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Theorem cntzrec 19367
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 3287 . . . 4 (∀𝑥𝑆𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ↔ ∀𝑦𝑇𝑥𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
2 eqcom 2742 . . . . 5 ((𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ↔ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
322ralbii 3126 . . . 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 2735 . . 3 (+g𝑀) = (+g𝑀)
8 cntzrec.z . . 3 𝑍 = (Cntz‘𝑀)
96, 7, 8sscntz 19357 . 2 ((𝑆𝐵𝑇𝐵) → (𝑆 ⊆ (𝑍𝑇) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
106, 7, 8sscntz 19357 . . 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 1537  wral 3059  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Cntzccntz 19346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-cntz 19348
This theorem is referenced by:  cntzrecd  19711  lsmcntzr  19713  cntzspan  19877  dprdfadd  20055
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