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Theorem p1val 14472
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 2965 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 p1val.t . . 3  |-  .1.  =  ( 1. `  K )
3 fveq2 5729 . . . . . 6  |-  ( k  =  K  ->  ( lub `  k )  =  ( lub `  K
) )
4 p1val.u . . . . . 6  |-  U  =  ( lub `  K
)
53, 4syl6eqr 2487 . . . . 5  |-  ( k  =  K  ->  ( lub `  k )  =  U )
6 fveq2 5729 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
7 p1val.b . . . . . 6  |-  B  =  ( Base `  K
)
86, 7syl6eqr 2487 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
95, 8fveq12d 5735 . . . 4  |-  ( k  =  K  ->  (
( lub `  k
) `  ( Base `  k ) )  =  ( U `  B
) )
10 df-p1 14470 . . . 4  |-  1.  =  ( k  e.  _V  |->  ( ( lub `  k
) `  ( Base `  k ) ) )
11 fvex 5743 . . . 4  |-  ( U `
 B )  e. 
_V
129, 10, 11fvmpt 5807 . . 3  |-  ( K  e.  _V  ->  ( 1. `  K )  =  ( U `  B
) )
132, 12syl5eq 2481 . 2  |-  ( K  e.  _V  ->  .1.  =  ( U `  B ) )
141, 13syl 16 1  |-  ( K  e.  V  ->  .1.  =  ( U `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2957   ` cfv 5455   Basecbs 13470   lubclub 14400   1.cp1 14468
This theorem is referenced by:  ple1  14474  clatp1ex  24195  xrsp1  24205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-p1 14470
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