Theorem cntziinsn 19465

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 2762	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		cntzrec.z	. . 3 ⊢ 𝑍 = (Cntz‘𝑀)
4	1, 2, 3	cntzval 19449	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
5		ssel2 3929	. . . . . 6 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
6	1, 2, 3	cntzsnval 19452	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (𝑍‘{𝑥}) = {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
7	5, 6	syl 18	. . . . 5 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝑆) → (𝑍‘{𝑥}) = {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
8	7	iineq2dv 4980	. . . 4 ⊢ (𝑆 ⊆ 𝐵 → ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥}) = ∩ 𝑥 ∈ 𝑆 {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
9	8	ineq2d 4169	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)}))
10		riinrab 5048	. . 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 2813	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
12	4, 11	eqtr4d 2800	1 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414   ∩ cin 3901   ⊆ wss 3902  {csn 4587  ∩ ciin 4955  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  +_gcplusg 17346  Cntzccntz 19443

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-cntz 19445

This theorem is used by: (None)