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Theorem elcntz 17736
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 17735 . . 3 (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
54eleq2d 2685 . 2 (𝑆𝐵 → (𝐴 ∈ (𝑍𝑆) ↔ 𝐴 ∈ {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6 oveq1 6642 . . . . 5 (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
7 oveq2 6643 . . . . 5 (𝑥 = 𝐴 → (𝑦 + 𝑥) = (𝑦 + 𝐴))
86, 7eqeq12d 2635 . . . 4 (𝑥 = 𝐴 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝐴 + 𝑦) = (𝑦 + 𝐴)))
98ralbidv 2983 . . 3 (𝑥 = 𝐴 → (∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
109elrab 3357 . 2 (𝐴 ∈ {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝐴𝐵 ∧ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
115, 10syl6bb 276 1 (𝑆𝐵 → (𝐴 ∈ (𝑍𝑆) ↔ (𝐴𝐵 ∧ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  {crab 2913  wss 3567  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  Cntzccntz 17729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-cntz 17731
This theorem is referenced by:  cntzel  17737  cntzi  17743  resscntz  17745  cntzsubm  17749  cntzmhm  17752  oppgcntz  17775  dprdfcntz  18395
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