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Theorem ltexprlemopl 7563
Description: The lower cut of our constructed difference is open. Lemma for ltexpri 7575. (Contributed by Jim Kingdon, 21-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
ltexprlemopl  |-  ( ( A  <P  B  /\  q  e.  Q.  /\  q  e.  ( 1st `  C
) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  C ) ) )
Distinct variable groups:    x, y, q, r, A    x, B, y, q, r    x, C, y, q, r

Proof of Theorem ltexprlemopl
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 ltexprlem.1 . . . . 5  |-  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 ) ) } >.
21ltexprlemell 7560 . . . 4  |-  ( q  e.  ( 1st `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
32simprbi 273 . . 3  |-  ( q  e.  ( 1st `  C
)  ->  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )
4 19.42v 1899 . . . . . . . 8  |-  ( E. y ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  <->  ( A  <P  B  /\  E. y
( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) ) )
5 19.42v 1899 . . . . . . . . 9  |-  ( E. y ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  <->  ( q  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
65anbi2i 454 . . . . . . . 8  |-  ( ( A  <P  B  /\  E. y ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  <->  ( A  <P  B  /\  ( q  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) ) )
74, 6bitri 183 . . . . . . 7  |-  ( E. y ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  <->  ( A  <P  B  /\  ( q  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) ) )
8 ltrelpr 7467 . . . . . . . . . . . . . 14  |-  <P  C_  ( P.  X.  P. )
98brel 4663 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
109simprd 113 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  B  e. 
P. )
11 prop 7437 . . . . . . . . . . . . 13  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
12 prnmaxl 7450 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  ( y  +Q  q
)  e.  ( 1st `  B ) )  ->  E. s  e.  ( 1st `  B ) ( y  +Q  q ) 
<Q  s )
1311, 12sylan 281 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  ( y  +Q  q
)  e.  ( 1st `  B ) )  ->  E. s  e.  ( 1st `  B ) ( y  +Q  q ) 
<Q  s )
1410, 13sylan 281 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( y  +Q  q
)  e.  ( 1st `  B ) )  ->  E. s  e.  ( 1st `  B ) ( y  +Q  q ) 
<Q  s )
1514adantrl 475 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  ->  E. s  e.  ( 1st `  B
) ( y  +Q  q )  <Q  s
)
1615adantrl 475 . . . . . . . . 9  |-  ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  ->  E. s  e.  ( 1st `  B ) ( y  +Q  q ) 
<Q  s )
179simpld 111 . . . . . . . . . . . . . . 15  |-  ( A 
<P  B  ->  A  e. 
P. )
1817ad2antrr 485 . . . . . . . . . . . . . 14  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  A  e.  P. )
19 simplrr 531 . . . . . . . . . . . . . . 15  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  q )  e.  ( 1st `  B
) ) )
2019simpld 111 . . . . . . . . . . . . . 14  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  y  e.  ( 2nd `  A
) )
21 prop 7437 . . . . . . . . . . . . . . 15  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
22 elprnqu 7444 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2321, 22sylan 281 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2418, 20, 23syl2anc 409 . . . . . . . . . . . . 13  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  y  e.  Q. )
25 simplrl 530 . . . . . . . . . . . . 13  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  q  e.  Q. )
26 ltaddnq 7369 . . . . . . . . . . . . 13  |-  ( ( y  e.  Q.  /\  q  e.  Q. )  ->  y  <Q  ( y  +Q  q ) )
2724, 25, 26syl2anc 409 . . . . . . . . . . . 12  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  y  <Q  ( y  +Q  q
) )
28 simprr 527 . . . . . . . . . . . 12  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  (
y  +Q  q ) 
<Q  s )
29 ltsonq 7360 . . . . . . . . . . . . 13  |-  <Q  Or  Q.
