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Theorem p0val 15798
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 3118 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 p0val.z . . 3  |-  .0.  =  ( 0. `  K )
3 fveq2 5872 . . . . . 6  |-  ( p  =  K  ->  ( glb `  p )  =  ( glb `  K
) )
4 p0val.g . . . . . 6  |-  G  =  ( glb `  K
)
53, 4syl6eqr 2516 . . . . 5  |-  ( p  =  K  ->  ( glb `  p )  =  G )
6 fveq2 5872 . . . . . 6  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
7 p0val.b . . . . . 6  |-  B  =  ( Base `  K
)
86, 7syl6eqr 2516 . . . . 5  |-  ( p  =  K  ->  ( Base `  p )  =  B )
95, 8fveq12d 5878 . . . 4  |-  ( p  =  K  ->  (
( glb `  p
) `  ( Base `  p ) )  =  ( G `  B
) )
10 df-p0 15796 . . . 4  |-  0.  =  ( p  e.  _V  |->  ( ( glb `  p
) `  ( Base `  p ) ) )
11 fvex 5882 . . . 4  |-  ( G `
 B )  e. 
_V
129, 10, 11fvmpt 5956 . . 3  |-  ( K  e.  _V  ->  ( 0. `  K )  =  ( G `  B
) )
132, 12syl5eq 2510 . 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 1395    e. wcel 1819   _Vcvv 3109   ` cfv 5594   Basecbs 14644   glbcglb 15699   0.cp0 15794
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-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  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-p0 15796
This theorem is referenced by:  p0le  15800  clatp0cl  27819  xrsp0  27829  op0cl  35052  atl0cl  35171
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