Theorem cntziinsn 19551

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 2760	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		cntzrec.z	. . 3 ⊢ 𝑍 = (Cntz‘𝑀)
4	1, 2, 3	cntzval 19535	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
5		ssel2 3903	. . . . . 6 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
6	1, 2, 3	cntzsnval 19538	. . . . . 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 4976	. . . 4 ⊢ (𝑆 ⊆ 𝐵 → ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥}) = ∩ 𝑥 ∈ 𝑆 {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
9	8	ineq2d 4117	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 {𝑦 ∈ 𝐵 ∣ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)}))
10		riinrab 5043	. . 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 2811	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)})
12	4, 11	eqtr4d 2798	1 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3895   ∩ cin 4089  {csn 4583  ∩ ciin 4951  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  Cntzccntz 19529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-cntz 19531

This theorem is used by: (None)