Theorem cntzun 33164

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 4129	. . . . . . 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 585	. . . . 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 683	. . . . 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 289	. . . 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 4122	. . . . 5 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) ↔ (𝑋 ∪ 𝑌) ⊆ 𝐵)
7		cntzun.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
8		eqid 2741	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
9		cntzun.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
10	7, 8, 9	elcntz 19292	. . . . 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 219	. . . 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 19292	. . . . 5 ⊢ (𝑋 ⊆ 𝐵 → (𝑥 ∈ (𝑍‘𝑋) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑋 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
13	7, 8, 9	elcntz 19292	. . . . 5 ⊢ (𝑌 ⊆ 𝐵 → (𝑥 ∈ (𝑍‘𝑌) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑌 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))))
14	12, 13	bi2anan9 645	. . . 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 313	. . 3 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → (𝑥 ∈ (𝑍‘(𝑋 ∪ 𝑌)) ↔ (𝑥 ∈ (𝑍‘𝑋) ∧ 𝑥 ∈ (𝑍‘𝑌))))
16		elin 3901	. . 3 ⊢ (𝑥 ∈ ((𝑍‘𝑋) ∩ (𝑍‘𝑌)) ↔ (𝑥 ∈ (𝑍‘𝑋) ∧ 𝑥 ∈ (𝑍‘𝑌)))
17	15, 16	bitr4di 291	. 2 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → (𝑥 ∈ (𝑍‘(𝑋 ∪ 𝑌)) ↔ 𝑥 ∈ ((𝑍‘𝑋) ∩ (𝑍‘𝑌))))
18	17	eqrdv 2739	1 ⊢ ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → (𝑍‘(𝑋 ∪ 𝑌)) = ((𝑍‘𝑋) ∩ (𝑍‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055   ∪ cun 3883   ∩ cin 3884   ⊆ wss 3885  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  +_gcplusg 17215  Cntzccntz 19285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-cntz 19287

This theorem is referenced by: (None)