Theorem cntzrecd 19648

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 2741	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgss 19098	. . . 4 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
6	4	subgss 19098	. . . 4 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
7		cntzrecd.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
8	4, 7	cntzrec 19306	. . . 4 ⊢ ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊆ (𝑍‘𝑈) ↔ 𝑈 ⊆ (𝑍‘𝑇)))
9	5, 6, 8	syl2an 603	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊆ (𝑍‘𝑈) ↔ 𝑈 ⊆ (𝑍‘𝑇)))
10	2, 3, 9	syl2anc 591	. 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 1548   ∈ wcel 2121   ⊆ wss 3885  ‘cfv 6489  Basecbs 17174  SubGrpcsubg 19091  Cntzccntz 19285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-subg 19094  df-cntz 19287

This theorem is referenced by:  subgdisj2  19662  pj2f  19668  pj1id  19669  dprdcntz2  20010  dmdprdsplit2lem  20017  dmdprdsplit2  20018