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Theorem ltexprlemopl 7521
Description: The lower cut of our constructed difference is open. Lemma for ltexpri 7533. (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 7518 . . . 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 1886 . . . . . . . 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 1886 . . . . . . . . 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 453 . . . . . . . 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 7425 . . . . . . . . . . . . . 14  |-  <P  C_  ( P.  X.  P. )
98brel 4638 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
109simprd 113 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  B  e. 
P. )
11 prop 7395 . . . . . . . . . . . . 13  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
12 prnmaxl 7408 . . . . . . . . . . . . 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 470 . . . . . . . . . 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 470 . . . . . . . . 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 480 . . . . . . . . . . . . . 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 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
) )  ->  (
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 7395 . . . . . . . . . . . . . . 15  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
22 elprnqu 7402 . . . . . . . . . . . . . . 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 525 . . . . . . . . . . . . 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 7327 . . . . . . . . . . . . 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 522 . . . . . . . . . . . 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 7318 . . . . . . . . . . . . 13  |-  <Q  Or  Q.
30 ltrelnq 7285 . . . . . . . . . . . . 13  |-  <Q  C_  ( Q.  X.  Q. )
3129, 30sotri 4981 . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 521 . . . . . . . . . . . . 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 7401 . . . . . . . . . . . . . 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 7328 . . . . . . . . . . . 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 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 ) )  ->  ( y  +Q  q )  <Q  s
)
42 simprr 522 . . . . . . . . . . . . . . 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 3992 . . . . . . . . . . . . . 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 521 . . . . . . . . . . . . . . 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 7320 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  Q.  /\  r  e.  Q.  /\  y  e.  Q. )  ->  (
q  <Q  r  <->  ( y  +Q  q )  <Q  (
y  +Q  r ) ) )
4844, 45, 46, 47syl3anc 1220 . . . . . . . . . . . . . 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 525 . . . . . . . . . . . . . . 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 2234 . . . . . . . . . . . . . 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 2559 . . . . . . . . . 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 2579 . . . . . . . 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 1580 . . . . . . 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 2735 . . . . . 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 1886 . . . . . . 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 1886 . . . . . . . 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 453 . . . . . . 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 2464 . . . . 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 7518 . . . . . 6  |-  ( r  e.  ( 1st `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )
7069anbi2i 453 . . . . 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 2464 . . . 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 1181 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 963    = wceq 1335   E.wex 1472    e. wcel 2128   E.wrex 2436   {crab 2439   <.cop 3563   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   1stc1st 6086   2ndc2nd 6087   Q.cnq 7200    +Q cplq 7202    <Q cltq 7205   P.cnp 7211    <P cltp 7215
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-ltnqqs 7273  df-inp 7386  df-iltp 7390
This theorem is referenced by:  ltexprlemrnd  7525
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