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Theorem p0val 14147
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 2796 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 p0val.z . . 3  |-  .0.  =  ( 0. `  K )
3 fveq2 5525 . . . . . 6  |-  ( p  =  K  ->  ( glb `  p )  =  ( glb `  K
) )
4 p0val.g . . . . . 6  |-  G  =  ( glb `  K
)
53, 4syl6eqr 2333 . . . . 5  |-  ( p  =  K  ->  ( glb `  p )  =  G )
6 fveq2 5525 . . . . . 6  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
7 p0val.b . . . . . 6  |-  B  =  ( Base `  K
)
86, 7syl6eqr 2333 . . . . 5  |-  ( p  =  K  ->  ( Base `  p )  =  B )
95, 8fveq12d 5531 . . . 4  |-  ( p  =  K  ->  (
( glb `  p
) `  ( Base `  p ) )  =  ( G `  B
) )
10 df-p0 14145 . . . 4  |-  0.  =  ( p  e.  _V  |->  ( ( glb `  p
) `  ( Base `  p ) ) )
11 fvex 5539 . . . 4  |-  ( G `
 B )  e. 
_V
129, 10, 11fvmpt 5602 . . 3  |-  ( K  e.  _V  ->  ( 0. `  K )  =  ( G `  B
) )
132, 12syl5eq 2327 . 2  |-  ( K  e.  _V  ->  .0.  =  ( G `  B ) )
141, 13syl 15 1  |-  ( K  e.  V  ->  .0.  =  ( G `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   ` cfv 5255   Basecbs 13148   glbcglb 14077   0.cp0 14143
This theorem is referenced by:  p0le  14149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-p0 14145
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