Theorem cntzrec 19378

Description: Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypotheses

Ref	Expression
cntzrec.b	⊢ 𝐵 = (Base‘𝑀)
cntzrec.z	⊢ 𝑍 = (Cntz‘𝑀)

Assertion

Ref	Expression
cntzrec	⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ 𝑇 ⊆ (𝑍‘𝑆)))

Proof of Theorem cntzrec

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ralcom 3292	. . . 4 ⊢ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ↔ ∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))
2		eqcom 2771	. . . . 5 ⊢ ((𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ↔ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
3	2	2ralbii 3139	. . . 4 ⊢ (∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ↔ ∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
4	1, 3	bitri 277	. . 3 ⊢ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ↔ ∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
5	4	a1i 11	. 2 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ↔ ∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)))
6		cntzrec.b	. . 3 ⊢ 𝐵 = (Base‘𝑀)
7		eqid 2764	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
8		cntzrec.z	. . 3 ⊢ 𝑍 = (Cntz‘𝑀)
9	6, 7, 8	sscntz 19368	. 2 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)))
10	6, 7, 8	sscntz 19368	. . 3 ⊢ ((𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑇 ⊆ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)))
11	10	ancoms 462	. 2 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑇 ⊆ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)))
12	5, 9, 11	3bitr4d 313	1 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ 𝑇 ⊆ (𝑍‘𝑆)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562  ∀wral 3078   ⊆ wss 3906  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  +_gcplusg 17288  Cntzccntz 19357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-cntz 19359

This theorem is referenced by:  cntzrecd  19720  lsmcntzr  19722  cntzspan  19886  dprdfadd  20064