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Theorem sscntz 18459
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 18454 . . . 4 (𝑇𝐵 → (𝑍𝑇) = {𝑥𝐵 ∣ ∀𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
54sseq2d 4002 . . 3 (𝑇𝐵 → (𝑆 ⊆ (𝑍𝑇) ↔ 𝑆 ⊆ {𝑥𝐵 ∣ ∀𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6 ssrab 4052 . . 3 (𝑆 ⊆ {𝑥𝐵 ∣ ∀𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
75, 6syl6bb 289 . 2 (𝑇𝐵 → (𝑆 ⊆ (𝑍𝑇) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))))
8 ibar 531 . . 3 (𝑆𝐵 → (∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))))
98bicomd 225 . 2 (𝑆𝐵 → ((𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
107, 9sylan9bbr 513 1 ((𝑆𝐵𝑇𝐵) → (𝑆 ⊆ (𝑍𝑇) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1536  wral 3141  {crab 3145  wss 3939  cfv 6358  (class class class)co 7159  Basecbs 16486  +gcplusg 16568  Cntzccntz 18448
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-cntz 18450
This theorem is referenced by:  cntz2ss  18466  cntzrec  18467  submcmn2  18962  mplcoe5lem  20251  symgcntz  30733
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