Theorem cntziinsn 18856

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	⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (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.

Step	Hyp	Ref	Expression
1		cntzrec.b	. . 3 ⊢ 𝐵 = (Base‘𝑀)
2		eqid 2738	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		cntzrec.z	. . 3 ⊢ 𝑍 = (Cntz‘𝑀)
4	1, 2, 3	cntzval 18842	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
5		ssel2 3912	. . . . . 6 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
6	1, 2, 3	cntzsnval 18845	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (𝑍‘{𝑥}) = {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
7	5, 6	syl 17	. . . . 5 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝑆) → (𝑍‘{𝑥}) = {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
8	7	iineq2dv 4946	. . . 4 ⊢ (𝑆 ⊆ 𝐵 → ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥}) = ∩ 𝑥 ∈ 𝑆 {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
9	8	ineq2d 4143	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)}))
10		riinrab 5009	. . 3 ⊢ (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)}) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)}
11	9, 10	eqtrdi 2795	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
12	4, 11	eqtr4d 2781	1 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067   ∩ cin 3882   ⊆ wss 3883  {csn 4558  ∩ ciin 4922  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Cntzccntz 18836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-cntz 18838

This theorem is referenced by: (None)