MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cntzel Structured version   Visualization version   GIF version

Theorem cntzel 17696
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 17695 . 2 (𝑆𝐵 → (𝑋 ∈ (𝑍𝑆) ↔ (𝑋𝐵 ∧ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
54baibd 947 1 ((𝑆𝐵𝑋𝐵) → (𝑋 ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wss 3560  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  Cntzccntz 17688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-cntz 17690
This theorem is referenced by:  cntzsubg  17709  cntzcmn  18185  cntzsubr  18752  cntzsdrg  37292
  Copyright terms: Public domain W3C validator