Theorem cntzun 33016

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 4168	. . . . . . 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 676	. . . . 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 287	. . . 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 4161	. . . . 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 19260	. . . . 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 217	. . . 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 19260	. . . . 5 ⊢ (𝑋 ⊆ 𝐵 → (𝑥 ∈ (𝑍‘𝑋) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑋 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
13	7, 8, 9	elcntz 19260	. . . . 5 ⊢ (𝑌 ⊆ 𝐵 → (𝑥 ∈ (𝑍‘𝑌) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑌 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
14	12, 13	bi2anan9 638	. . . 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 311	. . 3 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → (𝑥 ∈ (𝑍‘(𝑋 ∪ 𝑌)) ↔ (𝑥 ∈ (𝑍‘𝑋) ∧ 𝑥 ∈ (𝑍‘𝑌))))
16		elin 3938	. . 3 ⊢ (𝑥 ∈ ((𝑍‘𝑋) ∩ (𝑍‘𝑌)) ↔ (𝑥 ∈ (𝑍‘𝑋) ∧ 𝑥 ∈ (𝑍‘𝑌)))
17	15, 16	bitr4di 289	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3046   ∪ cun 3920   ∩ cin 3921   ⊆ wss 3922  ‘cfv 6519  (class class class)co 7394  Basecbs 17185  +_gcplusg 17226  Cntzccntz 19253

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-ov 7397  df-cntz 19255

This theorem is referenced by: (None)