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Theorem p1val 15522
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 3122 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 p1val.t . . 3  |-  .1.  =  ( 1. `  K )
3 fveq2 5864 . . . . . 6  |-  ( k  =  K  ->  ( lub `  k )  =  ( lub `  K
) )
4 p1val.u . . . . . 6  |-  U  =  ( lub `  K
)
53, 4syl6eqr 2526 . . . . 5  |-  ( k  =  K  ->  ( lub `  k )  =  U )
6 fveq2 5864 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
7 p1val.b . . . . . 6  |-  B  =  ( Base `  K
)
86, 7syl6eqr 2526 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
95, 8fveq12d 5870 . . . 4  |-  ( k  =  K  ->  (
( lub `  k
) `  ( Base `  k ) )  =  ( U `  B
) )
10 df-p1 15520 . . . 4  |-  1.  =  ( k  e.  _V  |->  ( ( lub `  k
) `  ( Base `  k ) ) )
11 fvex 5874 . . . 4  |-  ( U `
 B )  e. 
_V
129, 10, 11fvmpt 5948 . . 3  |-  ( K  e.  _V  ->  ( 1. `  K )  =  ( U `  B
) )
132, 12syl5eq 2520 . 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 1379    e. wcel 1767   _Vcvv 3113   ` cfv 5586   Basecbs 14483   lubclub 15422   1.cp1 15518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-p1 15520
This theorem is referenced by:  ple1  15524  clatp1cl  27319  xrsp1  27329  op1cl  33982
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