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Theorem ltexprlemopu 6759
Description: The upper cut of our constructed difference is open. Lemma for ltexpri 6769. (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
ltexprlemopu  |-  ( ( A  <P  B  /\  r  e.  Q.  /\  r  e.  ( 2nd `  C
) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  C ) ) )
Distinct variable groups:    x, y, q, r, A    x, B, y, q, r    x, C, y, q, r

Proof of Theorem ltexprlemopu
Dummy variables  s  t are mutually distinct and 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 ) ) } >.
21ltexprlemelu 6755 . . . 4  |-  ( r  e.  ( 2nd `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
32simprbi 264 . . 3  |-  ( r  e.  ( 2nd `  C
)  ->  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) )
4 19.42v 1802 . . . . . . . 8  |-  ( E. y ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  <->  ( A  <P  B  /\  E. y
( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) ) )
5 19.42v 1802 . . . . . . . . 9  |-  ( E. y ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) )  <->  ( r  e.  Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
65anbi2i 438 . . . . . . . 8  |-  ( ( A  <P  B  /\  E. y ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  <->  ( A  <P  B  /\  ( r  e.  Q.  /\  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) ) )
74, 6bitri 177 . . . . . . 7  |-  ( E. y ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  <->  ( A  <P  B  /\  ( r  e.  Q.  /\  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) ) )
8 ltrelpr 6661 . . . . . . . . . . . . . . 15  |-  <P  C_  ( P.  X.  P. )
98brel 4420 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
109simprd 111 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  B  e. 
P. )
11 prop 6631 . . . . . . . . . . . . 13  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
1210, 11syl 14 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
13 prnminu 6645 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  ( y  +Q  r
)  e.  ( 2nd `  B ) )  ->  E. s  e.  ( 2nd `  B ) s 
<Q  ( y  +Q  r
) )
1412, 13sylan 271 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( y  +Q  r
)  e.  ( 2nd `  B ) )  ->  E. s  e.  ( 2nd `  B ) s 
<Q  ( y  +Q  r
) )
1514adantrl 455 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) )  ->  E. s  e.  ( 2nd `  B
) s  <Q  (
y  +Q  r ) )
1615adantrl 455 . . . . . . . . 9  |-  ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  ->  E. s  e.  ( 2nd `  B ) s 
<Q  ( y  +Q  r
) )
17 ltdfpr 6662 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  E. t  e.  Q.  ( t  e.  ( 2nd `  A
)  /\  t  e.  ( 1st `  B ) ) ) )
1817biimpd 136 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  ->  E. t  e.  Q.  ( t  e.  ( 2nd `  A )  /\  t  e.  ( 1st `  B ) ) ) )
199, 18mpcom 36 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  E. t  e.  Q.  ( t  e.  ( 2nd `  A
)  /\  t  e.  ( 1st `  B ) ) )
2019ad2antrr 465 . . . . . . . . . . . 12  |-  ( ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  ->  E. t  e.  Q.  ( t  e.  ( 2nd `  A
)  /\  t  e.  ( 1st `  B ) ) )
219simpld 109 . . . . . . . . . . . . . . . 16  |-  ( A 
<P  B  ->  A  e. 
