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Theorem p1val 14164
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 2809 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 p1val.t . . 3  |-  .1.  =  ( 1. `  K )
3 fveq2 5541 . . . . . 6  |-  ( k  =  K  ->  ( lub `  k )  =  ( lub `  K
) )
4 p1val.u . . . . . 6  |-  U  =  ( lub `  K
)
53, 4syl6eqr 2346 . . . . 5  |-  ( k  =  K  ->  ( lub `  k )  =  U )
6 fveq2 5541 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
7 p1val.b . . . . . 6  |-  B  =  ( Base `  K
)
86, 7syl6eqr 2346 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
95, 8fveq12d 5547 . . . 4  |-  ( k  =  K  ->  (
( lub `  k
) `  ( Base `  k ) )  =  ( U `  B
) )
10 df-p1 14162 . . . 4  |-  1.  =  ( k  e.  _V  |->  ( ( lub `  k
) `  ( Base `  k ) ) )
11 fvex 5555 . . . 4  |-  ( U `
 B )  e. 
_V
129, 10, 11fvmpt 5618 . . 3  |-  ( K  e.  _V  ->  ( 1. `  K )  =  ( U `  B
) )
132, 12syl5eq 2340 . 2  |-  ( K  e.  _V  ->  .1.  =  ( U `  B ) )
141, 13syl 15 1  |-  ( K  e.  V  ->  .1.  =  ( U `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   ` cfv 5271   Basecbs 13164   lubclub 14092   1.cp1 14160
This theorem is referenced by:  ple1  14166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-p1 14162
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