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Theorem cntziinsn 19356
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 2736 . . 3 (+g𝑀) = (+g𝑀)
3 cntzrec.z . . 3 𝑍 = (Cntz‘𝑀)
41, 2, 3cntzval 19340 . 2 (𝑆𝐵 → (𝑍𝑆) = {𝑦𝐵 ∣ ∀𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
5 ssel2 3977 . . . . . 6 ((𝑆𝐵𝑥𝑆) → 𝑥𝐵)
61, 2, 3cntzsnval 19343 . . . . . 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 5016 . . . 4 (𝑆𝐵 𝑥𝑆 (𝑍‘{𝑥}) = 𝑥𝑆 {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
98ineq2d 4219 . . 3 (𝑆𝐵 → (𝐵 𝑥𝑆 (𝑍‘{𝑥})) = (𝐵 𝑥𝑆 {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)}))
10 riinrab 5083 . . 3 (𝐵 𝑥𝑆 {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)}) = {𝑦𝐵 ∣ ∀𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)}
119, 10eqtrdi 2792 . 2 (𝑆𝐵 → (𝐵 𝑥𝑆 (𝑍‘{𝑥})) = {𝑦𝐵 ∣ ∀𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
124, 11eqtr4d 2779 1 (𝑆𝐵 → (𝑍𝑆) = (𝐵 𝑥𝑆 (𝑍‘{𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  {crab 3435  cin 3949  wss 3950  {csn 4625   ciin 4991  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  Cntzccntz 19334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-cntz 19336
This theorem is referenced by: (None)
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