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Theorem cntzel 18067
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 18066 . 2 (𝑆𝐵 → (𝑋 ∈ (𝑍𝑆) ↔ (𝑋𝐵 ∧ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
54baibd 536 1 ((𝑆𝐵𝑋𝐵) → (𝑋 ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3090  wss 3770  cfv 6102  (class class class)co 6879  Basecbs 16183  +gcplusg 16266  Cntzccntz 18059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-ov 6882  df-cntz 18061
This theorem is referenced by:  cntzsubg  18080  cntzcmn  18559  cntzsubr  19129  cntzsdrg  38552
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