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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 486 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝐽 ∈ (TopOn‘𝑋)) |
| 4 | pi1val.2 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
| 5 | 4 | adantr 486 | . . . . 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 486 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝐵 = (Base‘𝐺)) |
| 9 | eqidd 2761 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (Base‘𝑂) = (Base‘𝑂)) | |
| 10 | 6, 3, 5, 1, 8, 9 | pi1buni 25268 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → ∪ 𝐵 = (Base‘𝑂)) |
| 11 | simprl 783 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀𝑅𝑁) | |
| 12 | pi1bas3.r | . . . . . . . . 9 ⊢ 𝑅 = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)) | |
| 13 | 12 | breqi 5109 | . . . . . . . 8 ⊢ (𝑀𝑅𝑁 ↔ 𝑀(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑁) |
| 14 | brinxp2 5733 | . . . . . . . 8 ⊢ (𝑀(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑁 ↔ ((𝑀 ∈ ∪ 𝐵 ∧ 𝑁 ∈ ∪ 𝐵) ∧ 𝑀( ≃ph‘𝐽)𝑁)) | |
| 15 | 13, 14 | bitri 278 | . . . . . . 7 ⊢ (𝑀𝑅𝑁 ↔ ((𝑀 ∈ ∪ 𝐵 ∧ 𝑁 ∈ ∪ 𝐵) ∧ 𝑀( ≃ph‘𝐽)𝑁)) |
| 16 | 11, 15 | sylib 221 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → ((𝑀 ∈ ∪ 𝐵 ∧ 𝑁 ∈ ∪ 𝐵) ∧ 𝑀( ≃ph‘𝐽)𝑁)) |
| 17 | 16 | simplld 780 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀 ∈ ∪ 𝐵) |
| 18 | simprr 785 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃𝑅𝑄) | |
| 19 | 12 | breqi 5109 | . . . . . . . 8 ⊢ (𝑃𝑅𝑄 ↔ 𝑃(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑄) |
| 20 | brinxp2 5733 | . . . . . . . 8 ⊢ (𝑃(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑄 ↔ ((𝑃 ∈ ∪ 𝐵 ∧ 𝑄 ∈ ∪ 𝐵) ∧ 𝑃( ≃ph‘𝐽)𝑄)) | |
| 21 | 19, 20 | bitri 278 | . . . . . . 7 ⊢ (𝑃𝑅𝑄 ↔ ((𝑃 ∈ ∪ 𝐵 ∧ 𝑄 ∈ ∪ 𝐵) ∧ 𝑃( ≃ph‘𝐽)𝑄)) |
| 22 | 18, 21 | sylib 221 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → ((𝑃 ∈ ∪ 𝐵 ∧ 𝑄 ∈ ∪ 𝐵) ∧ 𝑃( ≃ph‘𝐽)𝑄)) |
| 23 | 22 | simplld 780 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃 ∈ ∪ 𝐵) |
| 24 | 1, 3, 5, 10, 17, 23 | om1addcl 25261 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃) ∈ ∪ 𝐵) |
| 25 | 16 | simplrd 782 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑁 ∈ ∪ 𝐵) |
| 26 | 22 | simplrd 782 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑄 ∈ ∪ 𝐵) |
| 27 | 1, 3, 5, 10, 25, 26 | om1addcl 25261 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑁(*𝑝‘𝐽)𝑄) ∈ ∪ 𝐵) |
| 28 | 6, 3, 5, 8 | pi1eluni 25270 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 ∈ ∪ 𝐵 ↔ (𝑀 ∈ (II Cn 𝐽) ∧ (𝑀‘0) = 𝑌 ∧ (𝑀‘1) = 𝑌))) |
| 29 | 17, 28 | mpbid 235 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 ∈ (II Cn 𝐽) ∧ (𝑀‘0) = 𝑌 ∧ (𝑀‘1) = 𝑌)) |
| 30 | 29 | simp3d 1162 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀‘1) = 𝑌) |
| 31 | 6, 3, 5, 8 | pi1eluni 25270 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃 ∈ ∪ 𝐵 ↔ (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))) |
| 32 | 23, 31 | mpbid 235 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌)) |
| 33 | 32 | simp2d 1161 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃‘0) = 𝑌) |
| 34 | 30, 33 | eqtr4d 2798 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀‘1) = (𝑃‘0)) |
| 35 | 16 | simprd 501 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀( ≃ph‘𝐽)𝑁) |
| 36 | 22 | simprd 501 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃( ≃ph‘𝐽)𝑄) |
| 37 | 34, 35, 36 | pcohtpy 25248 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃)( ≃ph‘𝐽)(𝑁(*𝑝‘𝐽)𝑄)) |
| 38 | 12 | breqi 5109 | . . . . 5 ⊢ ((𝑀(*𝑝‘𝐽)𝑃)𝑅(𝑁(*𝑝‘𝐽)𝑄) ↔ (𝑀(*𝑝‘𝐽)𝑃)(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))(𝑁(*𝑝‘𝐽)𝑄)) |
| 39 | brinxp2 5733 | . . . . 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 278 | . . . 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 1363 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃)𝑅(𝑁(*𝑝‘𝐽)𝑄)) |
| 42 | 1, 3, 5 | om1plusg 25262 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (*𝑝‘𝐽) = (+g‘𝑂)) |
| 43 | pi1cpbl.a | . . . . 5 ⊢ + = (+g‘𝑂) | |
| 44 | 42, 43 | eqtr4di 2813 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (*𝑝‘𝐽) = + ) |
| 45 | 44 | oveqd 7430 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃) = (𝑀 + 𝑃)) |
| 46 | 44 | oveqd 7430 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑁(*𝑝‘𝐽)𝑄) = (𝑁 + 𝑄)) |
| 47 | 41, 45, 46 | 3brtr3d 5136 | . 2 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄)) |
| 48 | 47 | ex 418 | 1 ⊢ (𝜑 → ((𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄))) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∩ cin 3898 ∪ cuni 4867 class class class wbr 5103 × cxp 5653 ‘cfv 6533 (class class class)co 7413 0cc0 11124 1c1 11125 Basecbs 17301 +gcplusg 17342 TopOnctopon 23135 Cn ccn 23449 IIcii 25103 ≃phcphtpc 25197 *𝑝cpco 25228 Ω1 comi 25229 π1 cpi1 25231 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-ec 8698 df-qs 8702 df-map 8828 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 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 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-icc 13405 df-fz 13562 df-fzo 13710 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-qus 17595 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-cn 23452 df-cnp 23453 df-tx 23788 df-hmeo 23981 df-xms 24546 df-ms 24547 df-tms 24548 df-ii 25105 df-htpy 25198 df-phtpy 25199 df-phtpc 25220 df-pco 25233 df-om1 25234 df-pi1 25236 |
| This theorem is used by: pi1addf 25275 pi1addval 25276 pi1grplem 25277 |
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