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Theorem cntrval 19350
Description: Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cntrval.b 𝐵 = (Base‘𝑀)
cntrval.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntrval (𝑍𝐵) = (Cntr‘𝑀)

Proof of Theorem cntrval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . . . 6 (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀))
2 cntrval.z . . . . . 6 𝑍 = (Cntz‘𝑀)
31, 2eqtr4di 2793 . . . . 5 (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍)
4 fveq2 6907 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5 cntrval.b . . . . . 6 𝐵 = (Base‘𝑀)
64, 5eqtr4di 2793 . . . . 5 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
73, 6fveq12d 6914 . . . 4 (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍𝐵))
8 df-cntr 19349 . . . 4 Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
9 fvex 6920 . . . 4 (𝑍𝐵) ∈ V
107, 8, 9fvmpt 7016 . . 3 (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍𝐵))
1110eqcomd 2741 . 2 (𝑀 ∈ V → (𝑍𝐵) = (Cntr‘𝑀))
12 0fv 6951 . . 3 (∅‘𝐵) = ∅
13 fvprc 6899 . . . . 5 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
142, 13eqtrid 2787 . . . 4 𝑀 ∈ V → 𝑍 = ∅)
1514fveq1d 6909 . . 3 𝑀 ∈ V → (𝑍𝐵) = (∅‘𝐵))
16 fvprc 6899 . . 3 𝑀 ∈ V → (Cntr‘𝑀) = ∅)
1712, 15, 163eqtr4a 2801 . 2 𝑀 ∈ V → (𝑍𝐵) = (Cntr‘𝑀))
1811, 17pm2.61i 182 1 (𝑍𝐵) = (Cntr‘𝑀)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  cfv 6563  Basecbs 17245  Cntzccntz 19346  Cntrccntr 19347
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-sep 5302  ax-nul 5312  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-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  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-iota 6516  df-fun 6565  df-fv 6571  df-cntr 19349
This theorem is referenced by:  elcntr  19361  cntrss  19362  cntri  19363  cntrsubgnsg  19374  cntrnsg  19375  oppgcntr  19399  cmnbascntr  19838  cntrcmnd  19875  cntrabl  19876  primefld  20823  rng2idl1cntr  21333  cntrcrng  33056
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