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Theorem cntrval 18441
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 6663 . . . . . 6 (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀))
2 cntrval.z . . . . . 6 𝑍 = (Cntz‘𝑀)
31, 2syl6eqr 2872 . . . . 5 (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍)
4 fveq2 6663 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5 cntrval.b . . . . . 6 𝐵 = (Base‘𝑀)
64, 5syl6eqr 2872 . . . . 5 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
73, 6fveq12d 6670 . . . 4 (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍𝐵))
8 df-cntr 18440 . . . 4 Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
9 fvex 6676 . . . 4 (𝑍𝐵) ∈ V
107, 8, 9fvmpt 6761 . . 3 (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍𝐵))
1110eqcomd 2825 . 2 (𝑀 ∈ V → (𝑍𝐵) = (Cntr‘𝑀))
12 0fv 6702 . . 3 (∅‘𝐵) = ∅
13 fvprc 6656 . . . . 5 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
142, 13syl5eq 2866 . . . 4 𝑀 ∈ V → 𝑍 = ∅)
1514fveq1d 6665 . . 3 𝑀 ∈ V → (𝑍𝐵) = (∅‘𝐵))
16 fvprc 6656 . . 3 𝑀 ∈ V → (Cntr‘𝑀) = ∅)
1712, 15, 163eqtr4a 2880 . 2 𝑀 ∈ V → (𝑍𝐵) = (Cntr‘𝑀))
1811, 17pm2.61i 184 1 (𝑍𝐵) = (Cntr‘𝑀)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2107  Vcvv 3493  c0 4289  cfv 6348  Basecbs 16475  Cntzccntz 18437  Cntrccntr 18438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-cntr 18440
This theorem is referenced by:  cntrss  18452  cntri  18453  cntrsubgnsg  18463  cntrnsg  18464  oppgcntr  18485  cntrcmnd  18954  cntrabl  18955  primefld  19576  cntrcrng  30690
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