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Theorem cntrval 17944
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 6344 . . . . . 6 (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀))
2 cntrval.z . . . . . 6 𝑍 = (Cntz‘𝑀)
31, 2syl6eqr 2804 . . . . 5 (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍)
4 fveq2 6344 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5 cntrval.b . . . . . 6 𝐵 = (Base‘𝑀)
64, 5syl6eqr 2804 . . . . 5 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
73, 6fveq12d 6350 . . . 4 (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍𝐵))
8 df-cntr 17943 . . . 4 Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
9 fvex 6354 . . . 4 (𝑍𝐵) ∈ V
107, 8, 9fvmpt 6436 . . 3 (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍𝐵))
1110eqcomd 2758 . 2 (𝑀 ∈ V → (𝑍𝐵) = (Cntr‘𝑀))
12 0fv 6380 . . 3 (∅‘𝐵) = ∅
13 fvprc 6338 . . . . 5 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
142, 13syl5eq 2798 . . . 4 𝑀 ∈ V → 𝑍 = ∅)
1514fveq1d 6346 . . 3 𝑀 ∈ V → (𝑍𝐵) = (∅‘𝐵))
16 fvprc 6338 . . 3 𝑀 ∈ V → (Cntr‘𝑀) = ∅)
1712, 15, 163eqtr4a 2812 . 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 1624  wcel 2131  Vcvv 3332  c0 4050  cfv 6041  Basecbs 16051  Cntzccntz 17940  Cntrccntr 17941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-cntr 17943
This theorem is referenced by:  cntri  17955  cntrsubgnsg  17965  cntrnsg  17966  oppgcntr  17987
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