Theorem cntziinsn 19307

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 2741	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		cntzrec.z	. . 3 ⊢ 𝑍 = (Cntz‘𝑀)
4	1, 2, 3	cntzval 19291	. 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 19294	. . . . . 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 4950	. . . 4 ⊢ (𝑆 ⊆ 𝐵 → ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥}) = ∩ 𝑥 ∈ 𝑆 {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
9	8	ineq2d 4152	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)}))
10		riinrab 5016	. . 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 2792	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
12	4, 11	eqtr4d 2779	1 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  {crab 3393   ∩ cin 3884   ⊆ wss 3885  {csn 4558  ∩ ciin 4925  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  +_gcplusg 17215  Cntzccntz 19285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-cntz 19287

This theorem is referenced by: (None)