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Theorem cntzssv 19346
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 4400 . . 3 ∅ ⊆ 𝐵
2 sseq1 4009 . . 3 ((𝑍𝑆) = ∅ → ((𝑍𝑆) ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
31, 2mpbiri 258 . 2 ((𝑍𝑆) = ∅ → (𝑍𝑆) ⊆ 𝐵)
4 n0 4353 . . 3 ((𝑍𝑆) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑍𝑆))
5 cntzrcl.b . . . . . . 7 𝐵 = (Base‘𝑀)
6 cntzrcl.z . . . . . . 7 𝑍 = (Cntz‘𝑀)
75, 6cntzrcl 19345 . . . . . 6 (𝑥 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆𝐵))
8 eqid 2737 . . . . . . 7 (+g𝑀) = (+g𝑀)
95, 8, 6cntzval 19339 . . . . . 6 (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)})
107, 9simpl2im 503 . . . . 5 (𝑥 ∈ (𝑍𝑆) → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)})
11 ssrab2 4080 . . . . 5 {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)} ⊆ 𝐵
1210, 11eqsstrdi 4028 . . . 4 (𝑥 ∈ (𝑍𝑆) → (𝑍𝑆) ⊆ 𝐵)
1312exlimiv 1930 . . 3 (∃𝑥 𝑥 ∈ (𝑍𝑆) → (𝑍𝑆) ⊆ 𝐵)
144, 13sylbi 217 . 2 ((𝑍𝑆) ≠ ∅ → (𝑍𝑆) ⊆ 𝐵)
153, 14pm2.61ine 3025 1 (𝑍𝑆) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  {crab 3436  Vcvv 3480  wss 3951  c0 4333  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Cntzccntz 19333
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-cntz 19335
This theorem is referenced by:  cntrss  19349  cntzsgrpcl  19352  cntz2ss  19353  cntzsubm  19356  cntzsubg  19357  cntzidss  19358  cntzmhm  19359  cntzmhm2  19360  cntzcmn  19858  cntzspan  19862  cntzsubrng  20567  cntzsubr  20606  cntzsdrg  20803
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