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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 2610 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | subgss 17418 | . . 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 18022 | . . . 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 17418 | . . . 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 18068 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑍‘𝑈) = (Base‘𝐺)) |
13 | 7, 10, 12 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝑍‘𝑈) = (Base‘𝐺)) |
14 | 4, 13 | sseqtr4d 3605 | 1 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 Basecbs 15695 SubGrpcsubg 17411 Cntzccntz 17571 CMndccmn 18016 Abelcabl 18017 |
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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 df-reu 2903 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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-subg 17414 df-cntz 17573 df-cmn 18018 df-abl 18019 |
This theorem is referenced by: lsmsubg2 18085 ablfacrp2 18289 ablfac1b 18292 pgpfaclem1 18303 pgpfaclem2 18304 pj1lmhm 18921 pj1lmhm2 18922 lvecindp 18959 lvecindp2 18960 pjdm2 19874 pjf2 19877 pjfo 19878 lshpsmreu 33414 lshpkrlem5 33419 |
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