Theorem cntziinsn 18465

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 2821	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		cntzrec.z	. . 3 ⊢ 𝑍 = (Cntz‘𝑀)
4	1, 2, 3	cntzval 18451	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
5		ssel2 3962	. . . . . 6 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
6	1, 2, 3	cntzsnval 18454	. . . . . 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 4944	. . . 4 ⊢ (𝑆 ⊆ 𝐵 → ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥}) = ∩ 𝑥 ∈ 𝑆 {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
9	8	ineq2d 4189	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)}))
10		riinrab 5006	. . 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	syl6eq 2872	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
12	4, 11	eqtr4d 2859	1 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  {crab 3142   ∩ cin 3935   ⊆ wss 3936  {csn 4567  ∩ ciin 4920  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  +_gcplusg 16565  Cntzccntz 18445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-cntz 18447

This theorem is referenced by: (None)