P. )
2221ad2antrr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  ->  A  e.  P. )
2322adantr 265 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
t  e.  Q.  /\  ( t  e.  ( 2nd `  A )  /\  t  e.  ( 1st `  B ) ) ) )  ->  A  e.  P. )
24 simplrr 496 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  ->  (
y  e.  ( 1st `  A )  /\  (
y  +Q  r )  e.  ( 2nd `  B
) ) )
2524simpld 109 . . . . . . . . . . . . . . 15  |-  ( ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  ->  y  e.  ( 1st `  A
) )
2625adantr 265 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
t  e.  Q.  /\  ( t  e.  ( 2nd `  A )  /\  t  e.  ( 1st `  B ) ) ) )  -> 
y  e.  ( 1st `  A ) )
27 simprrl 499 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
t  e.  Q.  /\  ( t  e.  ( 2nd `  A )  /\  t  e.  ( 1st `  B ) ) ) )  -> 
t  e.  ( 2nd `  A ) )
28 prop 6631 . . . . . . . . . . . . . . 15  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
29 prltlu 6643 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A )  /\  t  e.  ( 2nd `  A
) )  ->  y  <Q  t )
3028, 29syl3an1 1179 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  y  e.  ( 1st `  A )  /\  t  e.  ( 2nd `  A
) )  ->  y  <Q  t )
3123, 26, 27, 30syl3anc 1146 . . . . . . . . . . . . 13  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
t  e.  Q.  /\  ( t  e.  ( 2nd `  A )  /\  t  e.  ( 1st `  B ) ) ) )  -> 
y  <Q  t )
32 simplll 493 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
t  e.  Q.  /\  ( t  e.  ( 2nd `  A )  /\  t  e.  ( 1st `  B ) ) ) )  ->  A  <P  B )
33 simprrr 500 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
t  e.  Q.  /\  ( t  e.  ( 2nd `  A )  /\  t  e.  ( 1st `  B ) ) ) )  -> 
t  e.  ( 1st `  B ) )
34 simplrl 495 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
t  e.  Q.  /\  ( t  e.  ( 2nd `  A )  /\  t  e.  ( 1st `  B ) ) ) )  -> 
s  e.  ( 2nd `  B ) )
35 prltlu 6643 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  t  e.  ( 1st `  B )  /\  s  e.  ( 2nd `  B
) )  ->  t  <Q  s )
3612, 35syl3an1 1179 . . . . . . . . . . . . . 14  |-  ( ( A  <P  B  /\  t  e.  ( 1st `  B )  /\  s  e.  ( 2nd `  B
) )  ->  t  <Q  s )
3732, 33, 34, 36syl3anc 1146 . . . . . . . . . . . . 13  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
t  e.  Q.  /\  ( t  e.  ( 2nd `  A )  /\  t  e.  ( 1st `  B ) ) ) )  -> 
t  <Q  s )
38 ltsonq 6554 . . . . . . . . . . . . . 14  |-  <Q  Or  Q.
39 ltrelnq 6521 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
4038, 39sotri 4748 . . . . . . . . . . . . 13  |-  ( ( y  <Q  t  /\  t  <Q  s )  -> 
y  <Q  s )
4131, 37, 40syl2anc 397 . . . . . . . . . . . 12  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
t  e.  Q.  /\  ( t  e.  ( 2nd `  A )  /\  t  e.  ( 1st `  B ) ) ) )  -> 
y  <Q  s )
4220, 41rexlimddv 2454 . . . . . . . . . . 11  |-  ( ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  ->  y  <Q  s )
43 elprnql 6637 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
4428, 43sylan 271 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
4522, 25, 44syl2anc 397 . . . . . . . . . . . 12  |-  ( ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  ->  y  e.  Q. )
46 elprnqu 6638 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  s  e.  ( 2nd `  B ) )  -> 
s  e.  Q. )
4712, 46sylan 271 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  s  e.  ( 2nd `  B ) )  -> 
s  e.  Q. )
4847ad2ant2r 486 . . . . . . . . . . . 12  |-  ( ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  ->  s  e.  Q. )
49 ltexnqq 6564 . . . . . . . . . . . 12  |-  ( ( y  e.  Q.  /\  s  e.  Q. )  ->  ( y  <Q  s  <->  E. q  e.  Q.  (
y  +Q  q )  =  s ) )
5045, 48, 49syl2anc 397 . . . . . . . . . . 11  |-  ( ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  ->  (
y  <Q  s  <->  E. q  e.  Q.  ( y  +Q  q )  =  s ) )
5142, 50mpbid 139 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  ->  E. q  e.  Q.  ( y  +Q  q )  =  s )
52 simprr 492 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  ( y  +Q  q )  =  s )
53 simplrr 496 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  s  <Q  ( y  +Q  r ) )
5452, 53eqbrtrd 3812 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  ( y  +Q  q )  <Q  (
y  +Q  r ) )
55 simprl 491 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  q  e.  Q. )
56 simplrl 495 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  ->  r  e.  Q. )
5756adantr 265 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  r  e.  Q. )
5845adantr 265 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  y  e.  Q. )
59 ltanqg 6556 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  Q.  /\  r  e.  Q.  /\  y  e.  Q. )  ->  (
q  <Q  r  <->  ( y  +Q  q )  <Q  (
y  +Q  r ) ) )
6055, 57, 58, 59syl3anc 1146 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  ( q  <Q  r  <->  ( y  +Q  q )  <Q  (
y  +Q  r ) ) )
6154, 60mpbird 160 . . . . . . . . . . . . 13  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  q  <Q  r )
6225adantr 265 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  y  e.  ( 1st `  A ) )
63 simplrl 495 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  s  e.  ( 2nd `  B ) )
6452, 63eqeltrd 2130 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  ( y  +Q  q )  e.  ( 2nd `  B ) )
6562, 64jca 294 . . . . . . . . . . . . 13  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) )
6661, 55, 65jca32 297 . . . . . . . . . . . 12  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  (
q  e.  Q.  /\  ( y  +Q  q
)  =  s ) )  ->  ( q  <Q  r  /\  ( q  e.  Q.  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  q )  e.  ( 2nd `  B
) ) ) ) )
6766expr 361 . . . . . . . . . . 11  |-  ( ( ( ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  /\  q  e.  Q. )  ->  (
( y  +Q  q
)  =  s  -> 
( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) ) )
6867reximdva 2438 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  ->  ( E. q  e.  Q.  ( y  +Q  q
)  =  s  ->  E. q  e.  Q.  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) ) )
6951, 68mpd 13 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  (
y  +Q  r ) ) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
7016, 69rexlimddv 2454 . . . . . . . 8  |-  ( ( A  <P  B  /\  ( r  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  ->  E. q  e.  Q.  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
7170eximi 1507 . . . . . . 7  |-  ( E. y ( A  <P  B  /\  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  ->  E. y E. q  e. 
