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Mirrors > Home > MPE Home > Th. List > qus0 | Structured version Visualization version GIF version |
Description: Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) |
Ref | Expression |
---|---|
qusgrp.h | ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) |
qus0.p | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
qus0 | ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [ 0 ](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsgsubg 18010 | . . . . . . 7 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
2 | subgrcl 17983 | . . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp) |
4 | eqid 2777 | . . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | qus0.p | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
6 | 4, 5 | grpidcl 17837 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺)) |
7 | 3, 6 | syl 17 | . . . . 5 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 0 ∈ (Base‘𝐺)) |
8 | qusgrp.h | . . . . . 6 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) | |
9 | eqid 2777 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
10 | eqid 2777 | . . . . . 6 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
11 | 8, 4, 9, 10 | qusadd 18035 | . . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 0 ∈ (Base‘𝐺) ∧ 0 ∈ (Base‘𝐺)) → ([ 0 ](𝐺 ~QG 𝑆)(+g‘𝐻)[ 0 ](𝐺 ~QG 𝑆)) = [( 0 (+g‘𝐺) 0 )](𝐺 ~QG 𝑆)) |
12 | 7, 7, 11 | mpd3an23 1536 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ([ 0 ](𝐺 ~QG 𝑆)(+g‘𝐻)[ 0 ](𝐺 ~QG 𝑆)) = [( 0 (+g‘𝐺) 0 )](𝐺 ~QG 𝑆)) |
13 | 4, 9, 5 | grplid 17839 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
14 | 3, 7, 13 | syl2anc 579 | . . . . 5 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
15 | 14 | eceq1d 8065 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [( 0 (+g‘𝐺) 0 )](𝐺 ~QG 𝑆) = [ 0 ](𝐺 ~QG 𝑆)) |
16 | 12, 15 | eqtrd 2813 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ([ 0 ](𝐺 ~QG 𝑆)(+g‘𝐻)[ 0 ](𝐺 ~QG 𝑆)) = [ 0 ](𝐺 ~QG 𝑆)) |
17 | 8 | qusgrp 18033 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp) |
18 | eqid 2777 | . . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
19 | 8, 4, 18 | quseccl 18034 | . . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 0 ∈ (Base‘𝐺)) → [ 0 ](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
20 | 7, 19 | mpdan 677 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [ 0 ](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
21 | eqid 2777 | . . . . 5 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
22 | 18, 10, 21 | grpid 17844 | . . . 4 ⊢ ((𝐻 ∈ Grp ∧ [ 0 ](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) → (([ 0 ](𝐺 ~QG 𝑆)(+g‘𝐻)[ 0 ](𝐺 ~QG 𝑆)) = [ 0 ](𝐺 ~QG 𝑆) ↔ (0g‘𝐻) = [ 0 ](𝐺 ~QG 𝑆))) |
23 | 17, 20, 22 | syl2anc 579 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (([ 0 ](𝐺 ~QG 𝑆)(+g‘𝐻)[ 0 ](𝐺 ~QG 𝑆)) = [ 0 ](𝐺 ~QG 𝑆) ↔ (0g‘𝐻) = [ 0 ](𝐺 ~QG 𝑆))) |
24 | 16, 23 | mpbid 224 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (0g‘𝐻) = [ 0 ](𝐺 ~QG 𝑆)) |
25 | 24 | eqcomd 2783 | 1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [ 0 ](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2106 ‘cfv 6135 (class class class)co 6922 [cec 8024 Basecbs 16255 +gcplusg 16338 0gc0g 16486 /s cqus 16551 Grpcgrp 17809 SubGrpcsubg 17972 NrmSGrpcnsg 17973 ~QG cqg 17974 |
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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 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 |
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 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 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-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-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-ec 8028 df-qs 8032 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 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-fz 12644 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-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-0g 16488 df-imas 16554 df-qus 16555 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-subg 17975 df-nsg 17976 df-eqg 17977 |
This theorem is referenced by: qusinv 18037 qustgphaus 22334 qusker 30421 |
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