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Theorem ltexprlemopl 7150
Description: The lower cut of our constructed difference is open. Lemma for ltexpri 7162. (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 7147 . . . 4  |-  ( q  e.  ( 1st `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
32simprbi 269 . . 3  |-  ( q  e.  ( 1st `  C
)  ->  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )
4 19.42v 1834 . . . . . . . 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 1834 . . . . . . . . 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 445 . . . . . . . 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 182 . . . . . . 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 7054 . . . . . . . . . . . . . 14  |-  <P  C_  ( P.  X.  P. )
98brel 4486 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
109simprd 112 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  B  e. 
P. )
11 prop 7024 . . . . . . . . . . . . 13  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
12 prnmaxl 7037 . . . . . . . . . . . . 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 277 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  ( y  +Q  q
)  e.  ( 1st `  B ) )  ->  E. s  e.  ( 1st `  B ) ( y  +Q  q ) 
<Q  s )
1410, 13sylan 277 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( y  +Q  q
)  e.  ( 1st `  B ) )  ->  E. s  e.  ( 1st `  B ) ( y  +Q  q ) 
<Q  s )
1514adantrl 462 . . . . . . . . . 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 462 . . . . . . . . 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 110 . . . . . . . . . . . . . . 15  |-  ( A 
<P  B  ->  A  e. 
P. )
1817ad2antrr 472 . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . 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 110 . . . . . . . . . . . . . 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 7024 . . . . . . . . . . . . . . 15  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
22 elprnqu 7031 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2321, 22sylan 277 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2418, 20, 23syl2anc 403 . . . . . . . . . . . . 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 502 . . . . . . . . . . . . 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 6956 . . . . . . . . . . . . 13  |-  ( ( y  e.  Q.  /\  q  e.  Q. )  ->  y  <Q  ( y  +Q  q ) )
2724, 25, 26syl2anc 403 . . . . . . . . . . . 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 499 . . . . . . . . . . . 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 6947 . . . . . . . . . . . . 13  |-  <Q  Or  Q.
30 ltrelnq 6914 . . . . . . . . . . . . 13  |-  <Q  C_  ( Q.  X.  Q. )
3129, 30sotri 4822 . . . . . . . . . . . 12  |-  ( ( y  <Q  ( y  +Q  q )  /\  (
y  +Q  q ) 
<Q  s )  ->  y  <Q  s )
3227, 28, 31syl2anc 403 . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 498 . . . . . . . . . . . . 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 7030 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  s  e.  ( 1st `  B ) )  -> 
s  e.  Q. )
3611, 35sylan 277 . . . . . . . . . . . . 13  |-  ( ( B  e.  P.  /\  s  e.  ( 1st `  B ) )  -> 
s  e.  Q. )
3733, 34, 36syl2anc 403 . . . . . . . . . . . 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 6957 . . . . . . . . . . . 12  |-  ( ( y  e.  Q.  /\  s  e.  Q. )  ->  ( y  <Q  s  <->  E. r  e.  Q.  (
y  +Q  r )  =  s ) )
3924, 37, 38syl2anc 403 . . . . . . . . . . 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 145 . . . . . . . . . 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 503 . . . . . . . . . . . . . . 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 499 . . . . . . . . . . . . . . 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 3869 . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . 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 498 . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . 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 6949 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  Q.  /\  r  e.  Q.  /\  y  e.  Q. )  ->  (
q  <Q  r  <->  ( y  +Q  q )  <Q  (
y  +Q  r ) ) )
4844, 45, 46, 47syl3anc 1174 . . . . . . . . . . . . . 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 165 . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 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 502 . . . . . . . . . . . . . . 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 2164 . . . . . . . . . . . . . 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 300 . . . . . . . . . . . . 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 303 . . . . . . . . . . . 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 367 . . . . . . . . . . 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 2475 . . . . . . . . . 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 2493 . . . . . . . 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 1536 . . . . . . 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 133 . . . . . 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 2642 . . . . . 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 132 . . . . 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 1834 . . . . . . 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 1834 . . . . . . . 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 445 . . . . . . 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 182 . . . . . 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 2385 . . . . 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 120 . . . 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 7147 . . . . . 6  |-  ( r  e.  ( 1st `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )
7069anbi2i 445 . . . . 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 2385 . . . 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 132 . . 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 397 . 2  |-  ( ( A  <P  B  /\  ( q  e.  Q.  /\  q  e.  ( 1st `  C ) ) )  ->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )
74733impb 1139 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 102    <-> wb 103    /\ w3a 924    = wceq 1289   E.wex 1426    e. wcel 1438   E.wrex 2360   {crab 2363   <.cop 3447   class class class wbr 3843   ` cfv 5010  (class class class)co 5644   1stc1st 5901   2ndc2nd 5902   Q.cnq 6829    +Q cplq 6831    <Q cltq 6834   P.cnp 6840    <P cltp 6844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-eprel 4114  df-id 4118  df-po 4121  df-iso 4122  df-iord 4191  df-on 4193  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-irdg 6127  df-1o 6173  df-oadd 6177  df-omul 6178  df-er 6282  df-ec 6284  df-qs 6288  df-ni 6853  df-pli 6854  df-mi 6855  df-lti 6856  df-plpq 6893  df-mpq 6894  df-enq 6896  df-nqqs 6897  df-plqqs 6898  df-mqqs 6899  df-1nqqs 6900  df-ltnqqs 6902  df-inp 7015  df-iltp 7019
This theorem is referenced by:  ltexprlemrnd  7154
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