Theorem cntzun 32480

Description: The centralizer of a union is the intersection of the centralizers. (Contributed by Thierry Arnoux, 27-Nov-2023.)

Hypotheses

Ref	Expression
cntzun.b	⊢ 𝐵 = (Base‘𝑀)
cntzun.z	⊢ 𝑍 = (Cntz‘𝑀)

Assertion

Ref	Expression
cntzun	⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → (𝑍‘(𝑋 ∪ 𝑌)) = ((𝑍‘𝑋) ∩ (𝑍‘𝑌)))

Proof of Theorem cntzun

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ralunb 4192	. . . . . . 7 ⊢ (∀𝑦 ∈ (𝑋 ∪ 𝑌)(𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ↔ (∀𝑦 ∈ 𝑋 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ∧ ∀𝑦 ∈ 𝑌 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)))
2	1	a1i 11	. . . . . 6 ⊢ (((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ (𝑋 ∪ 𝑌)(𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ↔ (∀𝑦 ∈ 𝑋 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ∧ ∀𝑦 ∈ 𝑌 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
3	2	pm5.32da 577	. . . . 5 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑋 ∪ 𝑌)(𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)) ↔ (𝑥 ∈ 𝐵 ∧ (∀𝑦 ∈ 𝑋 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ∧ ∀𝑦 ∈ 𝑌 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)))))
4		anandi 672	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ (∀𝑦 ∈ 𝑋 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ∧ ∀𝑦 ∈ 𝑌 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))) ↔ ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑋 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)) ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑌 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
5	3, 4	bitrdi 286	. . . 4 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑋 ∪ 𝑌)(𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)) ↔ ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑋 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)) ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑌 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)))))
6		unss 4185	. . . . 5 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) ↔ (𝑋 ∪ 𝑌) ⊆ 𝐵)
7		cntzun.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
8		eqid 2730	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
9		cntzun.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
10	7, 8, 9	elcntz 19229	. . . . 5 ⊢ ((𝑋 ∪ 𝑌) ⊆ 𝐵 → (𝑥 ∈ (𝑍‘(𝑋 ∪ 𝑌)) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑋 ∪ 𝑌)(𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
11	6, 10	sylbi 216	. . . 4 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → (𝑥 ∈ (𝑍‘(𝑋 ∪ 𝑌)) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑋 ∪ 𝑌)(𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
12	7, 8, 9	elcntz 19229	. . . . 5 ⊢ (𝑋 ⊆ 𝐵 → (𝑥 ∈ (𝑍‘𝑋) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑋 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
13	7, 8, 9	elcntz 19229	. . . . 5 ⊢ (𝑌 ⊆ 𝐵 → (𝑥 ∈ (𝑍‘𝑌) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑌 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
14	12, 13	bi2anan9 635	. . . 4 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → ((𝑥 ∈ (𝑍‘𝑋) ∧ 𝑥 ∈ (𝑍‘𝑌)) ↔ ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑋 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)) ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑌 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)))))
15	5, 11, 14	3bitr4d 310	. . 3 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → (𝑥 ∈ (𝑍‘(𝑋 ∪ 𝑌)) ↔ (𝑥 ∈ (𝑍‘𝑋) ∧ 𝑥 ∈ (𝑍‘𝑌))))
16		elin 3965	. . 3 ⊢ (𝑥 ∈ ((𝑍‘𝑋) ∩ (𝑍‘𝑌)) ↔ (𝑥 ∈ (𝑍‘𝑋) ∧ 𝑥 ∈ (𝑍‘𝑌)))
17	15, 16	bitr4di 288	. 2 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → (𝑥 ∈ (𝑍‘(𝑋 ∪ 𝑌)) ↔ 𝑥 ∈ ((𝑍‘𝑋) ∩ (𝑍‘𝑌))))
18	17	eqrdv 2728	1 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → (𝑍‘(𝑋 ∪ 𝑌)) = ((𝑍‘𝑋) ∩ (𝑍‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  ‘cfv 6544  (class class class)co 7413  Basecbs 17150  +_gcplusg 17203  Cntzccntz 19222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7416  df-cntz 19224

This theorem is referenced by: (None)