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Theorem cntziinsn 17688
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 2621 . . 3 (+g𝑀) = (+g𝑀)
3 cntzrec.z . . 3 𝑍 = (Cntz‘𝑀)
41, 2, 3cntzval 17675 . 2 (𝑆𝐵 → (𝑍𝑆) = {𝑦𝐵 ∣ ∀𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
5 ssel2 3578 . . . . . 6 ((𝑆𝐵𝑥𝑆) → 𝑥𝐵)
61, 2, 3cntzsnval 17678 . . . . . 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 4509 . . . 4 (𝑆𝐵 𝑥𝑆 (𝑍‘{𝑥}) = 𝑥𝑆 {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
98ineq2d 3792 . . 3 (𝑆𝐵 → (𝐵 𝑥𝑆 (𝑍‘{𝑥})) = (𝐵 𝑥𝑆 {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)}))
10 riinrab 4562 . . 3 (𝐵 𝑥𝑆 {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)}) = {𝑦𝐵 ∣ ∀𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)}
119, 10syl6eq 2671 . 2 (𝑆𝐵 → (𝐵 𝑥𝑆 (𝑍‘{𝑥})) = {𝑦𝐵 ∣ ∀𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
124, 11eqtr4d 2658 1 (𝑆𝐵 → (𝑍𝑆) = (𝐵 𝑥𝑆 (𝑍‘{𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  cin 3554  wss 3555  {csn 4148   ciin 4486  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Cntzccntz 17669
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
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 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-cntz 17671
This theorem is referenced by: (None)
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