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Theorem elcntz 18465
 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 18464 . . 3 (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
54eleq2d 2875 . 2 (𝑆𝐵 → (𝐴 ∈ (𝑍𝑆) ↔ 𝐴 ∈ {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6 oveq1 7152 . . . . 5 (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
7 oveq2 7153 . . . . 5 (𝑥 = 𝐴 → (𝑦 + 𝑥) = (𝑦 + 𝐴))
86, 7eqeq12d 2814 . . . 4 (𝑥 = 𝐴 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝐴 + 𝑦) = (𝑦 + 𝐴)))
98ralbidv 3162 . . 3 (𝑥 = 𝐴 → (∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
109elrab 3630 . 2 (𝐴 ∈ {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝐴𝐵 ∧ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
115, 10syl6bb 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 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110   ⊆ wss 3883  ‘cfv 6332  (class class class)co 7145  Basecbs 16495  +gcplusg 16577  Cntzccntz 18458 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-cntz 18460 This theorem is referenced by:  cntzel  18466  cntzi  18472  resscntz  18475  cntzsubm  18479  cntzmhm  18482  oppgcntz  18505  dprdfcntz  19151  cntzun  30794
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