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Mirrors > Home > MPE Home > Th. List > ablcntzd | Structured version Visualization version GIF version |
Description: All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
ablcntzd.z | ⊢ 𝑍 = (Cntz‘𝐺) |
ablcntzd.a | ⊢ (𝜑 → 𝐺 ∈ Abel) |
ablcntzd.t | ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) |
ablcntzd.u | ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) |
Ref | Expression |
---|---|
ablcntzd | ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablcntzd.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
2 | eqid 2821 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | subgss 18274 | . . 3 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺)) |
5 | ablcntzd.a | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
6 | ablcmn 18907 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
8 | ablcntzd.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
9 | 2 | subgss 18274 | . . . 4 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐺)) |
11 | ablcntzd.z | . . . 4 ⊢ 𝑍 = (Cntz‘𝐺) | |
12 | 2, 11 | cntzcmn 18954 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑍‘𝑈) = (Base‘𝐺)) |
13 | 7, 10, 12 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝑍‘𝑈) = (Base‘𝐺)) |
14 | 4, 13 | sseqtrrd 4007 | 1 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6349 Basecbs 16477 SubGrpcsubg 18267 Cntzccntz 18439 CMndccmn 18900 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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
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-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-subg 18270 df-cntz 18441 df-cmn 18902 df-abl 18903 |
This theorem is referenced by: lsmsubg2 18973 ablfacrp2 19183 ablfac1b 19186 pgpfaclem1 19197 pgpfaclem2 19198 pj1lmhm 19866 pj1lmhm2 19867 lvecindp 19904 lvecindp2 19905 pjdm2 20849 pjf2 20852 pjfo 20853 lshpsmreu 36239 lshpkrlem5 36244 |
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