MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cntrval Structured version   Unicode version

Theorem cntrval 15837
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 5691 . . . . . 6  |-  ( m  =  M  ->  (Cntz `  m )  =  (Cntz `  M ) )
2 cntrval.z . . . . . 6  |-  Z  =  (Cntz `  M )
31, 2syl6eqr 2493 . . . . 5  |-  ( m  =  M  ->  (Cntz `  m )  =  Z )
4 fveq2 5691 . . . . . 6  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
5 cntrval.b . . . . . 6  |-  B  =  ( Base `  M
)
64, 5syl6eqr 2493 . . . . 5  |-  ( m  =  M  ->  ( Base `  m )  =  B )
73, 6fveq12d 5697 . . . 4  |-  ( m  =  M  ->  (
(Cntz `  m ) `  ( Base `  m
) )  =  ( Z `  B ) )
8 df-cntr 15836 . . . 4  |- Cntr  =  ( m  e.  _V  |->  ( (Cntz `  m ) `  ( Base `  m
) ) )
9 fvex 5701 . . . 4  |-  ( Z `
 B )  e. 
_V
107, 8, 9fvmpt 5774 . . 3  |-  ( M  e.  _V  ->  (Cntr `  M )  =  ( Z `  B ) )
1110eqcomd 2448 . 2  |-  ( M  e.  _V  ->  ( Z `  B )  =  (Cntr `  M )
)
12 0fv 5723 . . 3  |-  ( (/) `  B )  =  (/)
13 fvprc 5685 . . . . 5  |-  ( -.  M  e.  _V  ->  (Cntz `  M )  =  (/) )
142, 13syl5eq 2487 . . . 4  |-  ( -.  M  e.  _V  ->  Z  =  (/) )
1514fveq1d 5693 . . 3  |-  ( -.  M  e.  _V  ->  ( Z `  B )  =  ( (/) `  B
) )
16 fvprc 5685 . . 3  |-  ( -.  M  e.  _V  ->  (Cntr `  M )  =  (/) )
1712, 15, 163eqtr4a 2501 . 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 1369    e. wcel 1756   _Vcvv 2972   (/)c0 3637   ` cfv 5418   Basecbs 14174  Cntzccntz 15833  Cntrccntr 15834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-cntr 15836
This theorem is referenced by:  cntri  15848  cntrsubgnsg  15858  cntrnsg  15859  oppgcntr  15880
  Copyright terms: Public domain W3C validator