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Theorem ablcntzd 19875
Description: All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
ablcntzd.z 𝑍 = (Cntz‘𝐺)
ablcntzd.a (𝜑𝐺 ∈ Abel)
ablcntzd.t (𝜑𝑇 ∈ (SubGrp‘𝐺))
ablcntzd.u (𝜑𝑈 ∈ (SubGrp‘𝐺))
Assertion
Ref Expression
ablcntzd (𝜑𝑇 ⊆ (𝑍𝑈))

Proof of Theorem ablcntzd
StepHypRef Expression
1 ablcntzd.t . . 3 (𝜑𝑇 ∈ (SubGrp‘𝐺))
2 eqid 2737 . . . 4 (Base‘𝐺) = (Base‘𝐺)
32subgss 19145 . . 3 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
41, 3syl 17 . 2 (𝜑𝑇 ⊆ (Base‘𝐺))
5 ablcntzd.a . . . 4 (𝜑𝐺 ∈ Abel)
6 ablcmn 19805 . . . 4 (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
75, 6syl 17 . . 3 (𝜑𝐺 ∈ CMnd)
8 ablcntzd.u . . . 4 (𝜑𝑈 ∈ (SubGrp‘𝐺))
92subgss 19145 . . . 4 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
108, 9syl 17 . . 3 (𝜑𝑈 ⊆ (Base‘𝐺))
11 ablcntzd.z . . . 4 𝑍 = (Cntz‘𝐺)
122, 11cntzcmn 19858 . . 3 ((𝐺 ∈ CMnd ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑍𝑈) = (Base‘𝐺))
137, 10, 12syl2anc 584 . 2 (𝜑 → (𝑍𝑈) = (Base‘𝐺))
144, 13sseqtrrd 4021 1 (𝜑𝑇 ⊆ (𝑍𝑈))
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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