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Theorem cntzssv 17682
Description: The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Hypotheses
Ref Expression
cntzrcl.b 𝐵 = (Base‘𝑀)
cntzrcl.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntzssv (𝑍𝑆) ⊆ 𝐵

Proof of Theorem cntzssv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 3944 . . 3 ∅ ⊆ 𝐵
2 sseq1 3605 . . 3 ((𝑍𝑆) = ∅ → ((𝑍𝑆) ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
31, 2mpbiri 248 . 2 ((𝑍𝑆) = ∅ → (𝑍𝑆) ⊆ 𝐵)
4 n0 3907 . . 3 ((𝑍𝑆) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑍𝑆))
5 cntzrcl.b . . . . . . . 8 𝐵 = (Base‘𝑀)
6 cntzrcl.z . . . . . . . 8 𝑍 = (Cntz‘𝑀)
75, 6cntzrcl 17681 . . . . . . 7 (𝑥 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆𝐵))
87simprd 479 . . . . . 6 (𝑥 ∈ (𝑍𝑆) → 𝑆𝐵)
9 eqid 2621 . . . . . . 7 (+g𝑀) = (+g𝑀)
105, 9, 6cntzval 17675 . . . . . 6 (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)})
118, 10syl 17 . . . . 5 (𝑥 ∈ (𝑍𝑆) → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)})
12 ssrab2 3666 . . . . 5 {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)} ⊆ 𝐵
1311, 12syl6eqss 3634 . . . 4 (𝑥 ∈ (𝑍𝑆) → (𝑍𝑆) ⊆ 𝐵)
1413exlimiv 1855 . . 3 (∃𝑥 𝑥 ∈ (𝑍𝑆) → (𝑍𝑆) ⊆ 𝐵)
154, 14sylbi 207 . 2 ((𝑍𝑆) ≠ ∅ → (𝑍𝑆) ⊆ 𝐵)
163, 15pm2.61ine 2873 1 (𝑍𝑆) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  {crab 2911  Vcvv 3186  wss 3555  c0 3891  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Cntzccntz 17669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-cntz 17671
This theorem is referenced by:  cntz2ss  17686  cntzsubm  17689  cntzsubg  17690  cntzidss  17691  cntzmhm  17692  cntzmhm2  17693  cntzcmn  18166  cntzspan  18168  cntzsubr  18733  cntzsdrg  37253
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