Theorem cntzrecd 19751

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 2770	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgss 19196	. . . 4 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
6	4	subgss 19196	. . . 4 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
7		cntzrecd.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
8	4, 7	cntzrec 19409	. . . 4 ⊢ ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊆ (𝑍‘𝑈) ↔ 𝑈 ⊆ (𝑍‘𝑇)))
9	5, 6, 8	syl2an 607	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊆ (𝑍‘𝑈) ↔ 𝑈 ⊆ (𝑍‘𝑇)))
10	2, 3, 9	syl2anc 595	. 2 ⊢ (𝜑 → (𝑇 ⊆ (𝑍‘𝑈) ↔ 𝑈 ⊆ (𝑍‘𝑇)))
11	1, 10	mpbid 235	1 ⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1568   ∈ wcel 2150   ⊆ wss 3913  ‘cfv 6540  Basecbs 17272  SubGrpcsubg 19189  Cntzccntz 19388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7417  df-subg 19192  df-cntz 19390

This theorem is referenced by:  subgdisj2  19765  pj2f  19771  pj1id  19772  dprdcntz2  20113  dmdprdsplit2lem  20120  dmdprdsplit2  20121