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Theorem xporderlem 6926
Description: Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
Hypothesis
Ref Expression
xporderlem.1  |-  T  =  { <. x ,  y
>.  |  ( (
x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x
) R ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x ) S ( 2nd `  y
) ) ) ) }
Assertion
Ref Expression
xporderlem  |-  ( <.
a ,  b >. T <. c ,  d
>. 
<->  ( ( ( a  e.  A  /\  c  e.  A )  /\  (
b  e.  B  /\  d  e.  B )
)  /\  ( a R c  \/  (
a  =  c  /\  b S d ) ) ) )
Distinct variable groups:    x, A, y    x, B, y    x, R, y    x, S, y   
x, a, y    x, b, y    x, c, y   
x, d, y
Allowed substitution hints:    A( a, b, c, d)    B( a, b, c, d)    R( a, b, c, d)    S( a, b, c, d)    T( x, y, a, b, c, d)

Proof of Theorem xporderlem
StepHypRef Expression
1 df-br 4396 . . 3  |-  ( <.
a ,  b >. T <. c ,  d
>. 
<-> 
<. <. a ,  b
>. ,  <. c ,  d >. >.  e.  T )
2 xporderlem.1 . . . 4  |-  T  =  { <. x ,  y
>.  |  ( (
x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x
) R ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x ) S ( 2nd `  y
) ) ) ) }
32eleq2i 2541 . . 3  |-  ( <. <. a ,  b >. ,  <. c ,  d
>. >.  e.  T  <->  <. <. a ,  b >. ,  <. c ,  d >. >.  e.  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B
) )  /\  (
( 1st `  x
) R ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x ) S ( 2nd `  y
) ) ) ) } )
41, 3bitri 257 . 2  |-  ( <.
a ,  b >. T <. c ,  d
>. 
<-> 
<. <. a ,  b
>. ,  <. c ,  d >. >.  e.  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x ) R ( 1st `  y
)  \/  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x ) S ( 2nd `  y ) ) ) ) } )
5 opex 4664 . . 3  |-  <. a ,  b >.  e.  _V
6 opex 4664 . . 3  |-  <. c ,  d >.  e.  _V
7 eleq1 2537 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( x  e.  ( A  X.  B
)  <->  <. a ,  b
>.  e.  ( A  X.  B ) ) )
8 opelxp 4869 . . . . . 6  |-  ( <.
a ,  b >.  e.  ( A  X.  B
)  <->  ( a  e.  A  /\  b  e.  B ) )
97, 8syl6bb 269 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( x  e.  ( A  X.  B
)  <->  ( a  e.  A  /\  b  e.  B ) ) )
109anbi1d 719 . . . 4  |-  ( x  =  <. a ,  b
>.  ->  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B
) )  <->  ( (
a  e.  A  /\  b  e.  B )  /\  y  e.  ( A  X.  B ) ) ) )
11 vex 3034 . . . . . . 7  |-  a  e. 
_V
12 vex 3034 . . . . . . 7  |-  b  e. 
_V
1311, 12op1std 6822 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( 1st `  x
)  =  a )
1413breq1d 4405 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( ( 1st `  x ) R ( 1st `  y )  <-> 
a R ( 1st `  y ) ) )
1513eqeq1d 2473 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( ( 1st `  x )  =  ( 1st `  y )  <-> 
a  =  ( 1st `  y ) ) )
1611, 12op2ndd 6823 . . . . . . 7  |-  ( x  =  <. a ,  b
>.  ->  ( 2nd `  x
)  =  b )
1716breq1d 4405 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( ( 2nd `  x ) S ( 2nd `  y )  <-> 
b S ( 2nd `  y ) ) )
1815, 17anbi12d 725 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x ) S ( 2nd `  y ) )  <->  ( a  =  ( 1st `  y
)  /\  b S
( 2nd `  y
) ) ) )
1914, 18orbi12d 724 . . . 4  |-  ( x  =  <. a ,  b
>.  ->  ( ( ( 1st `  x ) R ( 1st `  y
)  \/  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x ) S ( 2nd `  y ) ) )  <->  ( a R ( 1st `  y
)  \/  ( a  =  ( 1st `  y
)  /\  b S
( 2nd `  y
) ) ) ) )
2010, 19anbi12d 725 . . 3  |-  ( x  =  <. a ,  b
>.  ->  ( ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x
) R ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x ) S ( 2nd `  y
) ) ) )  <-> 
( ( ( a  e.  A  /\  b  e.  B )  /\  y  e.  ( A  X.  B
) )  /\  (
a R ( 1st `  y )  \/  (
a  =  ( 1st `  y )  /\  b S ( 2nd `  y
) ) ) ) ) )
21 eleq1 2537 . . . . . 6  |-  ( y  =  <. c ,  d
>.  ->  ( y  e.  ( A  X.  B
)  <->  <. c ,  d
>.  e.  ( A  X.  B ) ) )
22 opelxp 4869 . . . . . 6  |-  ( <.
