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Theorem cntrval 15118
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 5728 . . . . . 6  |-  ( m  =  M  ->  (Cntz `  m )  =  (Cntz `  M ) )
2 cntrval.z . . . . . 6  |-  Z  =  (Cntz `  M )
31, 2syl6eqr 2486 . . . . 5  |-  ( m  =  M  ->  (Cntz `  m )  =  Z )
4 fveq2 5728 . . . . . 6  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
5 cntrval.b . . . . . 6  |-  B  =  ( Base `  M
)
64, 5syl6eqr 2486 . . . . 5  |-  ( m  =  M  ->  ( Base `  m )  =  B )
73, 6fveq12d 5734 . . . 4  |-  ( m  =  M  ->  (
(Cntz `  m ) `  ( Base `  m
) )  =  ( Z `  B ) )
8 df-cntr 15117 . . . 4  |- Cntr  =  ( m  e.  _V  |->  ( (Cntz `  m ) `  ( Base `  m
) ) )
9 fvex 5742 . . . 4  |-  ( Z `
 B )  e. 
_V
107, 8, 9fvmpt 5806 . . 3  |-  ( M  e.  _V  ->  (Cntr `  M )  =  ( Z `  B ) )
1110eqcomd 2441 . 2  |-  ( M  e.  _V  ->  ( Z `  B )  =  (Cntr `  M )
)
12 fv01 5763 . . 3  |-  ( (/) `  B )  =  (/)
13 fvprc 5722 . . . . 5  |-  ( -.  M  e.  _V  ->  (Cntz `  M )  =  (/) )
142, 13syl5eq 2480 . . . 4  |-  ( -.  M  e.  _V  ->  Z  =  (/) )
1514fveq1d 5730 . . 3  |-  ( -.  M  e.  _V  ->  ( Z `  B )  =  ( (/) `  B
) )
16 fvprc 5722 . . 3  |-  ( -.  M  e.  _V  ->  (Cntr `  M )  =  (/) )
1712, 15, 163eqtr4a 2494 . 2  |-  ( -.  M  e.  _V  ->  ( Z `  B )  =  (Cntr `  M
) )
1811, 17pm2.61i 158 1  |-  ( Z `
 B )  =  (Cntr `  M )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   ` cfv 5454   Basecbs 13469  Cntzccntz 15114  Cntrccntr 15115
This theorem is referenced by:  cntri  15129  cntrsubgnsg  15139  cntrnsg  15140  oppgcntr  15161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-cntr 15117
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