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Theorem elcntz 19228
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 19227 . . 3 (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
54eleq2d 2818 . 2 (𝑆𝐵 → (𝐴 ∈ (𝑍𝑆) ↔ 𝐴 ∈ {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6 oveq1 7419 . . . . 5 (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
7 oveq2 7420 . . . . 5 (𝑥 = 𝐴 → (𝑦 + 𝑥) = (𝑦 + 𝐴))
86, 7eqeq12d 2747 . . . 4 (𝑥 = 𝐴 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝐴 + 𝑦) = (𝑦 + 𝐴)))
98ralbidv 3176 . . 3 (𝑥 = 𝐴 → (∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
109elrab 3683 . 2 (𝐴 ∈ {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝐴𝐵 ∧ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
115, 10bitrdi 287 1 (𝑆𝐵 → (𝐴 ∈ (𝑍𝑆) ↔ (𝐴𝐵 ∧ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wral 3060  {crab 3431  wss 3948  cfv 6543  (class class class)co 7412  Basecbs 17149  +gcplusg 17202  Cntzccntz 19221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-cntz 19223
This theorem is referenced by:  cntzel  19229  cntzi  19235  elcntr  19236  resscntz  19239  cntzsgrpcl  19240  cntzsubm  19244  cntzmhm  19247  oppgcntz  19273  dprdfcntz  19927  rng2idl1cntr  21065  cntzun  32483
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