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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 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝐽 ∈ (TopOn‘𝑋)) |
| 4 | pi1val.2 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
| 5 | 4 | adantr 481 | . . . . 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 481 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝐵 = (Base‘𝐺)) |
| 9 | eqidd 2740 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (Base‘𝑂) = (Base‘𝑂)) | |
| 10 | 6, 3, 5, 1, 8, 9 | pi1buni 25026 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → ∪ 𝐵 = (Base‘𝑂)) |
| 11 | simprl 776 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀𝑅𝑁) | |
| 12 | pi1bas3.r | . . . . . . . . 9 ⊢ 𝑅 = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)) | |
| 13 | 12 | breqi 5079 | . . . . . . . 8 ⊢ (𝑀𝑅𝑁 ↔ 𝑀(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑁) |
| 14 | brinxp2 5697 | . . . . . . . 8 ⊢ (𝑀(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑁 ↔ ((𝑀 ∈ ∪ 𝐵 ∧ 𝑁 ∈ ∪ 𝐵) ∧ 𝑀( ≃ph‘𝐽)𝑁)) | |
| 15 | 13, 14 | bitri 276 | . . . . . . 7 ⊢ (𝑀𝑅𝑁 ↔ ((𝑀 ∈ ∪ 𝐵 ∧ 𝑁 ∈ ∪ 𝐵) ∧ 𝑀( ≃ph‘𝐽)𝑁)) |
| 16 | 11, 15 | sylib 219 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → ((𝑀 ∈ ∪ 𝐵 ∧ 𝑁 ∈ ∪ 𝐵) ∧ 𝑀( ≃ph‘𝐽)𝑁)) |
| 17 | 16 | simplld 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀 ∈ ∪ 𝐵) |
| 18 | simprr 778 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃𝑅𝑄) | |
| 19 | 12 | breqi 5079 | . . . . . . . 8 ⊢ (𝑃𝑅𝑄 ↔ 𝑃(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑄) |
| 20 | brinxp2 5697 | . . . . . . . 8 ⊢ (𝑃(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑄 ↔ ((𝑃 ∈ ∪ 𝐵 ∧ 𝑄 ∈ ∪ 𝐵) ∧ 𝑃( ≃ph‘𝐽)𝑄)) | |
| 21 | 19, 20 | bitri 276 | . . . . . . 7 ⊢ (𝑃𝑅𝑄 ↔ ((𝑃 ∈ ∪ 𝐵 ∧ 𝑄 ∈ ∪ 𝐵) ∧ 𝑃( ≃ph‘𝐽)𝑄)) |
| 22 | 18, 21 | sylib 219 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → ((𝑃 ∈ ∪ 𝐵 ∧ 𝑄 ∈ ∪ 𝐵) ∧ 𝑃( ≃ph‘𝐽)𝑄)) |
| 23 | 22 | simplld 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃 ∈ ∪ 𝐵) |
| 24 | 1, 3, 5, 10, 17, 23 | om1addcl 25019 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃) ∈ ∪ 𝐵) |
| 25 | 16 | simplrd 775 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑁 ∈ ∪ 𝐵) |
| 26 | 22 | simplrd 775 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑄 ∈ ∪ 𝐵) |
| 27 | 1, 3, 5, 10, 25, 26 | om1addcl 25019 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑁(*𝑝‘𝐽)𝑄) ∈ ∪ 𝐵) |
| 28 | 6, 3, 5, 8 | pi1eluni 25028 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 ∈ ∪ 𝐵 ↔ (𝑀 ∈ (II Cn 𝐽) ∧ (𝑀‘0) = 𝑌 ∧ (𝑀‘1) = 𝑌))) |
| 29 | 17, 28 | mpbid 233 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 ∈ (II Cn 𝐽) ∧ (𝑀‘0) = 𝑌 ∧ (𝑀‘1) = 𝑌)) |
| 30 | 29 | simp3d 1150 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀‘1) = 𝑌) |
| 31 | 6, 3, 5, 8 | pi1eluni 25028 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃 ∈ ∪ 𝐵 ↔ (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))) |
| 32 | 23, 31 | mpbid 233 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌)) |
| 33 | 32 | simp2d 1149 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃‘0) = 𝑌) |
| 34 | 30, 33 | eqtr4d 2777 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀‘1) = (𝑃‘0)) |
| 35 | 16 | simprd 496 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀( ≃ph‘𝐽)𝑁) |
| 36 | 22 | simprd 496 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃( ≃ph‘𝐽)𝑄) |
| 37 | 34, 35, 36 | pcohtpy 25006 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃)( ≃ph‘𝐽)(𝑁(*𝑝‘𝐽)𝑄)) |
| 38 | 12 | breqi 5079 | . . . . 5 ⊢ ((𝑀(*𝑝‘𝐽)𝑃)𝑅(𝑁(*𝑝‘𝐽)𝑄) ↔ (𝑀(*𝑝‘𝐽)𝑃)(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))(𝑁(*𝑝‘𝐽)𝑄)) |
| 39 | brinxp2 5697 | . . . . 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 276 | . . . 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 1351 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃)𝑅(𝑁(*𝑝‘𝐽)𝑄)) |
| 42 | 1, 3, 5 | om1plusg 25020 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (*𝑝‘𝐽) = (+g‘𝑂)) |
| 43 | pi1cpbl.a | . . . . 5 ⊢ + = (+g‘𝑂) | |
| 44 | 42, 43 | eqtr4di 2792 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (*𝑝‘𝐽) = + ) |
| 45 | 44 | oveqd 7374 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃) = (𝑀 + 𝑃)) |
| 46 | 44 | oveqd 7374 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑁(*𝑝‘𝐽)𝑄) = (𝑁 + 𝑄)) |
| 47 | 41, 45, 46 | 3brtr3d 5104 | . 2 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄)) |
| 48 | 47 | ex 413 | 1 ⊢ (𝜑 → ((𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∩ cin 3882 ∪ cuni 4839 class class class wbr 5073 × cxp 5617 ‘cfv 6486 (class class class)co 7357 0cc0 11030 1c1 11031 Basecbs 17171 +gcplusg 17212 TopOnctopon 22894 Cn ccn 23208 IIcii 24861 ≃phcphtpc 24955 *𝑝cpco 24986 Ω1 comi 24987 π1 cpi1 24989 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 |
| This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-ec 8636 df-qs 8640 df-map 8766 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9855 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-q 12891 df-rp 12935 df-xneg 13055 df-xadd 13056 df-xmul 13057 df-ioo 13294 df-icc 13297 df-fz 13454 df-fzo 13601 df-seq 13956 df-exp 14016 df-hash 14285 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-starv 17227 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ds 17234 df-unif 17235 df-hom 17236 df-cco 17237 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17458 df-qtop 17463 df-imas 17464 df-qus 17465 df-xps 17466 df-mre 17540 df-mrc 17541 df-acs 17543 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18744 df-mulg 19036 df-cntz 19284 df-cmn 19749 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-cnfld 21349 df-top 22878 df-topon 22895 df-topsp 22917 df-bases 22930 df-cld 23003 df-cn 23211 df-cnp 23212 df-tx 23546 df-hmeo 23739 df-xms 24304 df-ms 24305 df-tms 24306 df-ii 24863 df-htpy 24956 df-phtpy 24957 df-phtpc 24978 df-pco 24991 df-om1 24992 df-pi1 24994 |
| This theorem is referenced by: pi1addf 25033 pi1addval 25034 pi1grplem 25035 |
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