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Theorem sltval 25515
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 2464 . . . . 5  |-  ( f  =  A  ->  (
f  e.  No  <->  A  e.  No ) )
21anbi1d 686 . . . 4  |-  ( f  =  A  ->  (
( f  e.  No  /\  g  e.  No )  <-> 
( A  e.  No  /\  g  e.  No ) ) )
3 fveq1 5686 . . . . . . . 8  |-  ( f  =  A  ->  (
f `  y )  =  ( A `  y ) )
43eqeq1d 2412 . . . . . . 7  |-  ( f  =  A  ->  (
( f `  y
)  =  ( g `
 y )  <->  ( A `  y )  =  ( g `  y ) ) )
54ralbidv 2686 . . . . . 6  |-  ( f  =  A  ->  ( A. y  e.  x  ( f `  y
)  =  ( g `
 y )  <->  A. y  e.  x  ( A `  y )  =  ( g `  y ) ) )
6 fveq1 5686 . . . . . . 7  |-  ( f  =  A  ->  (
f `  x )  =  ( A `  x ) )
76breq1d 4182 . . . . . 6  |-  ( f  =  A  ->  (
( f `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x )  <->  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) )
85, 7anbi12d 692 . . . . 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 2687 . . . 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 692 . . 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 2464 . . . . 5  |-  ( g  =  B  ->  (
g  e.  No  <->  B  e.  No ) )
1211anbi2d 685 . . . 4  |-  ( g  =  B  ->  (
( A  e.  No  /\  g  e.  No )  <-> 
( A  e.  No  /\  B  e.  No ) ) )
13 fveq1 5686 . . . . . . . 8  |-  ( g  =  B  ->  (
g `  y )  =  ( B `  y ) )
1413eqeq2d 2415 . . . . . . 7  |-  ( g  =  B  ->  (
( A `  y
)  =  ( g `
 y )  <->  ( A `  y )  =  ( B `  y ) ) )
1514ralbidv 2686 . . . . . 6  |-  ( g  =  B  ->  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  <->  A. y  e.  x  ( A `  y )  =  ( B `  y ) ) )
16 fveq1 5686 . . . . . . 7  |-  ( g  =  B  ->  (
g `  x )  =  ( B `  x ) )
1716breq2d 4184 . . . . . 6  |-  ( g  =  B  ->  (
( A `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x )  <->  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) )
1815, 17anbi12d 692 . . . . 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 2687 . . . 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 692 . . 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 25512 . . 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 4434 . 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 851 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   (/)c0 3588   {ctp 3776   <.cop 3777   class class class wbr 4172   Oncon0 4541   ` cfv 5413   1oc1o 6676   2oc2o 6677   Nocsur 25508   < scslt 25509
This theorem is referenced by:  sltval2  25524  sltres  25532  nodense  25557  nobndup  25568  nobnddown  25569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-iota 5377  df-fv 5421  df-slt 25512
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