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Theorem ltexprlemdisj 6762
Description: Our constructed difference is disjoint. Lemma for ltexpri 6769. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
Assertion
Ref Expression
ltexprlemdisj  |-  ( A 
<P  B  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) ) )
Distinct variable groups:    x, y, q, A    x, B, y, q    x, C, y, q

Proof of Theorem ltexprlemdisj
Dummy variables  z  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltsonq 6554 . . . . . 6  |-  <Q  Or  Q.
2 ltrelnq 6521 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
31, 2son2lpi 4749 . . . . 5  |-  -.  (
y  <Q  z  /\  z  <Q  y )
4 ltrelpr 6661 . . . . . . . . . . . . . . . 16  |-  <P  C_  ( P.  X.  P. )
54brel 4420 . . . . . . . . . . . . . . 15  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
65simprd 111 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  B  e. 
P. )
7 prop 6631 . . . . . . . . . . . . . 14  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
86, 7syl 14 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
9 prltlu 6643 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  ( y  +Q  q
)  e.  ( 1st `  B )  /\  (
z  +Q  q )  e.  ( 2nd `  B
) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  q
) )
108, 9syl3an1 1179 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( y  +Q  q
)  e.  ( 1st `  B )  /\  (
z  +Q  q )  e.  ( 2nd `  B
) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  q
) )
11103expb 1116 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( ( y  +Q  q )  e.  ( 1st `  B )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  q
) )
1211adantlr 454 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  +Q  q )  e.  ( 1st `  B )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  q
) )
1312adantrll 461 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  +Q  q )  e.  ( 2nd `  B
) ) )  -> 
( y  +Q  q
)  <Q  ( z  +Q  q ) )
1413adantrrl 463 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
( y  +Q  q
)  <Q  ( z  +Q  q ) )
15 ltanqg 6556 . . . . . . . . . 10  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
1615adantl 266 . . . . . . . . 9  |-  ( ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
175simpld 109 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  A  e. 
P. )
18 prop 6631 . . . . . . . . . . . . 13  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
1917, 18syl 14 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
20 elprnqu 6638 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2119, 20sylan 271 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2221ad2ant2r 486 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  ->  y  e.  Q. )
2322adantrr 456 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
y  e.  Q. )
24 elprnql 6637 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
2519, 24sylan 271 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
2625ad2ant2r 486 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( z  e.  ( 1st `  A )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  ->  z  e.  Q. )
2726adantrl 455 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
z  e.  Q. )
28 simplr 490 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
q  e.  Q. )
29 addcomnqg 6537 . . . . . . . . . 10  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3029adantl 266 . . . . . . . . 9  |-  ( ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  /\  ( f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
3116, 23, 27, 28, 30caovord2d 5698 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
( y  <Q  z  <->  ( y  +Q  q ) 
<Q  ( z  +Q  q
) ) )
3214, 31mpbird 160 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
y  <Q  z )
33 prltlu 6643 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  z  <Q  y )
3419, 33syl3an1 1179 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  z  <Q  y )
35343com23 1121 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  A
) )  ->  z  <Q  y )
36353expb 1116 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( y  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  A ) ) )  ->  z  <Q  y )
3736adantlr 454 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( y  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  A ) ) )  ->  z  <Q  y )
3837adantrlr 462 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  z  e.  ( 1st `  A
) ) )  -> 
z  <Q  y )
3938adantrrr 464 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
z  <Q  y )
4032, 39jca 294 . . . . . 6  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
( y  <Q  z  /\  z  <Q  y ) )
4140ex 112 . . . . 5  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  ->  (
y  <Q  z  /\  z  <Q  y ) ) )
423, 41mtoi 600 . . . 4  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  -.  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
4342alrimivv 1771 . . 3  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  A. y A. z  -.  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
44 ltexprlem.1 . . . . . . . . . . . 12  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
4544ltexprlemell 6754 . . . . . . . . . . 11  |-  ( q  e.  ( 1st `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
4644ltexprlemelu 6755 . . . . . . . . . . 11  |-  ( q  e.  ( 2nd `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )
4745, 46anbi12i 441 . . . . . . . . . 10  |-  ( ( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C
) )  <->  ( (
q  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  /\  (
q  e.  Q.  /\  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
48 anandi 532 . . . . . . . . . 10  |-  ( ( q  e.  Q.  /\  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )  <->  ( (
q  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  /\  (
q  e.  Q.  /\  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
4947, 48bitr4i 180 . . . . . . . . 9  |-  ( ( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C
) )  <->  ( q  e.  Q.  /\  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
5049baib 839 . . . . . . . 8  |-  ( q  e.  Q.  ->  (
( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C ) )  <->  ( E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
51 eleq1 2116 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
y  e.  ( 1st `  A )  <->  z  e.  ( 1st `  A ) ) )
52 oveq1 5547 . . . . . . . . . . . 12  |-  ( y  =  z  ->  (
y  +Q  q )  =  ( z  +Q  q ) )
5352eleq1d 2122 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
( y  +Q  q
)  e.  ( 2nd `  B )  <->  ( z  +Q  q )  e.  ( 2nd `  B ) ) )
5451, 53anbi12d 450 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) )  <->  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
5554cbvexv 1811 . . . . . . . . 9  |-  ( E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) )  <->  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )
5655anbi2i 438 . . . . . . . 8  |-  ( ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) )  <->  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
5750, 56syl6bb 189 . . . . . . 7  |-  ( q  e.  Q.  ->  (
( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C ) )  <->  ( E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
58 eeanv 1823 . . . . . . 7  |-  ( E. y E. z ( ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  <->  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
5957, 58syl6bbr 191 . . . . . 6  |-  ( q  e.  Q.  ->  (
( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C ) )  <->  E. y E. z
( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
6059notbid 602 . . . . 5  |-  ( q  e.  Q.  ->  ( -.  ( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C ) )  <->  -.  E. y E. z ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
61 alnex 1404 . . . . . . 7  |-  ( A. z  -.  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  <->  -.  E. z
( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
6261albii 1375 . . . . . 6  |-  ( A. y A. z  -.  (
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  <->  A. y  -.  E. z ( ( y  e.  ( 2nd `  A )  /\  (
y  +Q  q )  e.  ( 1st `  B
) )  /\  (
z  e.  ( 1st `  A )  /\  (
z  +Q  q )  e.  ( 2nd `  B
) ) ) )
63 alnex 1404 . . . . . 6  |-  ( A. y  -.  E. z ( ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  <->  -.  E. y E. z ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
6462, 63bitri 177 . . . . 5  |-  ( A. y A. z  -.  (
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  <->  -.  E. y E. z ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
6560, 64syl6bbr 191 . . . 4  |-  ( q  e.  Q.  ->  ( -.  ( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C ) )  <->  A. y A. z  -.  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
6665adantl 266 . . 3  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) )  <->  A. y A. z  -.  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
6743, 66mpbird 160 . 2  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) ) )
6867ralrimiva 2409 1  |-  ( A 
<P  B  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   A.wral 2323   {crab 2327   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    +Q cplq 6438    <Q cltq 6441   P.cnp 6447    <P cltp 6451
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-in1 554  ax-in2 555  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-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  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-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-ltnqqs 6509  df-inp 6622  df-iltp 6626
This theorem is referenced by:  ltexprlempr  6764
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