Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ablnsg | Structured version Visualization version GIF version |
Description: Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
ablnsg | ⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2821 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | 1, 2 | ablcom 18918 | . . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦)) |
4 | 3 | 3expb 1116 | . . . . 5 ⊢ ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦)) |
5 | 4 | eleq1d 2897 | . . . 4 ⊢ ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g‘𝐺)𝑦) ∈ 𝑥)) |
6 | 5 | ralrimivva 3191 | . . 3 ⊢ (𝐺 ∈ Abel → ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g‘𝐺)𝑦) ∈ 𝑥)) |
7 | 1, 2 | isnsg 18301 | . . . 4 ⊢ (𝑥 ∈ (NrmSGrp‘𝐺) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g‘𝐺)𝑦) ∈ 𝑥))) |
8 | 7 | rbaib 541 | . . 3 ⊢ (∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g‘𝐺)𝑦) ∈ 𝑥) → (𝑥 ∈ (NrmSGrp‘𝐺) ↔ 𝑥 ∈ (SubGrp‘𝐺))) |
9 | 6, 8 | syl 17 | . 2 ⊢ (𝐺 ∈ Abel → (𝑥 ∈ (NrmSGrp‘𝐺) ↔ 𝑥 ∈ (SubGrp‘𝐺))) |
10 | 9 | eqrdv 2819 | 1 ⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 SubGrpcsubg 18267 NrmSGrpcnsg 18268 Abelcabl 18901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fv 6358 df-ov 7153 df-subg 18270 df-nsg 18271 df-cmn 18902 df-abl 18903 |
This theorem is referenced by: qusabl 18979 ablsimpnosubgd 19220 ablsimpgprmd 19231 qus1 20002 qusrhm 20004 quslmod 30918 quslmhm 30919 lidlnsg 30956 qusdimsum 31019 |
Copyright terms: Public domain | W3C validator |