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Theorem sscntz 19344
Description: A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cntzfval.b 𝐵 = (Base‘𝑀)
cntzfval.p + = (+g𝑀)
cntzfval.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
sscntz ((𝑆𝐵𝑇𝐵) → (𝑆 ⊆ (𝑍𝑇) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Distinct variable groups:   𝑥,𝑦, +   𝑥,𝐵   𝑥,𝑀,𝑦   𝑥,𝑇,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐵(𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem sscntz
StepHypRef Expression
1 cntzfval.b . . . . 5 𝐵 = (Base‘𝑀)
2 cntzfval.p . . . . 5 + = (+g𝑀)
3 cntzfval.z . . . . 5 𝑍 = (Cntz‘𝑀)
41, 2, 3cntzval 19339 . . . 4 (𝑇𝐵 → (𝑍𝑇) = {𝑥𝐵 ∣ ∀𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
54sseq2d 4016 . . 3 (𝑇𝐵 → (𝑆 ⊆ (𝑍𝑇) ↔ 𝑆 ⊆ {𝑥𝐵 ∣ ∀𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6 ssrab 4073 . . 3 (𝑆 ⊆ {𝑥𝐵 ∣ ∀𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
75, 6bitrdi 287 . 2 (𝑇𝐵 → (𝑆 ⊆ (𝑍𝑇) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))))
8 ibar 528 . . 3 (𝑆𝐵 → (∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))))
98bicomd 223 . 2 (𝑆𝐵 → ((𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
107, 9sylan9bbr 510 1 ((𝑆𝐵𝑇𝐵) → (𝑆 ⊆ (𝑍𝑇) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wral 3061  {crab 3436  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Cntzccntz 19333
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-cntz 19335
This theorem is referenced by:  cntz2ss  19353  cntzrec  19354  submcmn2  19857  mplcoe5lem  22057  symgcntz  33105
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