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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 17449 | . . . . . 6 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
2 | subgrcl 17422 | . . . . . 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 17290 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝑉) |
7 | 3, 6 | sylan 487 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝑉) |
8 | qusgrp.h | . . . . 5 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)) | |
9 | eqid 2610 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
10 | eqid 2610 | . . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
11 | 8, 4, 9, 10 | qusadd 17474 | . . . 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 1417 | . . 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 2610 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
14 | 4, 9, 13, 5 | grprinv 17292 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = (0g‘𝐺)) |
15 | 3, 14 | sylan 487 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = (0g‘𝐺)) |
16 | 15 | eceq1d 7670 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [(𝑋(+g‘𝐺)(𝐼‘𝑋))](𝐺 ~QG 𝑆) = [(0g‘𝐺)](𝐺 ~QG 𝑆)) |
17 | 8, 13 | qus0 17475 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
18 | 17 | adantr 480 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [(0g‘𝐺)](𝐺 ~QG 𝑆) = (0g‘𝐻)) |
19 | 12, 16, 18 | 3eqtrd 2648 | . 2 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → ([𝑋](𝐺 ~QG 𝑆)(+g‘𝐻)[(𝐼‘𝑋)](𝐺 ~QG 𝑆)) = (0g‘𝐻)) |
20 | 8 | qusgrp 17472 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp) |
21 | 20 | adantr 480 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → 𝐻 ∈ Grp) |
22 | eqid 2610 | . . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
23 | 8, 4, 22 | quseccl 17473 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
24 | 8, 4, 22 | quseccl 17473 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝐼‘𝑋) ∈ 𝑉) → [(𝐼‘𝑋)](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
25 | 7, 24 | syldan 486 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [(𝐼‘𝑋)](𝐺 ~QG 𝑆) ∈ (Base‘𝐻)) |
26 | eqid 2610 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
27 | qusinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐻) | |
28 | 22, 10, 26, 27 | grpinvid1 17293 | . . 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 1318 | . 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 246 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 [cec 7627 Basecbs 15695 +gcplusg 15768 0gc0g 15923 /s cqus 15988 Grpcgrp 17245 invgcminusg 17246 SubGrpcsubg 17411 NrmSGrpcnsg 17412 ~QG cqg 17413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-0g 15925 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-subg 17414 df-nsg 17415 df-eqg 17416 |
This theorem is referenced by: qussub 17477 |
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