MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  p0val Structured version   Unicode version

Theorem p0val 15216
Description: Value of poset zero. (Contributed by NM, 12-Oct-2011.)
Hypotheses
Ref Expression
p0val.b  |-  B  =  ( Base `  K
)
p0val.g  |-  G  =  ( glb `  K
)
p0val.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
p0val  |-  ( K  e.  V  ->  .0.  =  ( G `  B ) )

Proof of Theorem p0val
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 elex 2986 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 p0val.z . . 3  |-  .0.  =  ( 0. `  K )
3 fveq2 5696 . . . . . 6  |-  ( p  =  K  ->  ( glb `  p )  =  ( glb `  K
) )
4 p0val.g . . . . . 6  |-  G  =  ( glb `  K
)
53, 4syl6eqr 2493 . . . . 5  |-  ( p  =  K  ->  ( glb `  p )  =  G )
6 fveq2 5696 . . . . . 6  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
7 p0val.b . . . . . 6  |-  B  =  ( Base `  K
)
86, 7syl6eqr 2493 . . . . 5  |-  ( p  =  K  ->  ( Base `  p )  =  B )
95, 8fveq12d 5702 . . . 4  |-  ( p  =  K  ->  (
( glb `  p
) `  ( Base `  p ) )  =  ( G `  B
) )
10 df-p0 15214 . . . 4  |-  0.  =  ( p  e.  _V  |->  ( ( glb `  p
) `  ( Base `  p ) ) )
11 fvex 5706 . . . 4  |-  ( G `
 B )  e. 
_V
129, 10, 11fvmpt 5779 . . 3  |-  ( K  e.  _V  ->  ( 0. `  K )  =  ( G `  B
) )
132, 12syl5eq 2487 . 2  |-  ( K  e.  _V  ->  .0.  =  ( G `  B ) )
141, 13syl 16 1  |-  ( K  e.  V  ->  .0.  =  ( G `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2977   ` cfv 5423   Basecbs 14179   glbcglb 15118   0.cp0 15212
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-p0 15214
This theorem is referenced by:  p0le  15218  clatp0cl  26137  xrsp0  26147  op0cl  32834  atl0cl  32953
  Copyright terms: Public domain W3C validator