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Theorem cntzssv 19104
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 4355 . . 3 ∅ ⊆ 𝐵
2 sseq1 3968 . . 3 ((𝑍𝑆) = ∅ → ((𝑍𝑆) ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
31, 2mpbiri 257 . 2 ((𝑍𝑆) = ∅ → (𝑍𝑆) ⊆ 𝐵)
4 n0 4305 . . 3 ((𝑍𝑆) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑍𝑆))
5 cntzrcl.b . . . . . . 7 𝐵 = (Base‘𝑀)
6 cntzrcl.z . . . . . . 7 𝑍 = (Cntz‘𝑀)
75, 6cntzrcl 19103 . . . . . 6 (𝑥 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆𝐵))
8 eqid 2736 . . . . . . 7 (+g𝑀) = (+g𝑀)
95, 8, 6cntzval 19097 . . . . . 6 (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)})
107, 9simpl2im 504 . . . . 5 (𝑥 ∈ (𝑍𝑆) → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)})
11 ssrab2 4036 . . . . 5 {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)} ⊆ 𝐵
1210, 11eqsstrdi 3997 . . . 4 (𝑥 ∈ (𝑍𝑆) → (𝑍𝑆) ⊆ 𝐵)
1312exlimiv 1933 . . 3 (∃𝑥 𝑥 ∈ (𝑍𝑆) → (𝑍𝑆) ⊆ 𝐵)
144, 13sylbi 216 . 2 ((𝑍𝑆) ≠ ∅ → (𝑍𝑆) ⊆ 𝐵)
153, 14pm2.61ine 3027 1 (𝑍𝑆) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wex 1781  wcel 2106  wne 2942  wral 3063  {crab 3406  Vcvv 3444  wss 3909  c0 4281  cfv 6494  (class class class)co 7354  Basecbs 17080  +gcplusg 17130  Cntzccntz 19091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7357  df-cntz 19093
This theorem is referenced by:  cntrss  19106  cntz2ss  19109  cntzsubm  19112  cntzsubg  19113  cntzidss  19114  cntzmhm  19115  cntzmhm2  19116  cntzcmn  19614  cntzspan  19618  cntzsubr  20251  cntzsdrg  20265
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