30 ltrelnq 7327 . . . . . . . . . . . . 13  |-  <Q  C_  ( Q.  X.  Q. )
3129, 30sotri 5006 . . . . . . . . . . . 12  |-  ( ( y  <Q  ( y  +Q  q )  /\  (
y  +Q  q ) 
<Q  s )  ->  y  <Q  s )
3227, 28, 31syl2anc 409 . . . . . . . . . . 11  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  y  <Q  s )
3310ad2antrr 485 . . . . . . . . . . . . 13  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  B  e.  P. )
34 simprl 526 . . . . . . . . . . . . 13  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  s  e.  ( 1st `  B
) )
35 elprnql 7443 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  s  e.  ( 1st `  B ) )  -> 
s  e.  Q. )
3611, 35sylan 281 . . . . . . . . . . . . 13  |-  ( ( B  e.  P.  /\  s  e.  ( 1st `  B ) )  -> 
s  e.  Q. )
3733, 34, 36syl2anc 409 . . . . . . . . . . . 12  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  s  e.  Q. )
38 ltexnqq 7370 . . . . . . . . . . . 12  |-  ( ( y  e.  Q.  /\  s  e.  Q. )  ->  ( y  <Q  s  <->  E. r  e.  Q.  (
y  +Q  r )  =  s ) )
3924, 37, 38syl2anc 409 . . . . . . . . . . 11  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  (
y  <Q  s  <->  E. r  e.  Q.  ( y  +Q  r )  =  s ) )
4032, 39mpbid 146 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  E. r  e.  Q.  ( y  +Q  r )  =  s )
41 simplrr 531 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  ( y  +Q  q )  <Q  s
)
42 simprr 527 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  ( y  +Q  r )  =  s )
4341, 42breqtrrd 4017 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  ( y  +Q  q )  <Q  (
y  +Q  r ) )
4425adantr 274 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  q  e.  Q. )
45 simprl 526 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  r  e.  Q. )
4624adantr 274 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  y  e.  Q. )
47 ltanqg 7362 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  Q.  /\  r  e.  Q.  /\  y  e.  Q. )  ->  (
q  <Q  r  <->  ( y  +Q  q )  <Q  (
y  +Q  r ) ) )
4844, 45, 46, 47syl3anc 1233 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  ( q  <Q  r  <->  ( y  +Q  q )  <Q  (
y  +Q  r ) ) )
4943, 48mpbird 166 . . . . . . . . . . . . 13  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  q  <Q  r )
5020adantr 274 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  y  e.  ( 2nd `  A ) )
51 simplrl 530 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  s  e.  ( 1st `  B ) )
5242, 51eqeltrd 2247 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  ( y  +Q  r )  e.  ( 1st `  B ) )
5350, 52jca 304 . . . . . . . . . . . . 13  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) )
5449, 45, 53jca32 308 . . . . . . . . . . . 12  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  (
r  e.  Q.  /\  ( y  +Q  r
)  =  s ) )  ->  ( q  <Q  r  /\  ( r  e.  Q.  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  r )  e.  ( 1st `  B
) ) ) ) )
5554expr 373 . . . . . . . . . . 11  |-  ( ( ( ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  /\  r  e.  Q. )  ->  (
( y  +Q  r
)  =  s  -> 
( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) ) )
5655reximdva 2572 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  ( E. r  e.  Q.  ( y  +Q  r
)  =  s  ->  E. r  e.  Q.  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) ) )
5740, 56mpd 13 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  /\  ( s  e.  ( 1st `  B )  /\  ( y  +Q  q )  <Q  s
) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
5816, 57rexlimddv 2592 . . . . . . . 8  |-  ( ( A  <P  B  /\  ( q  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  ->  E. r  e.  Q.  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
5958eximi 1593 . . . . . . 7  |-  ( E. y ( A  <P  B  /\  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  ->  E. y E. r  e. 
Q.  ( q  <Q 
r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
607, 59sylbir 134 . . . . . 6  |-  ( ( A  <P  B  /\  ( q  e.  Q.  /\ 
E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  ->  E. y E. r  e. 
Q.  ( q  <Q 
r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
61 rexcom4 2753 . . . . . 6  |-  ( E. r  e.  Q.  E. y ( q  <Q 
r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )  <->  E. y E. r  e.  Q.  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
6260, 61sylibr 133 . . . . 5  |-  ( ( A  <P  B  /\  ( q  e.  Q.  /\ 
E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  ->  E. r  e.  Q.  E. y ( q  <Q 
r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
63 19.42v 1899 . . . . . . 7  |-  ( E. y ( q  <Q 
r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )  <->  ( q  <Q  r  /\  E. y
( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
64 19.42v 1899 . . . . . . . 8  |-  ( E. y ( r  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) )  <->  ( r  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )
6564anbi2i 454 . . . . . . 7  |-  ( ( q  <Q  r  /\  E. y ( r  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )  <->  ( q  <Q  r  /\  ( r  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
6663, 65bitri 183 . . . . . 6  |-  ( E. y ( q  <Q 
r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )  <->  ( q  <Q  r  /\  ( r  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
6766rexbii 2477 . . . . 5  |-  ( E. r  e.  Q.  E. y ( q  <Q 
r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )  <->  E. r  e.  Q.  ( q  <Q 
r  /\  ( r  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
6862, 67sylib 121 . . . 4  |-  ( ( A  <P  B  /\  ( q  e.  Q.  /\ 
E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  ->  E. r  e.  Q.  ( q  <Q  r  /\  ( r  e.  Q.  /\ 
E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
691ltexprlemell 7560 . . . . . 6  |-  ( r  e.  ( 1st `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )
7069anbi2i 454 . . . . 5  |-  ( ( q  <Q  r  /\  r  e.  ( 1st `  C ) )  <->  ( q  <Q  r  /\  ( r  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
7170rexbii 2477 . . . 4  |-  ( E. r  e.  Q.  (
q  <Q  r  /\  r  e.  ( 1st `  C
) )  <->  E. r  e.  Q.  ( q  <Q 
r  /\  ( r  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
7268, 71sylibr 133 . . 3  |-  ( ( A  <P  B  /\  ( q  e.  Q.  /\ 
E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )  ->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )
733, 72sylanr2 403 . 2  |-  ( ( A  <P  B  /\  ( q  e.  Q.  /\  q  e.  ( 1st `  C ) ) )  ->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )
74733impb 1194 1  |-  ( ( A  <P  B  /\  q  e.  Q.  /\  q  e.  ( 1st `  C
) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348   E.wex 1485    e. wcel 2141   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242    +Q cplq 7244    <Q cltq 7247   P.cnp 7253    <P cltp 7257
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-ltnqqs 7315  df-inp 7428  df-iltp 7432
This theorem is referenced by:  ltexprlemrnd  7567
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