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Theorem xporderlem 5880
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 3793 . . 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 2120 . . 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 177 . 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 2577 . . . 4  |-  a  e. 
_V
6 vex 2577 . . . 4  |-  b  e. 
_V
75, 6opex 3994 . . 3  |-  <. a ,  b >.  e.  _V
8 vex 2577 . . . 4  |-  c  e. 
_V
9 vex 2577 . . . 4  |-  d  e. 
_V
108, 9opex 3994 . . 3  |-  <. c ,  d >.  e.  _V
11 eleq1 2116 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( x  e.  ( A  X.  B
)  <->  <. a ,  b
>.  e.  ( A  X.  B ) ) )
12 opelxp 4402 . . . . . 6  |-  ( <.
a ,  b >.  e.  ( A  X.  B
)  <->  ( a  e.  A  /\  b  e.  B ) )
1311, 12syl6bb 189 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( x  e.  ( A  X.  B
)  <->  ( a  e.  A  /\  b  e.  B ) ) )
1413anbi1d 446 . . . 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 5803 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( 1st `  x
)  =  a )
1615breq1d 3802 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( ( 1st `  x ) R ( 1st `  y )  <-> 
a R ( 1st `  y ) ) )
1715eqeq1d 2064 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( ( 1st `  x )  =  ( 1st `  y )  <-> 
a  =  ( 1st `  y ) ) )
185, 6op2ndd 5804 . . . . . . 7  |-  ( x  =  <. a ,  b
>.  ->  ( 2nd `  x
)  =  b )
1918breq1d 3802 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( ( 2nd `  x ) S ( 2nd `  y )  <-> 
b S ( 2nd `  y ) ) )
2017, 19anbi12d 450 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x ) S ( 2nd `  y ) )  <->  ( a  =  ( 1st `  y
)  /\  b S
( 2nd `  y
) ) ) )
2116, 20orbi12d 717 . . . 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 450 . . 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 2116 . . . . . 6  |-  ( y  =  <. c ,  d
>.  ->  ( y  e.  ( A  X.  B
)  <->  <. c ,  d
>.  e.  ( A  X.  B ) ) )
24 opelxp 4402 . . . . . 6  |-  ( <.
c ,  d >.  e.  ( A  X.  B
)  <->  ( c  e.  A  /\  d  e.  B ) )
2523, 24syl6bb 189 . . . . 5  |-  ( y  =  <. c ,  d
>.  ->  ( y  e.  ( A  X.  B
)  <->  ( c  e.  A  /\  d  e.  B ) ) )
2625anbi2d 445 . . . 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 5803 . . . . . 6  |-  ( y  =  <. c ,  d
>.  ->  ( 1st `  y
)  =  c )
2827breq2d 3804 . . . . 5  |-  ( y  =  <. c ,  d
>.  ->  ( a R ( 1st `  y
)  <->  a R c ) )
2927eqeq2d 2067 . . . . . 6  |-  ( y  =  <. c ,  d
>.  ->  ( a  =  ( 1st `  y
)  <->  a  =  c ) )
308, 9op2ndd 5804 . . . . . . 7  |-  ( y  =  <. c ,  d
>.  ->  ( 2nd `  y
)  =  d )
3130breq2d 3804 . . . . . 6  |-  ( y  =  <. c ,  d
>.  ->  ( b S ( 2nd `  y
)  <->  b S d ) )
3229, 31anbi12d 450 . . . . 5  |-  ( y  =  <. c ,  d
>.  ->  ( ( a  =  ( 1st `  y
)  /\  b S
( 2nd `  y
) )  <->  ( a  =  c  /\  b S d ) ) )
3328, 32orbi12d 717 . . . 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 450 . . 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 4036 . 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 528 . . 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 439 . 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 199 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 101    <-> wb 102    \/ wo 639    = wceq 1259    e. wcel 1409   <.cop 3406   class class class wbr 3792   {copab 3845    X. cxp 4371   ` cfv 4930   1stc1st 5793   2ndc2nd 5794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fv 4938  df-1st 5795  df-2nd 5796
This theorem is referenced by:  poxp  5881
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