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Theorem cntzrcl 19345
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 4338 . . . 4 ¬ 𝑋 ∈ ∅
2 cntzrcl.z . . . . . . . 8 𝑍 = (Cntz‘𝑀)
3 fvprc 6898 . . . . . . . 8 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
42, 3eqtrid 2789 . . . . . . 7 𝑀 ∈ V → 𝑍 = ∅)
54fveq1d 6908 . . . . . 6 𝑀 ∈ V → (𝑍𝑆) = (∅‘𝑆))
6 0fv 6950 . . . . . 6 (∅‘𝑆) = ∅
75, 6eqtrdi 2793 . . . . 5 𝑀 ∈ V → (𝑍𝑆) = ∅)
87eleq2d 2827 . . . 4 𝑀 ∈ V → (𝑋 ∈ (𝑍𝑆) ↔ 𝑋 ∈ ∅))
91, 8mtbiri 327 . . 3 𝑀 ∈ V → ¬ 𝑋 ∈ (𝑍𝑆))
109con4i 114 . 2 (𝑋 ∈ (𝑍𝑆) → 𝑀 ∈ V)
11 cntzrcl.b . . . . . . . 8 𝐵 = (Base‘𝑀)
12 eqid 2737 . . . . . . . 8 (+g𝑀) = (+g𝑀)
1311, 12, 2cntzfval 19338 . . . . . . 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 5916 . . . . 5 (𝑋 ∈ (𝑍𝑆) → dom 𝑍 = dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}))
16 eqid 2737 . . . . . 6 (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}) = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)})
1716dmmptss 6261 . . . . 5 dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}) ⊆ 𝒫 𝐵
1815, 17eqsstrdi 4028 . . . 4 (𝑋 ∈ (𝑍𝑆) → dom 𝑍 ⊆ 𝒫 𝐵)
19 elfvdm 6943 . . . 4 (𝑋 ∈ (𝑍𝑆) → 𝑆 ∈ dom 𝑍)
2018, 19sseldd 3984 . . 3 (𝑋 ∈ (𝑍𝑆) → 𝑆 ∈ 𝒫 𝐵)
2120elpwid 4609 . 2 (𝑋 ∈ (𝑍𝑆) → 𝑆𝐵)
2210, 21jca 511 1 (𝑋 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951  c0 4333  𝒫 cpw 4600  cmpt 5225  dom cdm 5685  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:  cntzssv  19346  cntzi  19347  resscntz  19351  cntzmhm  19359  oppgcntz  19383
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