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Theorem cntrval 17673
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 6148 . . . . . 6 (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀))
2 cntrval.z . . . . . 6 𝑍 = (Cntz‘𝑀)
31, 2syl6eqr 2673 . . . . 5 (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍)
4 fveq2 6148 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5 cntrval.b . . . . . 6 𝐵 = (Base‘𝑀)
64, 5syl6eqr 2673 . . . . 5 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
73, 6fveq12d 6154 . . . 4 (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍𝐵))
8 df-cntr 17672 . . . 4 Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
9 fvex 6158 . . . 4 (𝑍𝐵) ∈ V
107, 8, 9fvmpt 6239 . . 3 (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍𝐵))
1110eqcomd 2627 . 2 (𝑀 ∈ V → (𝑍𝐵) = (Cntr‘𝑀))
12 0fv 6184 . . 3 (∅‘𝐵) = ∅
13 fvprc 6142 . . . . 5 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
142, 13syl5eq 2667 . . . 4 𝑀 ∈ V → 𝑍 = ∅)
1514fveq1d 6150 . . 3 𝑀 ∈ V → (𝑍𝐵) = (∅‘𝐵))
16 fvprc 6142 . . 3 𝑀 ∈ V → (Cntr‘𝑀) = ∅)
1712, 15, 163eqtr4a 2681 . 2 𝑀 ∈ V → (𝑍𝐵) = (Cntr‘𝑀))
1811, 17pm2.61i 176 1 (𝑍𝐵) = (Cntr‘𝑀)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3186  c0 3891  cfv 5847  Basecbs 15781  Cntzccntz 17669  Cntrccntr 17670
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-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
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-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-cntr 17672
This theorem is referenced by:  cntri  17684  cntrsubgnsg  17694  cntrnsg  17695  oppgcntr  17716
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