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Theorem ltexprlemdisj 7436
Description: Our constructed difference is disjoint. Lemma for ltexpri 7443. (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 7228 . . . . . 6  |-  <Q  Or  Q.
2 ltrelnq 7195 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
31, 2son2lpi 4941 . . . . 5  |-  -.  (
y  <Q  z  /\  z  <Q  y )
4 ltrelpr 7335 . . . . . . . . . . . . . . . 16  |-  <P  C_  ( P.  X.  P. )
54brel 4597 . . . . . . . . . . . . . . 15  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
65simprd 113 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  B  e. 
P. )
7 prop 7305 . . . . . . . . . . . . . 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 7317 . . . . . . . . . . . . 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 1250 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( y  +Q  q
)  e.  ( 1st `  B )  /\  (
z  +Q  q )  e.  ( 2nd `  B
) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  q
) )
11103expb 1183 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( ( y  +Q  q )  e.  ( 1st `  B )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  q
) )
1211adantlr 469 . . . . . . . . . 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 476 . . . . . . . . 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 478 . . . . . . . 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 7230 . . . . . . . . . 10  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
1615adantl 275 . . . . . . . . 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 111 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  A  e. 
P. )
18 prop 7305 . . . . . . . . . . . . 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 7312 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2119, 20sylan 281 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2221ad2ant2r 501 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  ->  y  e.  Q. )
2322adantrr 471 . . . . . . . . 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 7311 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
2519, 24sylan 281 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
2625ad2ant2r 501 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( z  e.  ( 1st `  A )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  ->  z  e.  Q. )
2726adantrl 470 . . . . . . . . 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 520 . . . . . . . . 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 7211 . . . . . . . . . 10  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3029adantl 275 . . . . . . . . 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 5946 . . . . . . . 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 166 . . . . . . 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 7317 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  z  <Q  y )
3419, 33syl3an1 1250 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  z  <Q  y )
35343com23 1188 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  A
) )  ->  z  <Q  y )
36353expb 1183 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( y  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  A ) ) )  ->  z  <Q  y )
3736adantlr 469 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( y  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  A ) ) )  ->  z  <Q  y )
3837adantrlr 477 . . . . . . . 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 479 . . . . . . 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 304 . . . . . 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 114 . . . . 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 654 . . . 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 1848 . . 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 7428 . . . . . . . . . . 11  |-  ( q  e.  ( 1st `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
4644ltexprlemelu 7429 . . . . . . . . . . 11  |-  ( q  e.  ( 2nd `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )
4745, 46anbi12i 456 . . . . . . . . . 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 580 . . . . . . . . . 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 186 . . . . . . . . 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 905 . . . . . . . 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 2203 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
y  e.  ( 1st `  A )  <->  z  e.  ( 1st `  A ) ) )
52 oveq1 5787 . . . . . . . . . . . 12  |-  ( y  =  z  ->  (
y  +Q  q )  =  ( z  +Q  q ) )
5352eleq1d 2209 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
( y  +Q  q
)  e.  ( 2nd `  B )  <->  ( z  +Q  q )  e.  ( 2nd `  B ) ) )
5451, 53anbi12d 465 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) )  <->  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
5554cbvexv 1891 . . . . . . . . 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 453 . . . . . . . 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 195 . . . . . . 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 1905 . . . . . . 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 197 . . . . . 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 657 . . . . 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 1476 . . . . . . 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 1447 . . . . . 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 1476 . . . . . 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 183 . . . . 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 197 . . . 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 275 . . 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 166 . 2  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) ) )
6867ralrimiva 2508 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 103    <-> wb 104    /\ w3a 963   A.wal 1330    = wceq 1332   E.wex 1469    e. wcel 1481   A.wral 2417   {crab 2421   <.cop 3533   class class class wbr 3935   ` cfv 5129  (class class class)co 5780   1stc1st 6042   2ndc2nd 6043   Q.cnq 7110    +Q cplq 7112    <Q cltq 7115   P.cnp 7121    <P cltp 7125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4049  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-iinf 4508
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-tr 4033  df-eprel 4217  df-id 4221  df-po 4224  df-iso 4225  df-iord 4294  df-on 4296  df-suc 4299  df-iom 4511  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-ov 5783  df-oprab 5784  df-mpo 5785  df-1st 6044  df-2nd 6045  df-recs 6208  df-irdg 6273  df-oadd 6323  df-omul 6324  df-er 6435  df-ec 6437  df-qs 6441  df-ni 7134  df-pli 7135  df-mi 7136  df-lti 7137  df-plpq 7174  df-enq 7177  df-nqqs 7178  df-plqqs 7179  df-ltnqqs 7183  df-inp 7296  df-iltp 7300
This theorem is referenced by:  ltexprlempr  7438
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