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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 2737 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 3 | 2 | subgss 19145 | . . 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 19805 | . . . 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 19145 | . . . 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 19858 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑍‘𝑈) = (Base‘𝐺)) |
| 13 | 7, 10, 12 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑍‘𝑈) = (Base‘𝐺)) |
| 14 | 4, 13 | sseqtrrd 4021 | 1 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 Basecbs 17247 SubGrpcsubg 19138 Cntzccntz 19333 CMndccmn 19798 Abelcabl 19799 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 |
| This theorem is referenced by: lsmsubg2 19877 ablfacrp2 20087 ablfac1b 20090 pgpfaclem1 20101 pgpfaclem2 20102 pj1lmhm 21099 pj1lmhm2 21100 lvecindp 21140 lvecindp2 21141 pjdm2 21731 pjf2 21734 pjfo 21735 lshpsmreu 39110 lshpkrlem5 39115 |
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