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Mirrors > Home > MPE Home > Th. List > qusghm | Structured version Visualization version GIF version |
Description: If 𝑌 is a normal subgroup of 𝐺, then the "natural map" from elements to their cosets is a group homomorphism from 𝐺 to 𝐺 / 𝑌. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 18-Sep-2015.) |
Ref | Expression |
---|---|
qusghm.x | ⊢ 𝑋 = (Base‘𝐺) |
qusghm.h | ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌)) |
qusghm.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) |
Ref | Expression |
---|---|
qusghm | ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusghm.x | . 2 ⊢ 𝑋 = (Base‘𝐺) | |
2 | eqid 2797 | . 2 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
3 | eqid 2797 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2797 | . 2 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
5 | nsgsubg 17936 | . . 3 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺)) | |
6 | subgrcl 17909 | . . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp) |
8 | qusghm.h | . . 3 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌)) | |
9 | 8 | qusgrp 17959 | . 2 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp) |
10 | 8, 1, 2 | quseccl 17960 | . . 3 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → [𝑥](𝐺 ~QG 𝑌) ∈ (Base‘𝐻)) |
11 | qusghm.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) | |
12 | 10, 11 | fmptd 6608 | . 2 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐹:𝑋⟶(Base‘𝐻)) |
13 | 8, 1, 3, 4 | qusadd 17961 | . . . 4 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ([𝑦](𝐺 ~QG 𝑌)(+g‘𝐻)[𝑧](𝐺 ~QG 𝑌)) = [(𝑦(+g‘𝐺)𝑧)](𝐺 ~QG 𝑌)) |
14 | 13 | 3expb 1150 | . . 3 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ([𝑦](𝐺 ~QG 𝑌)(+g‘𝐻)[𝑧](𝐺 ~QG 𝑌)) = [(𝑦(+g‘𝐺)𝑧)](𝐺 ~QG 𝑌)) |
15 | eceq1 8018 | . . . . . 6 ⊢ (𝑥 = 𝑦 → [𝑥](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌)) | |
16 | ovex 6908 | . . . . . . 7 ⊢ (𝐺 ~QG 𝑌) ∈ V | |
17 | ecexg 7984 | . . . . . . 7 ⊢ ((𝐺 ~QG 𝑌) ∈ V → [𝑥](𝐺 ~QG 𝑌) ∈ V) | |
18 | 16, 17 | ax-mp 5 | . . . . . 6 ⊢ [𝑥](𝐺 ~QG 𝑌) ∈ V |
19 | 15, 11, 18 | fvmpt3i 6510 | . . . . 5 ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = [𝑦](𝐺 ~QG 𝑌)) |
20 | 19 | ad2antrl 720 | . . . 4 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = [𝑦](𝐺 ~QG 𝑌)) |
21 | eceq1 8018 | . . . . . 6 ⊢ (𝑥 = 𝑧 → [𝑥](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌)) | |
22 | 21, 11, 18 | fvmpt3i 6510 | . . . . 5 ⊢ (𝑧 ∈ 𝑋 → (𝐹‘𝑧) = [𝑧](𝐺 ~QG 𝑌)) |
23 | 22 | ad2antll 721 | . . . 4 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = [𝑧](𝐺 ~QG 𝑌)) |
24 | 20, 23 | oveq12d 6894 | . . 3 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐹‘𝑦)(+g‘𝐻)(𝐹‘𝑧)) = ([𝑦](𝐺 ~QG 𝑌)(+g‘𝐻)[𝑧](𝐺 ~QG 𝑌))) |
25 | 1, 3 | grpcl 17743 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋) |
26 | 25 | 3expb 1150 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋) |
27 | 7, 26 | sylan 576 | . . . 4 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋) |
28 | eceq1 8018 | . . . . 5 ⊢ (𝑥 = (𝑦(+g‘𝐺)𝑧) → [𝑥](𝐺 ~QG 𝑌) = [(𝑦(+g‘𝐺)𝑧)](𝐺 ~QG 𝑌)) | |
29 | 28, 11, 18 | fvmpt3i 6510 | . . . 4 ⊢ ((𝑦(+g‘𝐺)𝑧) ∈ 𝑋 → (𝐹‘(𝑦(+g‘𝐺)𝑧)) = [(𝑦(+g‘𝐺)𝑧)](𝐺 ~QG 𝑌)) |
30 | 27, 29 | syl 17 | . . 3 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(+g‘𝐺)𝑧)) = [(𝑦(+g‘𝐺)𝑧)](𝐺 ~QG 𝑌)) |
31 | 14, 24, 30 | 3eqtr4rd 2842 | . 2 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(+g‘𝐺)𝑧)) = ((𝐹‘𝑦)(+g‘𝐻)(𝐹‘𝑧))) |
32 | 1, 2, 3, 4, 7, 9, 12, 31 | isghmd 17979 | 1 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3383 ↦ cmpt 4920 ‘cfv 6099 (class class class)co 6876 [cec 7978 Basecbs 16181 +gcplusg 16264 /s cqus 16477 Grpcgrp 17735 SubGrpcsubg 17898 NrmSGrpcnsg 17899 ~QG cqg 17900 GrpHom cghm 17967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-ec 7982 df-qs 7986 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-sup 8588 df-inf 8589 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-fz 12577 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-0g 16414 df-imas 16480 df-qus 16481 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-grp 17738 df-minusg 17739 df-subg 17901 df-nsg 17902 df-eqg 17903 df-ghm 17968 |
This theorem is referenced by: qusrhm 19557 |
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