Q.  ( q  <Q 
r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
727, 71sylbir 129 . . . . . 6  |-  ( ( A  <P  B  /\  ( r  e.  Q.  /\ 
E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  ->  E. y E. q  e. 
Q.  ( q  <Q 
r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
73 rexcom4 2594 . . . . . 6  |-  ( E. q  e.  Q.  E. y ( q  <Q 
r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )  <->  E. y E. q  e.  Q.  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
7472, 73sylibr 141 . . . . 5  |-  ( ( A  <P  B  /\  ( r  e.  Q.  /\ 
E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  ->  E. q  e.  Q.  E. y ( q  <Q 
r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
75 19.42v 1802 . . . . . . 7  |-  ( E. y ( q  <Q 
r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )  <->  ( q  <Q  r  /\  E. y
( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
76 19.42v 1802 . . . . . . . 8  |-  ( E. y ( q  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) )  <->  ( q  e.  Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )
7776anbi2i 438 . . . . . . 7  |-  ( ( q  <Q  r  /\  E. y ( q  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )  <->  ( q  <Q  r  /\  ( q  e.  Q.  /\  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
7875, 77bitri 177 . . . . . 6  |-  ( E. y ( q  <Q 
r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )  <->  ( q  <Q  r  /\  ( q  e.  Q.  /\  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
7978rexbii 2348 . . . . 5  |-  ( E. q  e.  Q.  E. y ( q  <Q 
r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )  <->  E. q  e.  Q.  ( q  <Q 
r  /\  ( q  e.  Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
8074, 79sylib 131 . . . 4  |-  ( ( A  <P  B  /\  ( r  e.  Q.  /\ 
E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  ->  E. q  e.  Q.  ( q  <Q  r  /\  ( q  e.  Q.  /\ 
E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
811ltexprlemelu 6755 . . . . . 6  |-  ( q  e.  ( 2nd `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )
8281anbi2i 438 . . . . 5  |-  ( ( q  <Q  r  /\  q  e.  ( 2nd `  C ) )  <->  ( q  <Q  r  /\  ( q  e.  Q.  /\  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
8382rexbii 2348 . . . 4  |-  ( E. q  e.  Q.  (
q  <Q  r  /\  q  e.  ( 2nd `  C
) )  <->  E. q  e.  Q.  ( q  <Q 
r  /\  ( q  e.  Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
8480, 83sylibr 141 . . 3  |-  ( ( A  <P  B  /\  ( r  e.  Q.  /\ 
E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )  ->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) )
853, 84sylanr2 391 . 2  |-  ( ( A  <P  B  /\  ( r  e.  Q.  /\  r  e.  ( 2nd `  C ) ) )  ->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) )
86853impb 1111 1  |-  ( ( A  <P  B  /\  r  e.  Q.  /\  r  e.  ( 2nd `  C
) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259   E.wex 1397    e. wcel 1409   E.wrex 2324   {crab 2327   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    +Q cplq 6438    <Q cltq 6441   P.cnp 6447    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-ltnqqs 6509  df-inp 6622  df-iltp 6626
This theorem is referenced by:  ltexprlemrnd  6761
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