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Theorem ltexprlemopl 7630
Description: The lower cut of our constructed difference is open. Lemma for ltexpri 7642. (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 7627 . . . 4  |-  ( q  e.  ( 1st `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
32simprbi 275 . . 3  |-  ( q  e.  ( 1st `  C
)  ->  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )
4 19.42v 1918 . . . . . . . 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 1918 . . . . . . . . 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 457 . . . . . . . 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 184 . . . . . . 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 7534 . . . . . . . . . . . . . 14  |-  <P  C_  ( P.  X.  P. )
98brel 4696 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
109simprd 114 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  B  e. 
P. )
11 prop 7504 . . . . . . . . . . . . 13  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
12 prnmaxl 7517 . . . . . . . . . . . . 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 283 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  ( y  +Q  q
)  e.  ( 1st `  B ) )  ->  E. s  e.  ( 1st `  B ) ( y  +Q  q ) 
<Q  s )
1410, 13sylan 283 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( y  +Q  q
)  e.  ( 1st `  B ) )  ->  E. s  e.  ( 1st `  B ) ( y  +Q  q ) 
<Q  s )
1514adantrl 478 . . . . . . . . . 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 478 . . . . . . . . 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 112 . . . . . . . . . . . . . . 15  |-  ( A 
<P  B  ->  A  e. 
P. )
1817ad2antrr 488 . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . 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 112 . . . . . . . . . . . . . 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 7504 . . . . . . . . . . . . . . 15  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
22 elprnqu 7511 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2321, 22sylan 283 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2418, 20, 23syl2anc 411 . . . . . . . . . . . . 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 535 . . . . . . . . . . . . 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 7436 . . . . . . . . . . . . 13  |-  ( ( y  e.  Q.  /\  q  e.  Q. )  ->  y  <Q  ( y  +Q  q ) )
2724, 25, 26syl2anc 411 . . . . . . . . . . . 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 531 . . . . . . . . . . . 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 7427 . . . . . . . . . . . . 13  |-  <Q  Or  Q.
30 ltrelnq 7394 . . . . . . . . . . . . 13  |-  <Q  C_  ( Q.  X.  Q. )
3129, 30sotri 5042 . . . . . . . . . . . 12  |-  ( ( y  <Q  ( y  +Q  q )  /\  (
y  +Q  q ) 
<Q  s )  ->  y  <Q  s )
3227, 28, 31syl2anc 411 . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 529 . . . . . . . . . . . . 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 7510 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  s  e.  ( 1st `  B ) )  -> 
s  e.  Q. )
3611, 35sylan 283 . . . . . . . . . . . . 13  |-  ( ( B  e.  P.  /\  s  e.  ( 1st `  B ) )  -> 
s  e.  Q. )
3733, 34, 36syl2anc 411 . . . . . . . . . . . 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 7437 . . . . . . . . . . . 12  |-  ( ( y  e.  Q.  /\  s  e.  Q. )  ->  ( y  <Q  s  <->  E. r  e.  Q.  (
y  +Q  r )  =  s ) )
3924, 37, 38syl2anc 411 . . . . . . . . . . 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 147 . . . . . . . . . 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 536 . . . . . . . . . . . . . . 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 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  r )  =  s )
4341, 42breqtrrd 4046 . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . 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 7429 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  Q.  /\  r  e.  Q.  /\  y  e.  Q. )  ->  (
q  <Q  r  <->  ( y  +Q  q )  <Q  (
y  +Q  r ) ) )
4844, 45, 46, 47syl3anc 1249 . . . . . . . . . . . . . 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 167 . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . 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 2266 . . . . . . . . . . . . . 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 306 . . . . . . . . . . . . 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 310 . . . . . . . . . . . 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 375 . . . . . . . . . . 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 2592 . . . . . . . . . 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 2612 . . . . . . . 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 1611 . . . . . . 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 135 . . . . . 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 2775 . . . . . 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 134 . . . . 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 1918 . . . . . . 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 1918 . . . . . . . 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 457 . . . . . . 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 184 . . . . . 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 2497 . . . . 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 122 . . . 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 7627 . . . . . 6  |-  ( r  e.  ( 1st `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )
7069anbi2i 457 . . . . 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 2497 . . . 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 134 . . 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 405 . 2  |-  ( ( A  <P  B  /\  ( q  e.  Q.  /\  q  e.  ( 1st `  C ) ) )  ->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )
74733impb 1201 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 104    <-> wb 105    /\ w3a 980    = wceq 1364   E.wex 1503    e. wcel 2160   E.wrex 2469   {crab 2472   <.cop 3610   class class class wbr 4018   ` cfv 5235  (class class class)co 5896   1stc1st 6163   2ndc2nd 6164   Q.cnq 7309    +Q cplq 7311    <Q cltq 7314   P.cnp 7320    <P cltp 7324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-1o 6441  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-pli 7334  df-mi 7335  df-lti 7336  df-plpq 7373  df-mpq 7374  df-enq 7376  df-nqqs 7377  df-plqqs 7378  df-mqqs 7379  df-1nqqs 7380  df-ltnqqs 7382  df-inp 7495  df-iltp 7499
This theorem is referenced by:  ltexprlemrnd  7634
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