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Theorem cntrval 16484
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 5872 . . . . . 6  |-  ( m  =  M  ->  (Cntz `  m )  =  (Cntz `  M ) )
2 cntrval.z . . . . . 6  |-  Z  =  (Cntz `  M )
31, 2syl6eqr 2516 . . . . 5  |-  ( m  =  M  ->  (Cntz `  m )  =  Z )
4 fveq2 5872 . . . . . 6  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
5 cntrval.b . . . . . 6  |-  B  =  ( Base `  M
)
64, 5syl6eqr 2516 . . . . 5  |-  ( m  =  M  ->  ( Base `  m )  =  B )
73, 6fveq12d 5878 . . . 4  |-  ( m  =  M  ->  (
(Cntz `  m ) `  ( Base `  m
) )  =  ( Z `  B ) )
8 df-cntr 16483 . . . 4  |- Cntr  =  ( m  e.  _V  |->  ( (Cntz `  m ) `  ( Base `  m
) ) )
9 fvex 5882 . . . 4  |-  ( Z `
 B )  e. 
_V
107, 8, 9fvmpt 5956 . . 3  |-  ( M  e.  _V  ->  (Cntr `  M )  =  ( Z `  B ) )
1110eqcomd 2465 . 2  |-  ( M  e.  _V  ->  ( Z `  B )  =  (Cntr `  M )
)
12 0fv 5905 . . 3  |-  ( (/) `  B )  =  (/)
13 fvprc 5866 . . . . 5  |-  ( -.  M  e.  _V  ->  (Cntz `  M )  =  (/) )
142, 13syl5eq 2510 . . . 4  |-  ( -.  M  e.  _V  ->  Z  =  (/) )
1514fveq1d 5874 . . 3  |-  ( -.  M  e.  _V  ->  ( Z `  B )  =  ( (/) `  B
) )
16 fvprc 5866 . . 3  |-  ( -.  M  e.  _V  ->  (Cntr `  M )  =  (/) )
1712, 15, 163eqtr4a 2524 . 2  |-  ( -.  M  e.  _V  ->  ( Z `  B )  =  (Cntr `  M
) )
1811, 17pm2.61i 164 1  |-  ( Z `
 B )  =  (Cntr `  M )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   ` cfv 5594   Basecbs 14644  Cntzccntz 16480  Cntrccntr 16481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-cntr 16483
This theorem is referenced by:  cntri  16495  cntrsubgnsg  16505  cntrnsg  16506  oppgcntr  16527
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