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Theorem cntzrec 19278
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 3281 . . . 4 (∀𝑥𝑆𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ↔ ∀𝑦𝑇𝑥𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
2 eqcom 2734 . . . . 5 ((𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ↔ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
322ralbii 3123 . . . 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 2727 . . 3 (+g𝑀) = (+g𝑀)
8 cntzrec.z . . 3 𝑍 = (Cntz‘𝑀)
96, 7, 8sscntz 19268 . 2 ((𝑆𝐵𝑇𝐵) → (𝑆 ⊆ (𝑍𝑇) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
106, 7, 8sscntz 19268 . . 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 205  wa 395   = wceq 1534  wral 3056  wss 3944  cfv 6542  (class class class)co 7414  Basecbs 17171  +gcplusg 17224  Cntzccntz 19257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-cntz 19259
This theorem is referenced by:  cntzrecd  19624  lsmcntzr  19626  cntzspan  19790  dprdfadd  19968
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