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Theorem cntzrecd 18795
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 2822 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
54subgss 18271 . . . 4 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
64subgss 18271 . . . 4 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
7 cntzrecd.z . . . . 5 𝑍 = (Cntz‘𝐺)
84, 7cntzrec 18455 . . . 4 ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊆ (𝑍𝑈) ↔ 𝑈 ⊆ (𝑍𝑇)))
95, 6, 8syl2an 598 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊆ (𝑍𝑈) ↔ 𝑈 ⊆ (𝑍𝑇)))
102, 3, 9syl2anc 587 . 2 (𝜑 → (𝑇 ⊆ (𝑍𝑈) ↔ 𝑈 ⊆ (𝑍𝑇)))
111, 10mpbid 235 1 (𝜑𝑈 ⊆ (𝑍𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2114  wss 3908  cfv 6334  Basecbs 16474  SubGrpcsubg 18264  Cntzccntz 18436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-subg 18267  df-cntz 18438
This theorem is referenced by:  subgdisj2  18809  pj2f  18815  pj1id  18816  dprdcntz2  19151  dmdprdsplit2lem  19158  dmdprdsplit2  19159
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