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Theorem cntzun 31044
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 ((𝑋𝐵𝑌𝐵) → (𝑍‘(𝑋𝑌)) = ((𝑍𝑋) ∩ (𝑍𝑌)))

Proof of Theorem cntzun
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ralunb 4110 . . . . . . 7 (∀𝑦 ∈ (𝑋𝑌)(𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ↔ (∀𝑦𝑋 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ∧ ∀𝑦𝑌 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
21a1i 11 . . . . . 6 (((𝑋𝐵𝑌𝐵) ∧ 𝑥𝐵) → (∀𝑦 ∈ (𝑋𝑌)(𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ↔ (∀𝑦𝑋 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ∧ ∀𝑦𝑌 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))))
32pm5.32da 582 . . . . 5 ((𝑋𝐵𝑌𝐵) → ((𝑥𝐵 ∧ ∀𝑦 ∈ (𝑋𝑌)(𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)) ↔ (𝑥𝐵 ∧ (∀𝑦𝑋 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ∧ ∀𝑦𝑌 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))))
4 anandi 676 . . . . 5 ((𝑥𝐵 ∧ (∀𝑦𝑋 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) ∧ ∀𝑦𝑌 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))) ↔ ((𝑥𝐵 ∧ ∀𝑦𝑋 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)) ∧ (𝑥𝐵 ∧ ∀𝑦𝑌 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))))
53, 4bitrdi 290 . . . 4 ((𝑋𝐵𝑌𝐵) → ((𝑥𝐵 ∧ ∀𝑦 ∈ (𝑋𝑌)(𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)) ↔ ((𝑥𝐵 ∧ ∀𝑦𝑋 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)) ∧ (𝑥𝐵 ∧ ∀𝑦𝑌 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))))
6 unss 4103 . . . . 5 ((𝑋𝐵𝑌𝐵) ↔ (𝑋𝑌) ⊆ 𝐵)
7 cntzun.b . . . . . 6 𝐵 = (Base‘𝑀)
8 eqid 2737 . . . . . 6 (+g𝑀) = (+g𝑀)
9 cntzun.z . . . . . 6 𝑍 = (Cntz‘𝑀)
107, 8, 9elcntz 18721 . . . . 5 ((𝑋𝑌) ⊆ 𝐵 → (𝑥 ∈ (𝑍‘(𝑋𝑌)) ↔ (𝑥𝐵 ∧ ∀𝑦 ∈ (𝑋𝑌)(𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))))
116, 10sylbi 220 . . . 4 ((𝑋𝐵𝑌𝐵) → (𝑥 ∈ (𝑍‘(𝑋𝑌)) ↔ (𝑥𝐵 ∧ ∀𝑦 ∈ (𝑋𝑌)(𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))))
127, 8, 9elcntz 18721 . . . . 5 (𝑋𝐵 → (𝑥 ∈ (𝑍𝑋) ↔ (𝑥𝐵 ∧ ∀𝑦𝑋 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))))
137, 8, 9elcntz 18721 . . . . 5 (𝑌𝐵 → (𝑥 ∈ (𝑍𝑌) ↔ (𝑥𝐵 ∧ ∀𝑦𝑌 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))))
1412, 13bi2anan9 639 . . . 4 ((𝑋𝐵𝑌𝐵) → ((𝑥 ∈ (𝑍𝑋) ∧ 𝑥 ∈ (𝑍𝑌)) ↔ ((𝑥𝐵 ∧ ∀𝑦𝑋 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)) ∧ (𝑥𝐵 ∧ ∀𝑦𝑌 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))))
155, 11, 143bitr4d 314 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑥 ∈ (𝑍‘(𝑋𝑌)) ↔ (𝑥 ∈ (𝑍𝑋) ∧ 𝑥 ∈ (𝑍𝑌))))
16 elin 3887 . . 3 (𝑥 ∈ ((𝑍𝑋) ∩ (𝑍𝑌)) ↔ (𝑥 ∈ (𝑍𝑋) ∧ 𝑥 ∈ (𝑍𝑌)))
1715, 16bitr4di 292 . 2 ((𝑋𝐵𝑌𝐵) → (𝑥 ∈ (𝑍‘(𝑋𝑌)) ↔ 𝑥 ∈ ((𝑍𝑋) ∩ (𝑍𝑌))))
1817eqrdv 2735 1 ((𝑋𝐵𝑌𝐵) → (𝑍‘(𝑋𝑌)) = ((𝑍𝑋) ∩ (𝑍𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061  cun 3869  cin 3870  wss 3871  cfv 6385  (class class class)co 7218  Basecbs 16765  +gcplusg 16807  Cntzccntz 18714
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 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5184  ax-sep 5197  ax-nul 5204  ax-pow 5263  ax-pr 5327
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3415  df-sbc 3700  df-csb 3817  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-iun 4911  df-br 5059  df-opab 5121  df-mpt 5141  df-id 5460  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-f1 6390  df-fo 6391  df-f1o 6392  df-fv 6393  df-ov 7221  df-cntz 18716
This theorem is referenced by: (None)
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