Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sltval Unicode version

Theorem sltval 24301
Description: The value of the surreal less than relationship. (Contributed by Scott Fenton, 14-Jun-2011.)
Assertion
Ref Expression
sltval  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
<->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) ) )
Distinct variable groups:    x, A, y    x, B, y

Proof of Theorem sltval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2343 . . . . 5  |-  ( f  =  A  ->  (
f  e.  No  <->  A  e.  No ) )
21anbi1d 685 . . . 4  |-  ( f  =  A  ->  (
( f  e.  No  /\  g  e.  No )  <-> 
( A  e.  No  /\  g  e.  No ) ) )
3 fveq1 5524 . . . . . . . 8  |-  ( f  =  A  ->  (
f `  y )  =  ( A `  y ) )
43eqeq1d 2291 . . . . . . 7  |-  ( f  =  A  ->  (
( f `  y
)  =  ( g `
 y )  <->  ( A `  y )  =  ( g `  y ) ) )
54ralbidv 2563 . . . . . 6  |-  ( f  =  A  ->  ( A. y  e.  x  ( f `  y
)  =  ( g `
 y )  <->  A. y  e.  x  ( A `  y )  =  ( g `  y ) ) )
6 fveq1 5524 . . . . . . 7  |-  ( f  =  A  ->  (
f `  x )  =  ( A `  x ) )
76breq1d 4033 . . . . . 6  |-  ( f  =  A  ->  (
( f `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x )  <->  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) )
85, 7anbi12d 691 . . . . 5  |-  ( f  =  A  ->  (
( A. y  e.  x  ( f `  y )  =  ( g `  y )  /\  ( f `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) )  <->  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x
) ) ) )
98rexbidv 2564 . . . 4  |-  ( f  =  A  ->  ( E. x  e.  On  ( A. y  e.  x  ( f `  y
)  =  ( g `
 y )  /\  ( f `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) )  <->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) ) )
102, 9anbi12d 691 . . 3  |-  ( f  =  A  ->  (
( ( f  e.  No  /\  g  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( f `  y
)  =  ( g `
 y )  /\  ( f `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) )  <-> 
( ( A  e.  No  /\  g  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x
) ) ) ) )
11 eleq1 2343 . . . . 5  |-  ( g  =  B  ->  (
g  e.  No  <->  B  e.  No ) )
1211anbi2d 684 . . . 4  |-  ( g  =  B  ->  (
( A  e.  No  /\  g  e.  No )  <-> 
( A  e.  No  /\  B  e.  No ) ) )
13 fveq1 5524 . . . . . . . 8  |-  ( g  =  B  ->  (
g `  y )  =  ( B `  y ) )
1413eqeq2d 2294 . . . . . . 7  |-  ( g  =  B  ->  (
( A `  y
)  =  ( g `
 y )  <->  ( A `  y )  =  ( B `  y ) ) )
1514ralbidv 2563 . . . . . 6  |-  ( g  =  B  ->  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  <->  A. y  e.  x  ( A `  y )  =  ( B `  y ) ) )
16 fveq1 5524 . . . . . . 7  |-  ( g  =  B  ->  (
g `  x )  =  ( B `  x ) )
1716breq2d 4035 . . . . . 6  |-  ( g  =  B  ->  (
( A `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x )  <->  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) )
1815, 17anbi12d 691 . . . . 5  |-  ( g  =  B  ->  (
( A. y  e.  x  ( A `  y )  =  ( g `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) )  <->  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) ) )
1918rexbidv 2564 . . . 4  |-  ( g  =  B  ->  ( E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x
) )  <->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) ) )
2012, 19anbi12d 691 . . 3  |-  ( g  =  B  ->  (
( ( A  e.  No  /\  g  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x
) ) )  <->  ( ( A  e.  No  /\  B  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) ) ) )
21 df-slt 24298 . . 3  |-  < s  =  { <. f ,  g
>.  |  ( (
f  e.  No  /\  g  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( f `  y
)  =  ( g `
 y )  /\  ( f `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) ) }
2210, 20, 21brabg 4284 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
<->  ( ( A  e.  No  /\  B  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) ) ) )
2322bianabs 850 1  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
<->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   (/)c0 3455   {ctp 3642   <.cop 3643   class class class wbr 4023   Oncon0 4392   ` cfv 5255   1oc1o 6472   2oc2o 6473   Nocsur 24294   < scslt 24295
This theorem is referenced by:  sltval2  24310  sltres  24318  nodense  24343  nobndup  24354  nobnddown  24355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263  df-slt 24298
  Copyright terms: Public domain W3C validator