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Theorem ltexprlemdisj 7315
Description: Our constructed difference is disjoint. Lemma for ltexpri 7322. (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 7107 . . . . . 6  |-  <Q  Or  Q.
2 ltrelnq 7074 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
31, 2son2lpi 4871 . . . . 5  |-  -.  (
y  <Q  z  /\  z  <Q  y )
4 ltrelpr 7214 . . . . . . . . . . . . . . . 16  |-  <P  C_  ( P.  X.  P. )
54brel 4529 . . . . . . . . . . . . . . 15  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
65simprd 113 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  B  e. 
P. )
7 prop 7184 . . . . . . . . . . . . . 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 7196 . . . . . . . . . . . . 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 1217 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( y  +Q  q
)  e.  ( 1st `  B )  /\  (
z  +Q  q )  e.  ( 2nd `  B
) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  q
) )
11103expb 1150 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( ( y  +Q  q )  e.  ( 1st `  B )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  q
) )
1211adantlr 464 . . . . . . . . . 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 471 . . . . . . . . 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 473 . . . . . . . 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 7109 . . . . . . . . . 10  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
1615adantl 273 . . . . . . . . 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 7184 . . . . . . . . . . . . 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 7191 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2119, 20sylan 279 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2221ad2ant2r 496 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  ->  y  e.  Q. )
2322adantrr 466 . . . . . . . . 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 7190 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
2519, 24sylan 279 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
2625ad2ant2r 496 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( z  e.  ( 1st `  A )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  ->  z  e.  Q. )
2726adantrl 465 . . . . . . . . 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 500 . . . . . . . . 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 7090 . . . . . . . . . 10  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3029adantl 273 . . . . . . . . 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 5872 . . . . . . . 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 7196 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  z  <Q  y )
3419, 33syl3an1 1217 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  z  <Q  y )
35343com23 1155 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  A
) )  ->  z  <Q  y )
36353expb 1150 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( y  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  A ) ) )  ->  z  <Q  y )
3736adantlr 464 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( y  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  A ) ) )  ->  z  <Q  y )
3837adantrlr 472 . . . . . . . 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 474 . . . . . . 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 302 . . . . . 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 631 . . . 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 1814 . . 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 7307 . . . . . . . . . . 11  |-  ( q  e.  ( 1st `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
4644ltexprlemelu 7308 . . . . . . . . . . 11  |-  ( q  e.  ( 2nd `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )
4745, 46anbi12i 451 . . . . . . . . . 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 560 . . . . . . . . . 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 872 . . . . . . . 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 2162 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
y  e.  ( 1st `  A )  <->  z  e.  ( 1st `  A ) ) )
52 oveq1 5713 . . . . . . . . . . . 12  |-  ( y  =  z  ->  (
y  +Q  q )  =  ( z  +Q  q ) )
5352eleq1d 2168 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
( y  +Q  q
)  e.  ( 2nd `  B )  <->  ( z  +Q  q )  e.  ( 2nd `  B ) ) )
5451, 53anbi12d 460 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) )  <->  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
5554cbvexv 1855 . . . . . . . . 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 448 . . . . . . . 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 1867 . . . . . . 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 633 . . . . 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 1443 . . . . . . 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 1414 . . . . . 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 1443 . . . . . 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 273 . . 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 2464 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 930   A.wal 1297    = wceq 1299   E.wex 1436    e. wcel 1448   A.wral 2375   {crab 2379   <.cop 3477   class class class wbr 3875   ` cfv 5059  (class class class)co 5706   1stc1st 5967   2ndc2nd 5968   Q.cnq 6989    +Q cplq 6991    <Q cltq 6994   P.cnp 7000    <P cltp 7004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-pli 7014  df-mi 7015  df-lti 7016  df-plpq 7053  df-enq 7056  df-nqqs 7057  df-plqqs 7058  df-ltnqqs 7062  df-inp 7175  df-iltp 7179
This theorem is referenced by:  ltexprlempr  7317
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