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Mirrors > Home > MPE Home > Th. List > Mathboxes > nsgqus0 | Structured version Visualization version GIF version |
Description: A normal subgroup 𝑁 is a member of all subgroups 𝐹 of the quotient group by 𝑁. In fact, it is the identity element of the quotient group. (Contributed by Thierry Arnoux, 27-Jul-2024.) |
Ref | Expression |
---|---|
nsgqus0.q | ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) |
Ref | Expression |
---|---|
nsgqus0 | ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . . 3 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺)) | |
2 | nsgsubg 18390 | . . 3 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺)) | |
3 | eqid 2758 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | eqid 2758 | . . . 4 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺) | |
5 | 3, 4 | lsm02 18878 | . . 3 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → ({(0g‘𝐺)} (LSSum‘𝐺)𝑁) = 𝑁) |
6 | 1, 2, 5 | 3syl 18 | . 2 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → ({(0g‘𝐺)} (LSSum‘𝐺)𝑁) = 𝑁) |
7 | nsgqus0.q | . . . . . 6 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) | |
8 | 7, 3 | qus0 18418 | . . . . 5 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → [(0g‘𝐺)](𝐺 ~QG 𝑁) = (0g‘𝑄)) |
9 | 8 | adantr 484 | . . . 4 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → [(0g‘𝐺)](𝐺 ~QG 𝑁) = (0g‘𝑄)) |
10 | eqid 2758 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
11 | 2 | adantr 484 | . . . . 5 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ (SubGrp‘𝐺)) |
12 | subgrcl 18364 | . . . . . . . 8 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
13 | 2, 12 | syl 17 | . . . . . . 7 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp) |
14 | 13 | adantr 484 | . . . . . 6 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → 𝐺 ∈ Grp) |
15 | 10, 3 | grpidcl 18211 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺)) |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → (0g‘𝐺) ∈ (Base‘𝐺)) |
17 | 10, 4, 11, 16 | quslsm 31126 | . . . 4 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → [(0g‘𝐺)](𝐺 ~QG 𝑁) = ({(0g‘𝐺)} (LSSum‘𝐺)𝑁)) |
18 | 9, 17 | eqtr3d 2795 | . . 3 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → (0g‘𝑄) = ({(0g‘𝐺)} (LSSum‘𝐺)𝑁)) |
19 | eqid 2758 | . . . . 5 ⊢ (0g‘𝑄) = (0g‘𝑄) | |
20 | 19 | subg0cl 18367 | . . . 4 ⊢ (𝐹 ∈ (SubGrp‘𝑄) → (0g‘𝑄) ∈ 𝐹) |
21 | 20 | adantl 485 | . . 3 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → (0g‘𝑄) ∈ 𝐹) |
22 | 18, 21 | eqeltrrd 2853 | . 2 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → ({(0g‘𝐺)} (LSSum‘𝐺)𝑁) ∈ 𝐹) |
23 | 6, 22 | eqeltrrd 2853 | 1 ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 ‘cfv 6340 (class class class)co 7156 [cec 8303 Basecbs 16554 0gc0g 16784 /s cqus 16849 Grpcgrp 18182 SubGrpcsubg 18353 NrmSGrpcnsg 18354 ~QG cqg 18355 LSSumclsm 18839 |
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 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-1st 7699 df-2nd 7700 df-tpos 7908 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-er 8305 df-ec 8307 df-qs 8311 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-sup 8952 df-inf 8953 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-nn 11688 df-2 11750 df-3 11751 df-4 11752 df-5 11753 df-6 11754 df-7 11755 df-8 11756 df-9 11757 df-n0 11948 df-z 12034 df-dec 12151 df-uz 12296 df-fz 12953 df-struct 16556 df-ndx 16557 df-slot 16558 df-base 16560 df-sets 16561 df-ress 16562 df-plusg 16649 df-mulr 16650 df-sca 16652 df-vsca 16653 df-ip 16654 df-tset 16655 df-ple 16656 df-ds 16658 df-0g 16786 df-imas 16852 df-qus 16853 df-mgm 17931 df-sgrp 17980 df-mnd 17991 df-submnd 18036 df-grp 18185 df-minusg 18186 df-subg 18356 df-nsg 18357 df-eqg 18358 df-oppg 18554 df-lsm 18841 |
This theorem is referenced by: nsgmgclem 31129 nsgmgc 31130 nsgqusf1olem2 31132 nsgqusf1olem3 31133 |
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