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Theorem cntziinsn 18456
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 2822 . . 3 (+g𝑀) = (+g𝑀)
3 cntzrec.z . . 3 𝑍 = (Cntz‘𝑀)
41, 2, 3cntzval 18442 . 2 (𝑆𝐵 → (𝑍𝑆) = {𝑦𝐵 ∣ ∀𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
5 ssel2 3937 . . . . . 6 ((𝑆𝐵𝑥𝑆) → 𝑥𝐵)
61, 2, 3cntzsnval 18445 . . . . . 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 4919 . . . 4 (𝑆𝐵 𝑥𝑆 (𝑍‘{𝑥}) = 𝑥𝑆 {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
98ineq2d 4163 . . 3 (𝑆𝐵 → (𝐵 𝑥𝑆 (𝑍‘{𝑥})) = (𝐵 𝑥𝑆 {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)}))
10 riinrab 4981 . . 3 (𝐵 𝑥𝑆 {𝑦𝐵 ∣ (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)}) = {𝑦𝐵 ∣ ∀𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)}
119, 10syl6eq 2873 . 2 (𝑆𝐵 → (𝐵 𝑥𝑆 (𝑍‘{𝑥})) = {𝑦𝐵 ∣ ∀𝑥𝑆 (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦)})
124, 11eqtr4d 2860 1 (𝑆𝐵 → (𝑍𝑆) = (𝐵 𝑥𝑆 (𝑍‘{𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  {crab 3134  cin 3907  wss 3908  {csn 4539   ciin 4895  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  Cntzccntz 18436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-cntz 18438
This theorem is referenced by: (None)
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