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Theorem p0val 16238
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 3096 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 p0val.z . . 3  |-  .0.  =  ( 0. `  K )
3 fveq2 5881 . . . . . 6  |-  ( p  =  K  ->  ( glb `  p )  =  ( glb `  K
) )
4 p0val.g . . . . . 6  |-  G  =  ( glb `  K
)
53, 4syl6eqr 2488 . . . . 5  |-  ( p  =  K  ->  ( glb `  p )  =  G )
6 fveq2 5881 . . . . . 6  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
7 p0val.b . . . . . 6  |-  B  =  ( Base `  K
)
86, 7syl6eqr 2488 . . . . 5  |-  ( p  =  K  ->  ( Base `  p )  =  B )
95, 8fveq12d 5887 . . . 4  |-  ( p  =  K  ->  (
( glb `  p
) `  ( Base `  p ) )  =  ( G `  B
) )
10 df-p0 16236 . . . 4  |-  0.  =  ( p  e.  _V  |->  ( ( glb `  p
) `  ( Base `  p ) ) )
11 fvex 5891 . . . 4  |-  ( G `
 B )  e. 
_V
129, 10, 11fvmpt 5964 . . 3  |-  ( K  e.  _V  ->  ( 0. `  K )  =  ( G `  B
) )
132, 12syl5eq 2482 . 2  |-  ( K  e.  _V  ->  .0.  =  ( G `  B ) )
141, 13syl 17 1  |-  ( K  e.  V  ->  .0.  =  ( G `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   _Vcvv 3087   ` cfv 5601   Basecbs 15084   glbcglb 16139   0.cp0 16234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-p0 16236
This theorem is referenced by:  p0le  16240  clatp0cl  28270  xrsp0  28280  op0cl  32458  atl0cl  32577
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