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Theorem xporderlem 5996
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 3846 . . 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 2154 . . 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 182 . 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 vex 2622 . . . 4  |-  a  e. 
_V
6 vex 2622 . . . 4  |-  b  e. 
_V
75, 6opex 4056 . . 3  |-  <. a ,  b >.  e.  _V
8 vex 2622 . . . 4  |-  c  e. 
_V
9 vex 2622 . . . 4  |-  d  e. 
_V
108, 9opex 4056 . . 3  |-  <. c ,  d >.  e.  _V
11 eleq1 2150 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( x  e.  ( A  X.  B
)  <->  <. a ,  b
>.  e.  ( A  X.  B ) ) )
12 opelxp 4467 . . . . . 6  |-  ( <.
a ,  b >.  e.  ( A  X.  B
)  <->  ( a  e.  A  /\  b  e.  B ) )
1311, 12syl6bb 194 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( x  e.  ( A  X.  B
)  <->  ( a  e.  A  /\  b  e.  B ) ) )
1413anbi1d 453 . . . 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 ) ) ) )
155, 6op1std 5919 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( 1st `  x
)  =  a )
1615breq1d 3855 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( ( 1st `  x ) R ( 1st `  y )  <-> 
a R ( 1st `  y ) ) )
1715eqeq1d 2096 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( ( 1st `  x )  =  ( 1st `  y )  <-> 
a  =  ( 1st `  y ) ) )
185, 6op2ndd 5920 . . . . . . 7  |-  ( x  =  <. a ,  b
>.  ->  ( 2nd `  x
)  =  b )
1918breq1d 3855 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( ( 2nd `  x ) S ( 2nd `  y )  <-> 
b S ( 2nd `  y ) ) )
2017, 19anbi12d 457 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x ) S ( 2nd `  y ) )  <->  ( a  =  ( 1st `  y
)  /\  b S
( 2nd `  y
) ) ) )
2116, 20orbi12d 742 . . . 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
) ) ) ) )
2214, 21anbi12d 457 . . 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
) ) ) ) ) )
23 eleq1 2150 . . . . . 6  |-  ( y  =  <. c ,  d
>.  ->  ( y  e.  ( A  X.  B
)  <->  <. c ,  d
>.  e.  ( A  X.  B ) ) )
24 opelxp 4467 . . . . . 6  |-  ( <.
c ,  d >.  e.  ( A  X.  B
)  <->  ( c  e.  A  /\  d  e.  B ) )
2523, 24syl6bb 194 . . . . 5  |-  ( y  =  <. c ,  d
>.  ->  ( y  e.  ( A  X.  B
)  <->  ( c  e.  A  /\  d  e.  B ) ) )
2625anbi2d 452 . . . 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 )
) ) )
278, 9op1std 5919 . . . . . 6  |-  ( y  =  <. c ,  d
>.  ->  ( 1st `  y
)  =  c )
2827breq2d 3857 . . . . 5  |-  ( y  =  <. c ,  d
>.  ->  ( a R ( 1st `  y
)  <->  a R c ) )
2927eqeq2d 2099 . . . . . 6  |-  ( y  =  <. c ,  d
>.  ->  ( a  =  ( 1st `  y
)  <->  a  =  c ) )
308, 9op2ndd 5920 . . . . . . 7  |-  ( y  =  <. c ,  d
>.  ->  ( 2nd `  y
)  =  d )
3130breq2d 3857 . . . . . 6  |-  ( y  =  <. c ,  d
>.  ->  ( b S ( 2nd `  y
)  <->  b S d ) )
3229, 31anbi12d 457 . . . . 5  |-  ( y  =  <. c ,  d
>.  ->  ( ( a  =  ( 1st `  y
)  /\  b S
( 2nd `  y
) )  <->  ( a  =  c  /\  b S d ) ) )
3328, 32orbi12d 742 . . . 4  |-  ( y  =  <. c ,  d
>.  ->  ( ( a R ( 1st `  y
)  \/  ( a  =  ( 1st `  y
)  /\  b S
( 2nd `  y
) ) )  <->  ( a R c  \/  (
a  =  c  /\  b S d ) ) ) )
3426, 33anbi12d 457 . . 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 ) ) ) ) )
357, 10, 22, 34opelopab 4098 . 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 553 . . 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 446 . 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 204 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 set class
Syntax hints:    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   <.cop 3449   class class class wbr 3845   {copab 3898    X. cxp 4436   ` cfv 5015   1stc1st 5909   2ndc2nd 5910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-1st 5911  df-2nd 5912
This theorem is referenced by:  poxp  5997
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