c ,  d >.  e.  ( A  X.  B
)  <->  ( c  e.  A  /\  d  e.  B ) )
2321, 22syl6bb 269 . . . . 5  |-  ( y  =  <. c ,  d
>.  ->  ( y  e.  ( A  X.  B
)  <->  ( c  e.  A  /\  d  e.  B ) ) )
2423anbi2d 718 . . . 4  |-  ( y  =  <. c ,  d
>.  ->  ( ( ( a  e.  A  /\  b  e.  B )  /\  y  e.  ( A  X.  B ) )  <-> 
( ( a  e.  A  /\  b  e.  B )  /\  (
c  e.  A  /\  d  e.  B )
) ) )
25 vex 3034 . . . . . . 7  |-  c  e. 
_V
26 vex 3034 . . . . . . 7  |-  d  e. 
_V
2725, 26op1std 6822 . . . . . 6  |-  ( y  =  <. c ,  d
>.  ->  ( 1st `  y
)  =  c )
2827breq2d 4407 . . . . 5  |-  ( y  =  <. c ,  d
>.  ->  ( a R ( 1st `  y
)  <->  a R c ) )
2927eqeq2d 2481 . . . . . 6  |-  ( y  =  <. c ,  d
>.  ->  ( a  =  ( 1st `  y
)  <->  a  =  c ) )
3025, 26op2ndd 6823 . . . . . . 7  |-  ( y  =  <. c ,  d
>.  ->  ( 2nd `  y
)  =  d )
3130breq2d 4407 . . . . . 6  |-  ( y  =  <. c ,  d
>.  ->  ( b S ( 2nd `  y
)  <->  b S d ) )
3229, 31anbi12d 725 . . . . 5  |-  ( y  =  <. c ,  d
>.  ->  ( ( a  =  ( 1st `  y
)  /\  b S
( 2nd `  y
) )  <->  ( a  =  c  /\  b S d ) ) )
3328, 32orbi12d 724 . . . 4  |-  ( y  =  <. c ,  d
>.  ->  ( ( a R ( 1st `  y
)  \/  ( a  =  ( 1st `  y
)  /\  b S
( 2nd `  y
) ) )  <->  ( a R c  \/  (
a  =  c  /\  b S d ) ) ) )
3424, 33anbi12d 725 . . 3  |-  ( y  =  <. c ,  d
>.  ->  ( ( ( ( a  e.  A  /\  b  e.  B
)  /\  y  e.  ( A  X.  B
) )  /\  (
a R ( 1st `  y )  \/  (
a  =  ( 1st `  y )  /\  b S ( 2nd `  y
) ) ) )  <-> 
( ( ( a  e.  A  /\  b  e.  B )  /\  (
c  e.  A  /\  d  e.  B )
)  /\  ( a R c  \/  (
a  =  c  /\  b S d ) ) ) ) )
355, 6, 20, 34opelopab 4723 . 2  |-  ( <. <. a ,  b >. ,  <. c ,  d
>. >.  e.  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x ) R ( 1st `  y
)  \/  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x ) S ( 2nd `  y ) ) ) ) }  <-> 
( ( ( a  e.  A  /\  b  e.  B )  /\  (
c  e.  A  /\  d  e.  B )
)  /\  ( a R c  \/  (
a  =  c  /\  b S d ) ) ) )
36 an4 840 . . 3  |-  ( ( ( a  e.  A  /\  b  e.  B
)  /\  ( c  e.  A  /\  d  e.  B ) )  <->  ( (
a  e.  A  /\  c  e.  A )  /\  ( b  e.  B  /\  d  e.  B
) ) )
3736anbi1i 709 . 2  |-  ( ( ( ( a  e.  A  /\  b  e.  B )  /\  (
c  e.  A  /\  d  e.  B )
)  /\  ( a R c  \/  (
a  =  c  /\  b S d ) ) )  <->  ( ( ( a  e.  A  /\  c  e.  A )  /\  ( b  e.  B  /\  d  e.  B
) )  /\  (
a R c  \/  ( a  =  c  /\  b S d ) ) ) )
384, 35, 373bitri 279 1  |-  ( <.
a ,  b >. T <. c ,  d
>. 
<->  ( ( ( a  e.  A  /\  c  e.  A )  /\  (
b  e.  B  /\  d  e.  B )
)  /\  ( a R c  \/  (
a  =  c  /\  b S d ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904   <.cop 3965   class class class wbr 4395   {copab 4453    X. cxp 4837   ` cfv 5589   1stc1st 6810   2ndc2nd 6811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-1st 6812  df-2nd 6813
This theorem is referenced by:  poxp  6927  soxp  6928
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