MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cntzrcl Structured version   Visualization version   GIF version

Theorem cntzrcl 19358
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 4344 . . . 4 ¬ 𝑋 ∈ ∅
2 cntzrcl.z . . . . . . . 8 𝑍 = (Cntz‘𝑀)
3 fvprc 6899 . . . . . . . 8 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
42, 3eqtrid 2787 . . . . . . 7 𝑀 ∈ V → 𝑍 = ∅)
54fveq1d 6909 . . . . . 6 𝑀 ∈ V → (𝑍𝑆) = (∅‘𝑆))
6 0fv 6951 . . . . . 6 (∅‘𝑆) = ∅
75, 6eqtrdi 2791 . . . . 5 𝑀 ∈ V → (𝑍𝑆) = ∅)
87eleq2d 2825 . . . 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 2735 . . . . . . . 8 (+g𝑀) = (+g𝑀)
1311, 12, 2cntzfval 19351 . . . . . . 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 5919 . . . . 5 (𝑋 ∈ (𝑍𝑆) → dom 𝑍 = dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}))
16 eqid 2735 . . . . . 6 (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}) = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)})
1716dmmptss 6263 . . . . 5 dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦𝐵 ∣ ∀𝑧𝑥 (𝑦(+g𝑀)𝑧) = (𝑧(+g𝑀)𝑦)}) ⊆ 𝒫 𝐵
1815, 17eqsstrdi 4050 . . . 4 (𝑋 ∈ (𝑍𝑆) → dom 𝑍 ⊆ 𝒫 𝐵)
19 elfvdm 6944 . . . 4 (𝑋 ∈ (𝑍𝑆) → 𝑆 ∈ dom 𝑍)
2018, 19sseldd 3996 . . 3 (𝑋 ∈ (𝑍𝑆) → 𝑆 ∈ 𝒫 𝐵)
2120elpwid 4614 . 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 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605  cmpt 5231  dom cdm 5689  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Cntzccntz 19346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-cntz 19348
This theorem is referenced by:  cntzssv  19359  cntzi  19360  resscntz  19364  cntzmhm  19372  oppgcntz  19398
  Copyright terms: Public domain W3C validator