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Theorem cntzrecd 18137
Description: Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
cntzrecd.z 𝑍 = (Cntz‘𝐺)
cntzrecd.t (𝜑𝑇 ∈ (SubGrp‘𝐺))
cntzrecd.u (𝜑𝑈 ∈ (SubGrp‘𝐺))
cntzrecd.s (𝜑𝑇 ⊆ (𝑍𝑈))
Assertion
Ref Expression
cntzrecd (𝜑𝑈 ⊆ (𝑍𝑇))

Proof of Theorem cntzrecd
StepHypRef Expression
1 cntzrecd.s . 2 (𝜑𝑇 ⊆ (𝑍𝑈))
2 cntzrecd.t . . 3 (𝜑𝑇 ∈ (SubGrp‘𝐺))
3 cntzrecd.u . . 3 (𝜑𝑈 ∈ (SubGrp‘𝐺))
4 eqid 2651 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
54subgss 17642 . . . 4 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
64subgss 17642 . . . 4 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
7 cntzrecd.z . . . . 5 𝑍 = (Cntz‘𝐺)
84, 7cntzrec 17812 . . . 4 ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊆ (𝑍𝑈) ↔ 𝑈 ⊆ (𝑍𝑇)))
95, 6, 8syl2an 493 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊆ (𝑍𝑈) ↔ 𝑈 ⊆ (𝑍𝑇)))
102, 3, 9syl2anc 694 . 2 (𝜑 → (𝑇 ⊆ (𝑍𝑈) ↔ 𝑈 ⊆ (𝑍𝑇)))
111, 10mpbid 222 1 (𝜑𝑈 ⊆ (𝑍𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wss 3607  cfv 5926  Basecbs 15904  SubGrpcsubg 17635  Cntzccntz 17794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-subg 17638  df-cntz 17796
This theorem is referenced by:  subgdisj2  18151  pj2f  18157  pj1id  18158  dprdcntz2  18483  dmdprdsplit2lem  18490  dmdprdsplit2  18491
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