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Theorem cntrval 19338
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 6905 . . . . . 6 (𝑚 = 𝑀 → (Cntz‘𝑚) = (Cntz‘𝑀))
2 cntrval.z . . . . . 6 𝑍 = (Cntz‘𝑀)
31, 2eqtr4di 2794 . . . . 5 (𝑚 = 𝑀 → (Cntz‘𝑚) = 𝑍)
4 fveq2 6905 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5 cntrval.b . . . . . 6 𝐵 = (Base‘𝑀)
64, 5eqtr4di 2794 . . . . 5 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
73, 6fveq12d 6912 . . . 4 (𝑚 = 𝑀 → ((Cntz‘𝑚)‘(Base‘𝑚)) = (𝑍𝐵))
8 df-cntr 19337 . . . 4 Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
9 fvex 6918 . . . 4 (𝑍𝐵) ∈ V
107, 8, 9fvmpt 7015 . . 3 (𝑀 ∈ V → (Cntr‘𝑀) = (𝑍𝐵))
1110eqcomd 2742 . 2 (𝑀 ∈ V → (𝑍𝐵) = (Cntr‘𝑀))
12 0fv 6949 . . 3 (∅‘𝐵) = ∅
13 fvprc 6897 . . . . 5 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
142, 13eqtrid 2788 . . . 4 𝑀 ∈ V → 𝑍 = ∅)
1514fveq1d 6907 . . 3 𝑀 ∈ V → (𝑍𝐵) = (∅‘𝐵))
16 fvprc 6897 . . 3 𝑀 ∈ V → (Cntr‘𝑀) = ∅)
1712, 15, 163eqtr4a 2802 . 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 1539  wcel 2107  Vcvv 3479  c0 4332  cfv 6560  Basecbs 17248  Cntzccntz 19334  Cntrccntr 19335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-cntr 19337
This theorem is referenced by:  elcntr  19349  cntrss  19350  cntri  19351  cntrsubgnsg  19362  cntrnsg  19363  oppgcntr  19385  cmnbascntr  19824  cntrcmnd  19861  cntrabl  19862  primefld  20807  rng2idl1cntr  21316  cntrcrng  33074
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