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Theorem cntzrcl 17692
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 3900 . . . 4 ¬ 𝑋 ∈ ∅
2 cntzrcl.z . . . . . . . 8 𝑍 = (Cntz‘𝑀)
3 fvprc 6147 . . . . . . . 8 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
42, 3syl5eq 2667 . . . . . . 7 𝑀 ∈ V → 𝑍 = ∅)
54fveq1d 6155 . . . . . 6 𝑀 ∈ V → (𝑍𝑆) = (∅‘𝑆))
6 0fv 6189 . . . . . 6 (∅‘𝑆) = ∅
75, 6syl6eq 2671 . . . . 5 𝑀 ∈ V → (𝑍𝑆) = ∅)
87eleq2d 2684 . . . 4 𝑀 ∈ V → (𝑋 ∈ (𝑍𝑆) ↔ 𝑋 ∈ ∅))
91, 8mtbiri 317 . . 3 𝑀 ∈ V → ¬ 𝑋 ∈ (𝑍𝑆))
109con4i 113 . 2 (𝑋 ∈ (𝑍𝑆) → 𝑀 ∈ V)
11 cntzrcl.b . . . . . . . 8 𝐵 = (Base‘𝑀)
12 eqid 2621 . . . . . . . 8 (+g𝑀) = (+g𝑀)
1311, 12, 2cntzfval 17685 . . . . . . 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 5291 . . . . 5 (𝑋 ∈ (𝑍𝑆) → dom 𝑍 = dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}))
16 eqid 2621 . . . . . 6 (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}) = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)})
1716dmmptss 5595 . . . . 5 dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}) ⊆ 𝒫 𝐵
1815, 17syl6eqss 3639 . . . 4 (𝑋 ∈ (𝑍𝑆) → dom 𝑍 ⊆ 𝒫 𝐵)
19 elfvdm 6182 . . . 4 (𝑋 ∈ (𝑍𝑆) → 𝑆 ∈ dom 𝑍)
2018, 19sseldd 3588 . . 3 (𝑋 ∈ (𝑍𝑆) → 𝑆 ∈ 𝒫 𝐵)
2120elpwid 4146 . 2 (𝑋 ∈ (𝑍𝑆) → 𝑆𝐵)
2210, 21jca 554 1 (𝑋 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3189  wss 3559  c0 3896  𝒫 cpw 4135  cmpt 4678  dom cdm 5079  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  Cntzccntz 17680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-cntz 17682
This theorem is referenced by:  cntzssv  17693  cntzi  17694  resscntz  17696  cntzmhm  17703  oppgcntz  17726
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