![]() |
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 18961 | . . . . . 6 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
2 | subgrcl 18934 | . . . . . 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 18799 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝑉) |
7 | 3, 6 | sylan 581 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝑉) |
8 | qusgrp.h | . . . . 5 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) | |
9 | eqid 2737 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
10 | eqid 2737 | . . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
11 | 8, 4, 9, 10 | qusadd 18988 | . . . 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 1463 | . . 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 2737 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
14 | 4, 9, 13, 5 | grprinv 18802 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = (0g‘𝐺)) |
15 | 3, 14 | sylan 581 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = (0g‘𝐺)) |
16 | 15 | eceq1d 8688 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [(𝑋(+g‘𝐺)(𝐼‘𝑋))](𝐺 ~QG 𝑆) = [(0g‘𝐺)](𝐺 ~QG 𝑆)) |
17 | 8, 13 | qus0 18989 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
18 | 17 | adantr 482 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [(0g‘𝐺)](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
19 | 12, 16, 18 | 3eqtrd 2781 | . 2 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → ([𝑋](𝐺 ~QG 𝑆)(+g‘𝐻)[(𝐼‘𝑋)](𝐺 ~QG 𝑆)) = (0g‘𝐻)) |
20 | 8 | qusgrp 18986 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp) |
21 | 20 | adantr 482 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → 𝐻 ∈ Grp) |
22 | eqid 2737 | . . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
23 | 8, 4, 22 | quseccl 18987 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
24 | 8, 4, 22 | quseccl 18987 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝐼‘𝑋) ∈ 𝑉) → [(𝐼‘𝑋)](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
25 | 7, 24 | syldan 592 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [(𝐼‘𝑋)](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
26 | eqid 2737 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
27 | qusinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐻) | |
28 | 22, 10, 26, 27 | grpinvid1 18803 | . . 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 1372 | . 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 257 | 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6497 (class class class)co 7358 [cec 8647 Basecbs 17084 +gcplusg 17134 0gc0g 17322 /s cqus 17388 Grpcgrp 18749 invgcminusg 18750 SubGrpcsubg 18923 NrmSGrpcnsg 18924 ~QG cqg 18925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-ec 8651 df-qs 8655 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-fz 13426 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-mulr 17148 df-sca 17150 df-vsca 17151 df-ip 17152 df-tset 17153 df-ple 17154 df-ds 17156 df-0g 17324 df-imas 17391 df-qus 17392 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-grp 18752 df-minusg 18753 df-subg 18926 df-nsg 18927 df-eqg 18928 |
This theorem is referenced by: qussub 18991 nsgmgclem 32192 nsgqusf1olem1 32194 |
Copyright terms: Public domain | W3C validator |