![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > qusinv | Structured version Visualization version GIF version |
Description: Value of the group inverse operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) |
Ref | Expression |
---|---|
qusgrp.h | ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) |
qusinv.v | ⊢ 𝑉 = (Base‘𝐺) |
qusinv.i | ⊢ 𝐼 = (invg‘𝐺) |
qusinv.n | ⊢ 𝑁 = (invg‘𝐻) |
Ref | Expression |
---|---|
qusinv | ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝑁‘[𝑋](𝐺 ~QG 𝑆)) = [(𝐼‘𝑋)](𝐺 ~QG 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsgsubg 18302 | . . . . . 6 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
2 | subgrcl 18276 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp) |
4 | qusinv.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝐺) | |
5 | qusinv.i | . . . . . 6 ⊢ 𝐼 = (invg‘𝐺) | |
6 | 4, 5 | grpinvcl 18143 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝑉) |
7 | 3, 6 | sylan 583 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝑉) |
8 | qusgrp.h | . . . . 5 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) | |
9 | eqid 2798 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
10 | eqid 2798 | . . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
11 | 8, 4, 9, 10 | qusadd 18329 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉 ∧ (𝐼‘𝑋) ∈ 𝑉) → ([𝑋](𝐺 ~QG 𝑆)(+g‘𝐻)[(𝐼‘𝑋)](𝐺 ~QG 𝑆)) = [(𝑋(+g‘𝐺)(𝐼‘𝑋))](𝐺 ~QG 𝑆)) |
12 | 7, 11 | mpd3an3 1459 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → ([𝑋](𝐺 ~QG 𝑆)(+g‘𝐻)[(𝐼‘𝑋)](𝐺 ~QG 𝑆)) = [(𝑋(+g‘𝐺)(𝐼‘𝑋))](𝐺 ~QG 𝑆)) |
13 | eqid 2798 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
14 | 4, 9, 13, 5 | grprinv 18145 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = (0g‘𝐺)) |
15 | 3, 14 | sylan 583 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = (0g‘𝐺)) |
16 | 15 | eceq1d 8311 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [(𝑋(+g‘𝐺)(𝐼‘𝑋))](𝐺 ~QG 𝑆) = [(0g‘𝐺)](𝐺 ~QG 𝑆)) |
17 | 8, 13 | qus0 18330 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
18 | 17 | adantr 484 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [(0g‘𝐺)](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
19 | 12, 16, 18 | 3eqtrd 2837 | . 2 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → ([𝑋](𝐺 ~QG 𝑆)(+g‘𝐻)[(𝐼‘𝑋)](𝐺 ~QG 𝑆)) = (0g‘𝐻)) |
20 | 8 | qusgrp 18327 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp) |
21 | 20 | adantr 484 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → 𝐻 ∈ Grp) |
22 | eqid 2798 | . . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
23 | 8, 4, 22 | quseccl 18328 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
24 | 8, 4, 22 | quseccl 18328 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝐼‘𝑋) ∈ 𝑉) → [(𝐼‘𝑋)](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
25 | 7, 24 | syldan 594 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [(𝐼‘𝑋)](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
26 | eqid 2798 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
27 | qusinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐻) | |
28 | 22, 10, 26, 27 | grpinvid1 18146 | . . 3 ⊢ ((𝐻 ∈ Grp ∧ [𝑋](𝐺 ~QG 𝑆) ∈ (Base‘𝐻) ∧ [(𝐼‘𝑋)](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) → ((𝑁‘[𝑋](𝐺 ~QG 𝑆)) = [(𝐼‘𝑋)](𝐺 ~QG 𝑆) ↔ ([𝑋](𝐺 ~QG 𝑆)(+g‘𝐻)[(𝐼‘𝑋)](𝐺 ~QG 𝑆)) = (0g‘𝐻))) |
29 | 21, 23, 25, 28 | syl3anc 1368 | . 2 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → ((𝑁‘[𝑋](𝐺 ~QG 𝑆)) = [(𝐼‘𝑋)](𝐺 ~QG 𝑆) ↔ ([𝑋](𝐺 ~QG 𝑆)(+g‘𝐻)[(𝐼‘𝑋)](𝐺 ~QG 𝑆)) = (0g‘𝐻))) |
30 | 19, 29 | mpbird 260 | 1 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝑁‘[𝑋](𝐺 ~QG 𝑆)) = [(𝐼‘𝑋)](𝐺 ~QG 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 [cec 8270 Basecbs 16475 +gcplusg 16557 0gc0g 16705 /s cqus 16770 Grpcgrp 18095 invgcminusg 18096 SubGrpcsubg 18265 NrmSGrpcnsg 18266 ~QG cqg 18267 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-ec 8274 df-qs 8278 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-0g 16707 df-imas 16773 df-qus 16774 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-subg 18268 df-nsg 18269 df-eqg 18270 |
This theorem is referenced by: qussub 18332 |
Copyright terms: Public domain | W3C validator |