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Theorem cntzrec 19355
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 ((𝑆𝐵𝑇𝐵) → (𝑆 ⊆ (𝑍𝑇) ↔ 𝑇 ⊆ (𝑍𝑆)))

Proof of Theorem cntzrec
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ralcom 3288 . . . 4 (∀𝑥𝑆𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ↔ ∀𝑦𝑇𝑥𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
2 eqcom 2743 . . . . 5 ((𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ↔ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
322ralbii 3127 . . . 4 (∀𝑦𝑇𝑥𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ↔ ∀𝑦𝑇𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
41, 3bitri 275 . . 3 (∀𝑥𝑆𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ↔ ∀𝑦𝑇𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
54a1i 11 . 2 ((𝑆𝐵𝑇𝐵) → (∀𝑥𝑆𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ↔ ∀𝑦𝑇𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)))
6 cntzrec.b . . 3 𝐵 = (Base‘𝑀)
7 eqid 2736 . . 3 (+g𝑀) = (+g𝑀)
8 cntzrec.z . . 3 𝑍 = (Cntz‘𝑀)
96, 7, 8sscntz 19345 . 2 ((𝑆𝐵𝑇𝐵) → (𝑆 ⊆ (𝑍𝑇) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
106, 7, 8sscntz 19345 . . 3 ((𝑇𝐵𝑆𝐵) → (𝑇 ⊆ (𝑍𝑆) ↔ ∀𝑦𝑇𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)))
1110ancoms 458 . 2 ((𝑆𝐵𝑇𝐵) → (𝑇 ⊆ (𝑍𝑆) ↔ ∀𝑦𝑇𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)))
125, 9, 113bitr4d 311 1 ((𝑆𝐵𝑇𝐵) → (𝑆 ⊆ (𝑍𝑇) ↔ 𝑇 ⊆ (𝑍𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wral 3060  wss 3950  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  Cntzccntz 19334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-cntz 19336
This theorem is referenced by:  cntzrecd  19697  lsmcntzr  19699  cntzspan  19863  dprdfadd  20041
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