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Theorem cntziinsn 17813
Description: Express any centralizer as an intersection of singleton centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cntzrec.b 𝐵 = (Base‘𝑀)
cntzrec.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntziinsn (𝑆𝐵 → (𝑍𝑆) = (𝐵 𝑥𝑆 (𝑍‘{𝑥})))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑀   𝑥,𝑆   𝑥,𝑍

Proof of Theorem cntziinsn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cntzrec.b . . 3 𝐵 = (Base‘𝑀)
2 eqid 2651 . . 3 (+g𝑀) = (+g𝑀)
3 cntzrec.z . . 3 𝑍 = (Cntz‘𝑀)
41, 2, 3cntzval 17800 . 2 (𝑆𝐵 → (𝑍𝑆) = {𝑦𝐵 ∣ ∀𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
5 ssel2 3631 . . . . . 6 ((𝑆𝐵𝑥𝑆) → 𝑥𝐵)
61, 2, 3cntzsnval 17803 . . . . . 6 (𝑥𝐵 → (𝑍‘{𝑥}) = {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
75, 6syl 17 . . . . 5 ((𝑆𝐵𝑥𝑆) → (𝑍‘{𝑥}) = {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
87iineq2dv 4575 . . . 4 (𝑆𝐵 𝑥𝑆 (𝑍‘{𝑥}) = 𝑥𝑆 {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
98ineq2d 3847 . . 3 (𝑆𝐵 → (𝐵 𝑥𝑆 (𝑍‘{𝑥})) = (𝐵 𝑥𝑆 {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)}))
10 riinrab 4628 . . 3 (𝐵 𝑥𝑆 {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)}) = {𝑦𝐵 ∣ ∀𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)}
119, 10syl6eq 2701 . 2 (𝑆𝐵 → (𝐵 𝑥𝑆 (𝑍‘{𝑥})) = {𝑦𝐵 ∣ ∀𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
124, 11eqtr4d 2688 1 (𝑆𝐵 → (𝑍𝑆) = (𝐵 𝑥𝑆 (𝑍‘{𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  cin 3606  wss 3607  {csn 4210   ciin 4553  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Cntzccntz 17794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-cntz 17796
This theorem is referenced by: (None)
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