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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 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝐽 ∈ (TopOn‘𝑋)) |
4 | pi1val.2 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
5 | 4 | adantr 484 | . . . . 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 484 | . . . . . 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 23968 | . . . . 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 5075 | . . . . . . . 8 ⊢ (𝑀𝑅𝑁 ↔ 𝑀(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑁) |
14 | brinxp2 5643 | . . . . . . . 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 768 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀 ∈ ∪ 𝐵) |
18 | simprr 773 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃𝑅𝑄) | |
19 | 12 | breqi 5075 | . . . . . . . 8 ⊢ (𝑃𝑅𝑄 ↔ 𝑃(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑄) |
20 | brinxp2 5643 | . . . . . . . 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 768 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃 ∈ ∪ 𝐵) |
24 | 1, 3, 5, 10, 17, 23 | om1addcl 23961 | . . . 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 23961 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑁(*𝑝‘𝐽)𝑄) ∈ ∪ 𝐵) |
28 | 6, 3, 5, 8 | pi1eluni 23970 | . . . . . . . 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 1146 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀‘1) = 𝑌) |
31 | 6, 3, 5, 8 | pi1eluni 23970 | . . . . . . . 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 1145 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑃‘0) = 𝑌) |
34 | 30, 33 | eqtr4d 2782 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀‘1) = (𝑃‘0)) |
35 | 16 | simprd 499 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑀( ≃ph‘𝐽)𝑁) |
36 | 22 | simprd 499 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → 𝑃( ≃ph‘𝐽)𝑄) |
37 | 34, 35, 36 | pcohtpy 23948 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃)( ≃ph‘𝐽)(𝑁(*𝑝‘𝐽)𝑄)) |
38 | 12 | breqi 5075 | . . . . 5 ⊢ ((𝑀(*𝑝‘𝐽)𝑃)𝑅(𝑁(*𝑝‘𝐽)𝑄) ↔ (𝑀(*𝑝‘𝐽)𝑃)(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))(𝑁(*𝑝‘𝐽)𝑄)) |
39 | brinxp2 5643 | . . . . 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 1346 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃)𝑅(𝑁(*𝑝‘𝐽)𝑄)) |
42 | 1, 3, 5 | om1plusg 23962 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (*𝑝‘𝐽) = (+g‘𝑂)) |
43 | pi1cpbl.a | . . . . 5 ⊢ + = (+g‘𝑂) | |
44 | 42, 43 | eqtr4di 2798 | . . . 4 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (*𝑝‘𝐽) = + ) |
45 | 44 | oveqd 7251 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀(*𝑝‘𝐽)𝑃) = (𝑀 + 𝑃)) |
46 | 44 | oveqd 7251 | . . 3 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑁(*𝑝‘𝐽)𝑄) = (𝑁 + 𝑄)) |
47 | 41, 45, 46 | 3brtr3d 5100 | . 2 ⊢ ((𝜑 ∧ (𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄)) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄)) |
48 | 47 | ex 416 | 1 ⊢ (𝜑 → ((𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∩ cin 3882 ∪ cuni 4835 class class class wbr 5069 × cxp 5566 ‘cfv 6400 (class class class)co 7234 0cc0 10758 1c1 10759 Basecbs 16792 +gcplusg 16834 TopOnctopon 21838 Cn ccn 22152 IIcii 23803 ≃phcphtpc 23897 *𝑝cpco 23928 Ω1 comi 23929 π1 cpi1 23931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10814 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 ax-pre-mulgt0 10835 ax-pre-sup 10836 ax-mulf 10838 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-int 4876 df-iun 4922 df-iin 4923 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-se 5527 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-isom 6409 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-of 7490 df-om 7666 df-1st 7782 df-2nd 7783 df-supp 7927 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-1o 8225 df-2o 8226 df-er 8414 df-ec 8416 df-qs 8420 df-map 8533 df-ixp 8602 df-en 8650 df-dom 8651 df-sdom 8652 df-fin 8653 df-fsupp 9015 df-fi 9056 df-sup 9087 df-inf 9088 df-oi 9155 df-card 9584 df-pnf 10898 df-mnf 10899 df-xr 10900 df-ltxr 10901 df-le 10902 df-sub 11093 df-neg 11094 df-div 11519 df-nn 11860 df-2 11922 df-3 11923 df-4 11924 df-5 11925 df-6 11926 df-7 11927 df-8 11928 df-9 11929 df-n0 12120 df-z 12206 df-dec 12323 df-uz 12468 df-q 12574 df-rp 12616 df-xneg 12733 df-xadd 12734 df-xmul 12735 df-ioo 12968 df-icc 12971 df-fz 13125 df-fzo 13268 df-seq 13606 df-exp 13667 df-hash 13929 df-cj 14694 df-re 14695 df-im 14696 df-sqrt 14830 df-abs 14831 df-struct 16732 df-sets 16749 df-slot 16767 df-ndx 16777 df-base 16793 df-ress 16817 df-plusg 16847 df-mulr 16848 df-starv 16849 df-sca 16850 df-vsca 16851 df-ip 16852 df-tset 16853 df-ple 16854 df-ds 16856 df-unif 16857 df-hom 16858 df-cco 16859 df-rest 16959 df-topn 16960 df-0g 16978 df-gsum 16979 df-topgen 16980 df-pt 16981 df-prds 16984 df-xrs 17039 df-qtop 17044 df-imas 17045 df-qus 17046 df-xps 17047 df-mre 17121 df-mrc 17122 df-acs 17124 df-mgm 18146 df-sgrp 18195 df-mnd 18206 df-submnd 18251 df-mulg 18521 df-cntz 18743 df-cmn 19204 df-psmet 20387 df-xmet 20388 df-met 20389 df-bl 20390 df-mopn 20391 df-cnfld 20396 df-top 21822 df-topon 21839 df-topsp 21861 df-bases 21874 df-cld 21947 df-cn 22155 df-cnp 22156 df-tx 22490 df-hmeo 22683 df-xms 23249 df-ms 23250 df-tms 23251 df-ii 23805 df-htpy 23898 df-phtpy 23899 df-phtpc 23920 df-pco 23933 df-om1 23934 df-pi1 23936 |
This theorem is referenced by: pi1addf 23975 pi1addval 23976 pi1grplem 23977 |
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