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Theorem cntzrcl 17806
Description: Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Hypotheses
Ref Expression
cntzrcl.b 𝐵 = (Base‘𝑀)
cntzrcl.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntzrcl (𝑋 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆𝐵))

Proof of Theorem cntzrcl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noel 3952 . . . 4 ¬ 𝑋 ∈ ∅
2 cntzrcl.z . . . . . . . 8 𝑍 = (Cntz‘𝑀)
3 fvprc 6223 . . . . . . . 8 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
42, 3syl5eq 2697 . . . . . . 7 𝑀 ∈ V → 𝑍 = ∅)
54fveq1d 6231 . . . . . 6 𝑀 ∈ V → (𝑍𝑆) = (∅‘𝑆))
6 0fv 6265 . . . . . 6 (∅‘𝑆) = ∅
75, 6syl6eq 2701 . . . . 5 𝑀 ∈ V → (𝑍𝑆) = ∅)
87eleq2d 2716 . . . 4 𝑀 ∈ V → (𝑋 ∈ (𝑍𝑆) ↔ 𝑋 ∈ ∅))
91, 8mtbiri 316 . . 3 𝑀 ∈ V → ¬ 𝑋 ∈ (𝑍𝑆))
109con4i 113 . 2 (𝑋 ∈ (𝑍𝑆) → 𝑀 ∈ V)
11 cntzrcl.b . . . . . . . 8 𝐵 = (Base‘𝑀)
12 eqid 2651 . . . . . . . 8 (+g𝑀) = (+g𝑀)
1311, 12, 2cntzfval 17799 . . . . . . 7 (𝑀 ∈ V → 𝑍 = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}))
1410, 13syl 17 . . . . . 6 (𝑋 ∈ (𝑍𝑆) → 𝑍 = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}))
1514dmeqd 5358 . . . . 5 (𝑋 ∈ (𝑍𝑆) → dom 𝑍 = dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}))
16 eqid 2651 . . . . . 6 (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}) = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)})
1716dmmptss 5669 . . . . 5 dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}) ⊆ 𝒫 𝐵
1815, 17syl6eqss 3688 . . . 4 (𝑋 ∈ (𝑍𝑆) → dom 𝑍 ⊆ 𝒫 𝐵)
19 elfvdm 6258 . . . 4 (𝑋 ∈ (𝑍𝑆) → 𝑆 ∈ dom 𝑍)
2018, 19sseldd 3637 . . 3 (𝑋 ∈ (𝑍𝑆) → 𝑆 ∈ 𝒫 𝐵)
2120elpwid 4203 . 2 (𝑋 ∈ (𝑍𝑆) → 𝑆𝐵)
2210, 21jca 553 1 (𝑋 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  wss 3607  c0 3948  𝒫 cpw 4191  cmpt 4762  dom cdm 5143  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Cntzccntz 17794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-cntz 17796
This theorem is referenced by:  cntzssv  17807  cntzi  17808  resscntz  17810  cntzmhm  17817  oppgcntz  17840
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