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Theorem sscntz 17805
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 17800 . . . 4 (𝑇𝐵 → (𝑍𝑇) = {𝑥𝐵 ∣ ∀𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
54sseq2d 3666 . . 3 (𝑇𝐵 → (𝑆 ⊆ (𝑍𝑇) ↔ 𝑆 ⊆ {𝑥𝐵 ∣ ∀𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6 ssrab 3713 . . 3 (𝑆 ⊆ {𝑥𝐵 ∣ ∀𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
75, 6syl6bb 276 . 2 (𝑇𝐵 → (𝑆 ⊆ (𝑍𝑇) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))))
8 ibar 524 . . 3 (𝑆𝐵 → (∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))))
98bicomd 213 . 2 (𝑆𝐵 → ((𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
107, 9sylan9bbr 737 1 ((𝑆𝐵𝑇𝐵) → (𝑆 ⊆ (𝑍𝑇) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wral 2941  {crab 2945  wss 3607  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Cntzccntz 17794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-cntz 17796
This theorem is referenced by:  cntz2ss  17811  cntzrec  17812  submcmn2  18290  mplcoe5lem  19515
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