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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 2772 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | subgss 18054 | . . 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 18662 | . . . 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 18054 | . . . 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 18708 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑍‘𝑈) = (Base‘𝐺)) |
13 | 7, 10, 12 | syl2anc 576 | . 2 ⊢ (𝜑 → (𝑍‘𝑈) = (Base‘𝐺)) |
14 | 4, 13 | sseqtr4d 3894 | 1 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2048 ⊆ wss 3825 ‘cfv 6182 Basecbs 16329 SubGrpcsubg 18047 Cntzccntz 18206 CMndccmn 18656 Abelcabl 18657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-subg 18050 df-cntz 18208 df-cmn 18658 df-abl 18659 |
This theorem is referenced by: lsmsubg2 18725 ablfacrp2 18929 ablfac1b 18932 pgpfaclem1 18943 pgpfaclem2 18944 pj1lmhm 19584 pj1lmhm2 19585 lvecindp 19622 lvecindp2 19623 pjdm2 20547 pjf2 20550 pjfo 20551 lshpsmreu 35638 lshpkrlem5 35643 |
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