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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 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝐽 ∈ (TopOn‘𝑋)) |
| 4 | pi1val.2 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
| 5 | 4 | adantr 480 | . . . . 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 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝐵 = (Base‘𝐺)) |
| 9 | eqidd 2737 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (Base‘𝑂) = (Base‘𝑂)) | |
| 10 | 6, 3, 5, 1, 8, 9 | pi1buni 25007 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → ∪ 𝐵 = (Base‘𝑂)) |
| 11 | simprl 771 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀𝑅𝑁) | |
| 12 | pi1bas3.r | . . . . . . . . 9 ⊢ 𝑅 = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)) | |
| 13 | 12 | breqi 5091 | . . . . . . . 8 ⊢ (𝑀𝑅𝑁 ↔ 𝑀(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑁) |
| 14 | brinxp2 5709 | . . . . . . . 8 ⊢ (𝑀(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑁 ↔ ((𝑀 ∈ ∪ 𝐵 ∧ 𝑁 ∈ ∪ 𝐵) ∧ 𝑀( ≃ph‘𝐽)𝑁)) | |
| 15 | 13, 14 | bitri 275 | . . . . . . 7 ⊢ (𝑀𝑅𝑁 ↔ ((𝑀 ∈ ∪ 𝐵 ∧ 𝑁 ∈ ∪ 𝐵) ∧ 𝑀( ≃ph‘𝐽)𝑁)) |
| 16 | 11, 15 | sylib 218 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → ((𝑀 ∈ ∪ 𝐵 ∧ 𝑁 ∈ ∪ 𝐵) ∧ 𝑀( ≃ph‘𝐽)𝑁)) |
| 17 | 16 | simplld 768 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀 ∈ ∪ 𝐵) |
| 18 | simprr 773 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃𝑅𝑄) | |
| 19 | 12 | breqi 5091 | . . . . . . . 8 ⊢ (𝑃𝑅𝑄 ↔ 𝑃(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑄) |
| 20 | brinxp2 5709 | . . . . . . . 8 ⊢ (𝑃(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑄 ↔ ((𝑃 ∈ ∪ 𝐵 ∧ 𝑄 ∈ ∪ 𝐵) ∧ 𝑃( ≃ph‘𝐽)𝑄)) | |
| 21 | 19, 20 | bitri 275 | . . . . . . 7 ⊢ (𝑃𝑅𝑄 ↔ ((𝑃 ∈ ∪ 𝐵 ∧ 𝑄 ∈ ∪ 𝐵) ∧ 𝑃( ≃ph‘𝐽)𝑄)) |
| 22 | 18, 21 | sylib 218 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → ((𝑃 ∈ ∪ 𝐵 ∧ 𝑄 ∈ ∪ 𝐵) ∧ 𝑃( ≃ph‘𝐽)𝑄)) |
| 23 | 22 | simplld 768 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃 ∈ ∪ 𝐵) |
| 24 | 1, 3, 5, 10, 17, 23 | om1addcl 25000 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃) ∈ ∪ 𝐵) |
| 25 | 16 | simplrd 770 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑁 ∈ ∪ 𝐵) |
| 26 | 22 | simplrd 770 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑄 ∈ ∪ 𝐵) |
| 27 | 1, 3, 5, 10, 25, 26 | om1addcl 25000 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑁(*𝑝‘𝐽)𝑄) ∈ ∪ 𝐵) |
| 28 | 6, 3, 5, 8 | pi1eluni 25009 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 ∈ ∪ 𝐵 ↔ (𝑀 ∈ (II Cn 𝐽) ∧ (𝑀‘0) = 𝑌 ∧ (𝑀‘1) = 𝑌))) |
| 29 | 17, 28 | mpbid 232 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 ∈ (II Cn 𝐽) ∧ (𝑀‘0) = 𝑌 ∧ (𝑀‘1) = 𝑌)) |
| 30 | 29 | simp3d 1145 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀‘1) = 𝑌) |
| 31 | 6, 3, 5, 8 | pi1eluni 25009 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃 ∈ ∪ 𝐵 ↔ (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))) |
| 32 | 23, 31 | mpbid 232 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌)) |
| 33 | 32 | simp2d 1144 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃‘0) = 𝑌) |
| 34 | 30, 33 | eqtr4d 2774 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀‘1) = (𝑃‘0)) |
| 35 | 16 | simprd 495 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀( ≃ph‘𝐽)𝑁) |
| 36 | 22 | simprd 495 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃( ≃ph‘𝐽)𝑄) |
| 37 | 34, 35, 36 | pcohtpy 24987 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃)( ≃ph‘𝐽)(𝑁(*𝑝‘𝐽)𝑄)) |
| 38 | 12 | breqi 5091 | . . . . 5 ⊢ ((𝑀(*𝑝‘𝐽)𝑃)𝑅(𝑁(*𝑝‘𝐽)𝑄) ↔ (𝑀(*𝑝‘𝐽)𝑃)(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))(𝑁(*𝑝‘𝐽)𝑄)) |
| 39 | brinxp2 5709 | . . . . 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 275 | . . . 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 1346 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃)𝑅(𝑁(*𝑝‘𝐽)𝑄)) |
| 42 | 1, 3, 5 | om1plusg 25001 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (*𝑝‘𝐽) = (+g‘𝑂)) |
| 43 | pi1cpbl.a | . . . . 5 ⊢ + = (+g‘𝑂) | |
| 44 | 42, 43 | eqtr4di 2789 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (*𝑝‘𝐽) = + ) |
| 45 | 44 | oveqd 7384 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃) = (𝑀 + 𝑃)) |
| 46 | 44 | oveqd 7384 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑁(*𝑝‘𝐽)𝑄) = (𝑁 + 𝑄)) |
| 47 | 41, 45, 46 | 3brtr3d 5116 | . 2 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄)) |
| 48 | 47 | ex 412 | 1 ⊢ (𝜑 → ((𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∩ cin 3888 ∪ cuni 4850 class class class wbr 5085 × cxp 5629 ‘cfv 6498 (class class class)co 7367 0cc0 11038 1c1 11039 Basecbs 17179 +gcplusg 17220 TopOnctopon 22875 Cn ccn 23189 IIcii 24842 ≃phcphtpc 24936 *𝑝cpco 24967 Ω1 comi 24968 π1 cpi1 24970 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-icc 13305 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-qus 17473 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-cn 23192 df-cnp 23193 df-tx 23527 df-hmeo 23720 df-xms 24285 df-ms 24286 df-tms 24287 df-ii 24844 df-htpy 24937 df-phtpy 24938 df-phtpc 24959 df-pco 24972 df-om1 24973 df-pi1 24975 |
| This theorem is referenced by: pi1addf 25014 pi1addval 25015 pi1grplem 25016 |
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