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Theorem p0val 16336
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 3066 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 p0val.z . . 3  |-  .0.  =  ( 0. `  K )
3 fveq2 5888 . . . . . 6  |-  ( p  =  K  ->  ( glb `  p )  =  ( glb `  K
) )
4 p0val.g . . . . . 6  |-  G  =  ( glb `  K
)
53, 4syl6eqr 2514 . . . . 5  |-  ( p  =  K  ->  ( glb `  p )  =  G )
6 fveq2 5888 . . . . . 6  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
7 p0val.b . . . . . 6  |-  B  =  ( Base `  K
)
86, 7syl6eqr 2514 . . . . 5  |-  ( p  =  K  ->  ( Base `  p )  =  B )
95, 8fveq12d 5894 . . . 4  |-  ( p  =  K  ->  (
( glb `  p
) `  ( Base `  p ) )  =  ( G `  B
) )
10 df-p0 16334 . . . 4  |-  0.  =  ( p  e.  _V  |->  ( ( glb `  p
) `  ( Base `  p ) ) )
11 fvex 5898 . . . 4  |-  ( G `
 B )  e. 
_V
129, 10, 11fvmpt 5971 . . 3  |-  ( K  e.  _V  ->  ( 0. `  K )  =  ( G `  B
) )
132, 12syl5eq 2508 . 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 1455    e. wcel 1898   _Vcvv 3057   ` cfv 5601   Basecbs 15170   glbcglb 16237   0.cp0 16332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-p0 16334
This theorem is referenced by:  p0le  16338  clatp0cl  28482  xrsp0  28492  op0cl  32795  atl0cl  32914
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