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Theorem cntzrecd 18085
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 2621 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
54subgss 17589 . . . 4 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
64subgss 17589 . . . 4 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
7 cntzrecd.z . . . . 5 𝑍 = (Cntz‘𝐺)
84, 7cntzrec 17760 . . . 4 ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊆ (𝑍𝑈) ↔ 𝑈 ⊆ (𝑍𝑇)))
95, 6, 8syl2an 494 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊆ (𝑍𝑈) ↔ 𝑈 ⊆ (𝑍𝑇)))
102, 3, 9syl2anc 693 . 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 1482  wcel 1989  wss 3572  cfv 5886  Basecbs 15851  SubGrpcsubg 17582  Cntzccntz 17742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-subg 17585  df-cntz 17744
This theorem is referenced by:  subgdisj2  18099  pj2f  18105  pj1id  18106  dprdcntz2  18431  dmdprdsplit2lem  18438  dmdprdsplit2  18439
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