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Theorem p1val 15871
Description: Value of poset zero. (Contributed by NM, 22-Oct-2011.)
Hypotheses
Ref Expression
p1val.b  |-  B  =  ( Base `  K
)
p1val.u  |-  U  =  ( lub `  K
)
p1val.t  |-  .1.  =  ( 1. `  K )
Assertion
Ref Expression
p1val  |-  ( K  e.  V  ->  .1.  =  ( U `  B ) )

Proof of Theorem p1val
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 3115 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 p1val.t . . 3  |-  .1.  =  ( 1. `  K )
3 fveq2 5848 . . . . . 6  |-  ( k  =  K  ->  ( lub `  k )  =  ( lub `  K
) )
4 p1val.u . . . . . 6  |-  U  =  ( lub `  K
)
53, 4syl6eqr 2513 . . . . 5  |-  ( k  =  K  ->  ( lub `  k )  =  U )
6 fveq2 5848 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
7 p1val.b . . . . . 6  |-  B  =  ( Base `  K
)
86, 7syl6eqr 2513 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
95, 8fveq12d 5854 . . . 4  |-  ( k  =  K  ->  (
( lub `  k
) `  ( Base `  k ) )  =  ( U `  B
) )
10 df-p1 15869 . . . 4  |-  1.  =  ( k  e.  _V  |->  ( ( lub `  k
) `  ( Base `  k ) ) )
11 fvex 5858 . . . 4  |-  ( U `
 B )  e. 
_V
129, 10, 11fvmpt 5931 . . 3  |-  ( K  e.  _V  ->  ( 1. `  K )  =  ( U `  B
) )
132, 12syl5eq 2507 . 2  |-  ( K  e.  _V  ->  .1.  =  ( U `  B ) )
141, 13syl 16 1  |-  ( K  e.  V  ->  .1.  =  ( U `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   _Vcvv 3106   ` cfv 5570   Basecbs 14716   lubclub 15770   1.cp1 15867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-p1 15869
This theorem is referenced by:  ple1  15873  clatp1cl  27894  xrsp1  27904  op1cl  35307
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