MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elcntz Structured version   Visualization version   GIF version

Theorem elcntz 18716
Description: Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
cntzfval.b 𝐵 = (Base‘𝑀)
cntzfval.p + = (+g𝑀)
cntzfval.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
elcntz (𝑆𝐵 → (𝐴 ∈ (𝑍𝑆) ↔ (𝐴𝐵 ∧ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴))))
Distinct variable groups:   𝑦, +   𝑦,𝐴   𝑦,𝑀   𝑦,𝑆
Allowed substitution hints:   𝐵(𝑦)   𝑍(𝑦)

Proof of Theorem elcntz
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cntzfval.b . . . 4 𝐵 = (Base‘𝑀)
2 cntzfval.p . . . 4 + = (+g𝑀)
3 cntzfval.z . . . 4 𝑍 = (Cntz‘𝑀)
41, 2, 3cntzval 18715 . . 3 (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
54eleq2d 2823 . 2 (𝑆𝐵 → (𝐴 ∈ (𝑍𝑆) ↔ 𝐴 ∈ {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6 oveq1 7220 . . . . 5 (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
7 oveq2 7221 . . . . 5 (𝑥 = 𝐴 → (𝑦 + 𝑥) = (𝑦 + 𝐴))
86, 7eqeq12d 2753 . . . 4 (𝑥 = 𝐴 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝐴 + 𝑦) = (𝑦 + 𝐴)))
98ralbidv 3118 . . 3 (𝑥 = 𝐴 → (∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
109elrab 3602 . 2 (𝐴 ∈ {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝐴𝐵 ∧ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
115, 10bitrdi 290 1 (𝑆𝐵 → (𝐴 ∈ (𝑍𝑆) ↔ (𝐴𝐵 ∧ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061  {crab 3065  wss 3866  cfv 6380  (class class class)co 7213  Basecbs 16760  +gcplusg 16802  Cntzccntz 18709
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 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-cntz 18711
This theorem is referenced by:  cntzel  18717  cntzi  18723  resscntz  18726  cntzsubm  18730  cntzmhm  18733  oppgcntz  18756  dprdfcntz  19402  cntzun  31039
  Copyright terms: Public domain W3C validator