Theorem cntzun 33140

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 4148	. . . . . . 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 677	. . . . 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 4141	. . . . 5 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) ↔ (𝑋 ∪ 𝑌) ⊆ 𝐵)
7		cntzun.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
8		eqid 2735	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
9		cntzun.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
10	7, 8, 9	elcntz 19253	. . . . 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 19253	. . . . 5 ⊢ (𝑋 ⊆ 𝐵 → (𝑥 ∈ (𝑍‘𝑋) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑋 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
13	7, 8, 9	elcntz 19253	. . . . 5 ⊢ (𝑌 ⊆ 𝐵 → (𝑥 ∈ (𝑍‘𝑌) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑌 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
14	12, 13	bi2anan9 639	. . . 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 3916	. . 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 2733	1 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → (𝑍‘(𝑋 ∪ 𝑌)) = ((𝑍‘𝑋) ∩ (𝑍‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3050   ∪ cun 3898   ∩ cin 3899   ⊆ wss 3900  ‘cfv 6491  (class class class)co 7358  Basecbs 17138  +_gcplusg 17179  Cntzccntz 19246

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-cntz 19248

This theorem is referenced by: (None)