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Theorem cntzssv 17584
 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 3924 . . 3 ∅ ⊆ 𝐵
2 sseq1 3589 . . 3 ((𝑍𝑆) = ∅ → ((𝑍𝑆) ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
31, 2mpbiri 247 . 2 ((𝑍𝑆) = ∅ → (𝑍𝑆) ⊆ 𝐵)
4 n0 3890 . . 3 ((𝑍𝑆) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑍𝑆))
5 cntzrcl.b . . . . . . . 8 𝐵 = (Base‘𝑀)
6 cntzrcl.z . . . . . . . 8 𝑍 = (Cntz‘𝑀)
75, 6cntzrcl 17583 . . . . . . 7 (𝑥 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆𝐵))
87simprd 478 . . . . . 6 (𝑥 ∈ (𝑍𝑆) → 𝑆𝐵)
9 eqid 2610 . . . . . . 7 (+g𝑀) = (+g𝑀)
105, 9, 6cntzval 17577 . . . . . 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 3650 . . . . 5 {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)} ⊆ 𝐵
1311, 12syl6eqss 3618 . . . 4 (𝑥 ∈ (𝑍𝑆) → (𝑍𝑆) ⊆ 𝐵)
1413exlimiv 1845 . . 3 (∃𝑥 𝑥 ∈ (𝑍𝑆) → (𝑍𝑆) ⊆ 𝐵)
154, 14sylbi 206 . 2 ((𝑍𝑆) ≠ ∅ → (𝑍𝑆) ⊆ 𝐵)
163, 15pm2.61ine 2865 1 (𝑍𝑆) ⊆ 𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Cntzccntz 17571 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-cntz 17573 This theorem is referenced by:  cntz2ss  17588  cntzsubm  17591  cntzsubg  17592  cntzidss  17593  cntzmhm  17594  cntzmhm2  17595  cntzcmn  18068  cntzspan  18070  cntzsubr  18635  cntzsdrg  36791
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