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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 2724 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | subgss 19046 | . . 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 19699 | . . . 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 19046 | . . . 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 19752 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑍‘𝑈) = (Base‘𝐺)) |
13 | 7, 10, 12 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝑍‘𝑈) = (Base‘𝐺)) |
14 | 4, 13 | sseqtrrd 4016 | 1 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3941 ‘cfv 6534 Basecbs 17145 SubGrpcsubg 19039 Cntzccntz 19223 CMndccmn 19692 Abelcabl 19693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-subg 19042 df-cntz 19225 df-cmn 19694 df-abl 19695 |
This theorem is referenced by: lsmsubg2 19771 ablfacrp2 19981 ablfac1b 19984 pgpfaclem1 19995 pgpfaclem2 19996 pj1lmhm 20940 pj1lmhm2 20941 lvecindp 20981 lvecindp2 20982 pjdm2 21576 pjf2 21579 pjfo 21580 lshpsmreu 38473 lshpkrlem5 38478 |
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