Theorem cntzun 31902

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 4151	. . . . . . 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 579	. . . . 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 674	. . . . 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 4144	. . . . 5 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) ↔ (𝑋 ∪ 𝑌) ⊆ 𝐵)
7		cntzun.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
8		eqid 2736	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
9		cntzun.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
10	7, 8, 9	elcntz 19102	. . . . 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 19102	. . . . 5 ⊢ (𝑋 ⊆ 𝐵 → (𝑥 ∈ (𝑍‘𝑋) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑋 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
13	7, 8, 9	elcntz 19102	. . . . 5 ⊢ (𝑌 ⊆ 𝐵 → (𝑥 ∈ (𝑍‘𝑌) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑌 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
14	12, 13	bi2anan9 637	. . . 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 3926	. . 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 2734	1 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → (𝑍‘(𝑋 ∪ 𝑌)) = ((𝑍‘𝑋) ∩ (𝑍‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  Cntzccntz 19095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-cntz 19097

This theorem is referenced by: (None)