⊢ 𝑅 = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)) | |
| 13 | 12 | breqi 4892 | . . . . . . . 8 ⊢ (𝑀𝑅𝑁 ↔ 𝑀(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑁) |
| 14 | brinxp2 5426 | . . . . . . . 8 ⊢ (𝑀(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑁 ↔ ((𝑀 ∈ ∪ 𝐵 ∧ 𝑁 ∈ ∪ 𝐵) ∧ 𝑀( ≃ph‘𝐽)𝑁)) | |
| 15 | 13, 14 | bitri 267 | . . . . . . 7 ⊢ (𝑀𝑅𝑁 ↔ ((𝑀 ∈ ∪ 𝐵 ∧ 𝑁 ∈ ∪ 𝐵) ∧ 𝑀( ≃ph‘𝐽)𝑁)) |
| 16 | 11, 15 | sylib 210 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → ((𝑀 ∈ ∪ 𝐵 ∧ 𝑁 ∈ ∪ 𝐵) ∧ 𝑀( ≃ph‘𝐽)𝑁)) |
| 17 | 16 | simplld 758 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀 ∈ ∪ 𝐵) |
| 18 | simprr 763 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃𝑅𝑄) | |
| 19 | 12 | breqi 4892 | . . . . . . . 8 ⊢ (𝑃𝑅𝑄 ↔ 𝑃(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑄) |
| 20 | brinxp2 5426 | . . . . . . . 8 ⊢ (𝑃(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑄 ↔ ((𝑃 ∈ ∪ 𝐵 ∧ 𝑄 ∈ ∪ 𝐵) ∧ 𝑃( ≃ph‘𝐽)𝑄)) | |
| 21 | 19, 20 | bitri 267 | . . . . . . 7 ⊢ (𝑃𝑅𝑄 ↔ ((𝑃 ∈ ∪ 𝐵 ∧ 𝑄 ∈ ∪ 𝐵) ∧ 𝑃( ≃ph‘𝐽)𝑄)) |
| 22 | 18, 21 | sylib 210 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → ((𝑃 ∈ ∪ 𝐵 ∧ 𝑄 ∈ ∪ 𝐵) ∧ 𝑃( ≃ph‘𝐽)𝑄)) |
| 23 | 22 | simplld 758 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃 ∈ ∪ 𝐵) |
| 24 | 1, 3, 5, 10, 17, 23 | om1addcl 23240 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃) ∈ ∪ 𝐵) |
| 25 | 16 | simplrd 760 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑁 ∈ ∪ 𝐵) |
| 26 | 22 | simplrd 760 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑄 ∈ ∪ 𝐵) |
| 27 | 1, 3, 5, 10, 25, 26 | om1addcl 23240 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑁(*𝑝‘𝐽)𝑄) ∈ ∪ 𝐵) |
| 28 | 6, 3, 5, 8 | pi1eluni 23249 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 ∈ ∪ 𝐵 ↔ (𝑀 ∈ (II Cn 𝐽) ∧ (𝑀‘0) = 𝑌 ∧ (𝑀‘1) = 𝑌))) |
| 29 | 17, 28 | mpbid 224 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 ∈ (II Cn 𝐽) ∧ (𝑀‘0) = 𝑌 ∧ (𝑀‘1) = 𝑌)) |
| 30 | 29 | simp3d 1135 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀‘1) = 𝑌) |
| 31 | 6, 3, 5, 8 | pi1eluni 23249 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃 ∈ ∪ 𝐵 ↔ (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))) |
| 32 | 23, 31 | mpbid 224 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌)) |
| 33 | 32 | simp2d 1134 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃‘0) = 𝑌) |
| 34 | 30, 33 | eqtr4d 2817 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀‘1) = (𝑃‘0)) |
| 35 | 16 | simprd 491 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀( ≃ph‘𝐽)𝑁) |
| 36 | 22 | simprd 491 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃( ≃ph‘𝐽)𝑄) |
| 37 | 34, 35, 36 | pcohtpy 23227 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃)( ≃ph‘𝐽)(𝑁(*𝑝‘𝐽)𝑄)) |
| 38 | 12 | breqi 4892 | . . . . 5 ⊢ ((𝑀(*𝑝‘𝐽)𝑃)𝑅(𝑁(*𝑝‘𝐽)𝑄) ↔ (𝑀(*𝑝‘𝐽)𝑃)(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))(𝑁(*𝑝‘𝐽)𝑄)) |
| 39 | brinxp2 5426 | . . . . 5 ⊢ ((𝑀(*𝑝‘𝐽)𝑃)(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))(𝑁(*𝑝‘𝐽)𝑄) ↔ (((𝑀(*𝑝‘𝐽)𝑃) ∈ ∪ 𝐵 ∧ (𝑁(*𝑝‘𝐽)𝑄) ∈ ∪ 𝐵) ∧ (𝑀(*𝑝‘𝐽)𝑃)( ≃ph‘𝐽)(𝑁(*𝑝‘𝐽)𝑄))) | |
| 40 | 38, 39 | bitri 267 | . . . 4 ⊢ ((𝑀(*𝑝‘𝐽)𝑃)𝑅(𝑁(*𝑝‘𝐽)𝑄) ↔ (((𝑀(*𝑝‘𝐽)𝑃) ∈ ∪ 𝐵 ∧ (𝑁(*𝑝‘𝐽)𝑄) ∈ ∪ 𝐵) ∧ (𝑀(*𝑝‘𝐽)𝑃)( ≃ph‘𝐽)(𝑁(*𝑝‘𝐽)𝑄))) |
| 41 | 24, 27, 37, 40 | syl21anbrc 1401 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃)𝑅(𝑁(*𝑝‘𝐽)𝑄)) |
| 42 | 1, 3, 5 | om1plusg 23241 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (*𝑝‘𝐽) = (+g‘𝑂)) |
| 43 | pi1cpbl.a | . . . . 5 ⊢ + = (+g‘𝑂) | |
| 44 | 42, 43 | syl6eqr 2832 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (*𝑝‘𝐽) = + ) |
| 45 | 44 | oveqd 6939 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃) = (𝑀 + 𝑃)) |
| 46 | 44 | oveqd 6939 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑁(*𝑝‘𝐽)𝑄) = (𝑁 + 𝑄)) |
| 47 | 41, 45, 46 | 3brtr3d 4917 | . 2 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄)) |
| 48 | 47 | ex 403 | 1 ⊢ (𝜑 → ((𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∩ cin 3791 ∪ cuni 4671 class class class wbr 4886 × cxp 5353 ‘cfv 6135 (class class class)co 6922 0cc0 10272 1c1 10273 Basecbs 16255 +gcplusg 16338 TopOnctopon 21122 Cn ccn 21436 IIcii 23086 ≃phcphtpc 23176 *𝑝cpco 23207 Ω1 comi 23208 π1 cpi1 23210 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-mulf 10352 |
| This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-ec 8028 df-qs 8032 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-icc 12494 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-qus 16555 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-cn 21439 df-cnp 21440 df-tx 21774 df-hmeo 21967 df-xms 22533 df-ms 22534 df-tms 22535 df-ii 23088 df-htpy 23177 df-phtpy 23178 df-phtpc 23199 df-pco 23212 df-om1 23213 df-pi1 23215 |
| This theorem is referenced by: pi1addf 23254 pi1addval 23255 pi1grplem 23256 |
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