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Theorem sscntz 19314
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 19309 . . . 4 (𝑇𝐵 → (𝑍𝑇) = {𝑥𝐵 ∣ ∀𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
54sseq2d 4012 . . 3 (𝑇𝐵 → (𝑆 ⊆ (𝑍𝑇) ↔ 𝑆 ⊆ {𝑥𝐵 ∣ ∀𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6 ssrab 4067 . . 3 (𝑆 ⊆ {𝑥𝐵 ∣ ∀𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
75, 6bitrdi 286 . 2 (𝑇𝐵 → (𝑆 ⊆ (𝑍𝑇) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))))
8 ibar 527 . . 3 (𝑆𝐵 → (∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))))
98bicomd 222 . 2 (𝑆𝐵 → ((𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
107, 9sylan9bbr 509 1 ((𝑆𝐵𝑇𝐵) → (𝑆 ⊆ (𝑍𝑇) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wral 3051  {crab 3420  wss 3947  cfv 6544  (class class class)co 7414  Basecbs 17206  +gcplusg 17259  Cntzccntz 19303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7417  df-cntz 19305
This theorem is referenced by:  cntz2ss  19323  cntzrec  19324  submcmn2  19831  mplcoe5lem  22040  symgcntz  32965
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