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Theorem cntzel 18574
Description: Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Hypotheses
Ref Expression
cntzfval.b 𝐵 = (Base‘𝑀)
cntzfval.p + = (+g𝑀)
cntzfval.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntzel ((𝑆𝐵𝑋𝐵) → (𝑋 ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))
Distinct variable groups:   𝑦, +   𝑦,𝑀   𝑦,𝑆   𝑦,𝑋
Allowed substitution hints:   𝐵(𝑦)   𝑍(𝑦)

Proof of Theorem cntzel
StepHypRef Expression
1 cntzfval.b . . 3 𝐵 = (Base‘𝑀)
2 cntzfval.p . . 3 + = (+g𝑀)
3 cntzfval.z . . 3 𝑍 = (Cntz‘𝑀)
41, 2, 3elcntz 18573 . 2 (𝑆𝐵 → (𝑋 ∈ (𝑍𝑆) ↔ (𝑋𝐵 ∧ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
54baibd 543 1 ((𝑆𝐵𝑋𝐵) → (𝑋 ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wral 3054  wss 3844  cfv 6340  (class class class)co 7173  Basecbs 16589  +gcplusg 16671  Cntzccntz 18566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-cntz 18568
This theorem is referenced by:  cntzsubg  18588  cntzcmn  19082  cntzsubr  19690  cntzsdrg  19703
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