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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 18528 | . . . . . . 7 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
2 | subgrcl 18502 | . . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp) |
4 | eqid 2736 | . . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | qus0.p | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
6 | 4, 5 | grpidcl 18349 | . . . . . 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 2736 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
10 | eqid 2736 | . . . . . 6 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
11 | 8, 4, 9, 10 | qusadd 18555 | . . . . 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 1465 | . . . 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 18351 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
14 | 3, 7, 13 | syl2anc 587 | . . . . 5 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
15 | 14 | eceq1d 8408 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [( 0 (+g‘𝐺) 0 )](𝐺 ~QG 𝑆) = [ 0 ](𝐺 ~QG 𝑆)) |
16 | 12, 15 | eqtrd 2771 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ([ 0 ](𝐺 ~QG 𝑆)(+g‘𝐻)[ 0 ](𝐺 ~QG 𝑆)) = [ 0 ](𝐺 ~QG 𝑆)) |
17 | 8 | qusgrp 18553 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp) |
18 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
19 | 8, 4, 18 | quseccl 18554 | . . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 0 ∈ (Base‘𝐺)) → [ 0 ](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
20 | 7, 19 | mpdan 687 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [ 0 ](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
21 | eqid 2736 | . . . . 5 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
22 | 18, 10, 21 | grpid 18357 | . . . 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 587 | . . 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 235 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (0g‘𝐻) = [ 0 ](𝐺 ~QG 𝑆)) |
25 | 24 | eqcomd 2742 | 1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [ 0 ](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 [cec 8367 Basecbs 16666 +gcplusg 16749 0gc0g 16898 /s cqus 16964 Grpcgrp 18319 SubGrpcsubg 18491 NrmSGrpcnsg 18492 ~QG cqg 18493 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-ec 8371 df-qs 8375 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-0g 16900 df-imas 16967 df-qus 16968 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-minusg 18323 df-subg 18494 df-nsg 18495 df-eqg 18496 |
This theorem is referenced by: qusinv 18557 qustgphaus 22974 qusker 31217 nsgqus0 31263 qsidomlem1 31296 qsidomlem2 31297 |
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