Theorem cntzrecd 18798

Description: Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
cntzrecd.z	⊢ 𝑍 = (Cntz‘𝐺)
cntzrecd.t	⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
cntzrecd.u	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
cntzrecd.s	⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))

Assertion

Ref	Expression
cntzrecd	⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇))

Proof of Theorem cntzrecd

Step	Hyp	Ref	Expression
1		cntzrecd.s	. 2 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))
2		cntzrecd.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
3		cntzrecd.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
4		eqid 2821	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgss 18274	. . . 4 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
6	4	subgss 18274	. . . 4 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
7		cntzrecd.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
8	4, 7	cntzrec 18458	. . . 4 ⊢ ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊆ (𝑍‘𝑈) ↔ 𝑈 ⊆ (𝑍‘𝑇)))
9	5, 6, 8	syl2an 597	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊆ (𝑍‘𝑈) ↔ 𝑈 ⊆ (𝑍‘𝑇)))
10	2, 3, 9	syl2anc 586	. 2 ⊢ (𝜑 → (𝑇 ⊆ (𝑍‘𝑈) ↔ 𝑈 ⊆ (𝑍‘𝑇)))
11	1, 10	mpbid 234	1 ⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110   ⊆ wss 3935  ‘cfv 6349  Basecbs 16477  SubGrpcsubg 18267  Cntzccntz 18439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-subg 18270  df-cntz 18441

This theorem is referenced by:  subgdisj2  18812  pj2f  18818  pj1id  18819  dprdcntz2  19154  dmdprdsplit2lem  19161  dmdprdsplit2  19162