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Theorem p1val 15212
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 2981 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 p1val.t . . 3  |-  .1.  =  ( 1. `  K )
3 fveq2 5691 . . . . . 6  |-  ( k  =  K  ->  ( lub `  k )  =  ( lub `  K
) )
4 p1val.u . . . . . 6  |-  U  =  ( lub `  K
)
53, 4syl6eqr 2493 . . . . 5  |-  ( k  =  K  ->  ( lub `  k )  =  U )
6 fveq2 5691 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
7 p1val.b . . . . . 6  |-  B  =  ( Base `  K
)
86, 7syl6eqr 2493 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
95, 8fveq12d 5697 . . . 4  |-  ( k  =  K  ->  (
( lub `  k
) `  ( Base `  k ) )  =  ( U `  B
) )
10 df-p1 15210 . . . 4  |-  1.  =  ( k  e.  _V  |->  ( ( lub `  k
) `  ( Base `  k ) ) )
11 fvex 5701 . . . 4  |-  ( U `
 B )  e. 
_V
129, 10, 11fvmpt 5774 . . 3  |-  ( K  e.  _V  ->  ( 1. `  K )  =  ( U `  B
) )
132, 12syl5eq 2487 . 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 1369    e. wcel 1756   _Vcvv 2972   ` cfv 5418   Basecbs 14174   lubclub 15112   1.cp1 15208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-p1 15210
This theorem is referenced by:  ple1  15214  clatp1cl  26133  xrsp1  26143  op1cl  32830
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