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Theorem cntrval 14795
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  |-  ( Z `
 B )  =  (Cntr `  M )

Proof of Theorem cntrval
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . . 6  |-  ( m  =  M  ->  (Cntz `  m )  =  (Cntz `  M ) )
2 cntrval.z . . . . . 6  |-  Z  =  (Cntz `  M )
31, 2syl6eqr 2333 . . . . 5  |-  ( m  =  M  ->  (Cntz `  m )  =  Z )
4 fveq2 5525 . . . . . 6  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
5 cntrval.b . . . . . 6  |-  B  =  ( Base `  M
)
64, 5syl6eqr 2333 . . . . 5  |-  ( m  =  M  ->  ( Base `  m )  =  B )
73, 6fveq12d 5531 . . . 4  |-  ( m  =  M  ->  (
(Cntz `  m ) `  ( Base `  m
) )  =  ( Z `  B ) )
8 df-cntr 14794 . . . 4  |- Cntr  =  ( m  e.  _V  |->  ( (Cntz `  m ) `  ( Base `  m
) ) )
9 fvex 5539 . . . 4  |-  ( Z `
 B )  e. 
_V
107, 8, 9fvmpt 5602 . . 3  |-  ( M  e.  _V  ->  (Cntr `  M )  =  ( Z `  B ) )
1110eqcomd 2288 . 2  |-  ( M  e.  _V  ->  ( Z `  B )  =  (Cntr `  M )
)
12 fv01 5559 . . 3  |-  ( (/) `  B )  =  (/)
13 fvprc 5519 . . . . 5  |-  ( -.  M  e.  _V  ->  (Cntz `  M )  =  (/) )
142, 13syl5eq 2327 . . . 4  |-  ( -.  M  e.  _V  ->  Z  =  (/) )
1514fveq1d 5527 . . 3  |-  ( -.  M  e.  _V  ->  ( Z `  B )  =  ( (/) `  B
) )
16 fvprc 5519 . . 3  |-  ( -.  M  e.  _V  ->  (Cntr `  M )  =  (/) )
1712, 15, 163eqtr4a 2341 . 2  |-  ( -.  M  e.  _V  ->  ( Z `  B )  =  (Cntr `  M
) )
1811, 17pm2.61i 156 1  |-  ( Z `
 B )  =  (Cntr `  M )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ` cfv 5255   Basecbs 13148  Cntzccntz 14791  Cntrccntr 14792
This theorem is referenced by:  cntri  14806  cntrsubgnsg  14816  cntrnsg  14817  oppgcntr  14838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-cntr 14794
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