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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 18310 | . . . . . 6 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
2 | subgrcl 18284 | . . . . . 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 18151 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝑉) |
7 | 3, 6 | sylan 582 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝑉) |
8 | qusgrp.h | . . . . 5 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) | |
9 | eqid 2821 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
10 | eqid 2821 | . . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
11 | 8, 4, 9, 10 | qusadd 18337 | . . . 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 1458 | . . 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 2821 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
14 | 4, 9, 13, 5 | grprinv 18153 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = (0g‘𝐺)) |
15 | 3, 14 | sylan 582 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = (0g‘𝐺)) |
16 | 15 | eceq1d 8328 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [(𝑋(+g‘𝐺)(𝐼‘𝑋))](𝐺 ~QG 𝑆) = [(0g‘𝐺)](𝐺 ~QG 𝑆)) |
17 | 8, 13 | qus0 18338 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
18 | 17 | adantr 483 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [(0g‘𝐺)](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
19 | 12, 16, 18 | 3eqtrd 2860 | . 2 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → ([𝑋](𝐺 ~QG 𝑆)(+g‘𝐻)[(𝐼‘𝑋)](𝐺 ~QG 𝑆)) = (0g‘𝐻)) |
20 | 8 | qusgrp 18335 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp) |
21 | 20 | adantr 483 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → 𝐻 ∈ Grp) |
22 | eqid 2821 | . . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
23 | 8, 4, 22 | quseccl 18336 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
24 | 8, 4, 22 | quseccl 18336 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝐼‘𝑋) ∈ 𝑉) → [(𝐼‘𝑋)](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
25 | 7, 24 | syldan 593 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [(𝐼‘𝑋)](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
26 | eqid 2821 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
27 | qusinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐻) | |
28 | 22, 10, 26, 27 | grpinvid1 18154 | . . 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 1367 | . 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 259 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 [cec 8287 Basecbs 16483 +gcplusg 16565 0gc0g 16713 /s cqus 16778 Grpcgrp 18103 invgcminusg 18104 SubGrpcsubg 18273 NrmSGrpcnsg 18274 ~QG cqg 18275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-ec 8291 df-qs 8295 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-0g 16715 df-imas 16781 df-qus 16782 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-subg 18276 df-nsg 18277 df-eqg 18278 |
This theorem is referenced by: qussub 18340 |
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