ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltexprlemopu Unicode version

Theorem ltexprlemopu 7430
Description: The upper cut of our constructed difference is open. Lemma for ltexpri 7440. (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 7426 . . . 4  |-  ( r  e.  ( 2nd `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
32simprbi 273 . . 3  |-  ( r  e.  ( 2nd `  C
)  ->  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) )
4 19.42v 1878 . . . . . . . 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 1878 . . . . . . . . 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 452 . . . . . . . 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 183 . . . . . . 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 7332 . . . . . . . . . . . . . . 15  |-  <P  C_  ( P.  X.  P. )
98brel 4594 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
109simprd 113 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  B  e. 
P. )
11 prop 7302 . . . . . . . . . . . . 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 7316 . . . . . . . . . . . 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 281 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( y  +Q  r
)  e.  ( 2nd `  B ) )  ->  E. s  e.  ( 2nd `  B ) s 
<Q  ( y  +Q  r
) )
1514adantrl 469 . . . . . . . . . 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 469 . . . . . . . . 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 7333 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  E. t  e.  Q.  ( t  e.  ( 2nd `  A
)  /\  t  e.  ( 1st `  B ) ) ) )
1817biimpd 143 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 111 . . . . . . . . . . . . . . . 16  |-  ( A 
<P  B  ->  A  e. 
P. )
2221ad2antrr 479 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 525 . . . . . . . . . . . . . . . 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 111 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 528 . . . . . . . . . . . . . 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 7302 . . . . . . . . . . . . . . 15  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
29 prltlu 7314 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A )  /\  t  e.  ( 2nd `  A
) )  ->  y  <Q  t )
3028, 29syl3an1 1249 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  y  e.  ( 1st `  A )  /\  t  e.  ( 2nd `  A
) )  ->  y  <Q  t )
3123, 26, 27, 30syl3anc 1216 . . . . . . . . . . . . 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 522 . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . 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 524 . . . . . . . . . . . . . 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 7314 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  t  e.  ( 1st `  B )  /\  s  e.  ( 2nd `  B
) )  ->  t  <Q  s )
3612, 35syl3an1 1249 . . . . . . . . . . . . . 14  |-  ( ( A  <P  B  /\  t  e.  ( 1st `  B )  /\  s  e.  ( 2nd `  B
) )  ->  t  <Q  s )
3732, 33, 34, 36syl3anc 1216 . . . . . . . . . . . . 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 7225 . . . . . . . . . . . . . 14  |-  <Q  Or  Q.
39 ltrelnq 7192 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
4038, 39sotri 4937 . . . . . . . . . . . . 13  |-  ( ( y  <Q  t  /\  t  <Q  s )  -> 
y  <Q  s )
4131, 37, 40syl2anc 408 . . . . . . . . . . . 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 2554 . . . . . . . . . . 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 7308 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
4428, 43sylan 281 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
4522, 25, 44syl2anc 408 . . . . . . . . . . . 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 7309 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  s  e.  ( 2nd `  B ) )  -> 
s  e.  Q. )
4712, 46sylan 281 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  s  e.  ( 2nd `  B ) )  -> 
s  e.  Q. )
4847ad2ant2r 500 . . . . . . . . . . . 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 7235 . . . . . . . . . . . 12  |-  ( ( y  e.  Q.  /\  s  e.  Q. )  ->  ( y  <Q  s  <->  E. q  e.  Q.  (
y  +Q  q )  =  s ) )
5045, 48, 49syl2anc 408 . . . . . . . . . . 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 146 . . . . . . . . . 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 521 . . . . . . . . . . . . . . 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 525 . . . . . . . . . . . . . . 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 3953 . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . 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 524 . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . 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 7227 . . . . . . . . . . . . . . 15  |-  ( ( q  e.  Q.  /\  r  e.  Q.  /\  y  e.  Q. )  ->  (
q  <Q  r  <->  ( y  +Q  q )  <Q  (
y  +Q  r ) ) )
6055, 57, 58, 59syl3anc 1216 . . . . . . . . . . . . . 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 166 . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 524 . . . . . . . . . . . . . . 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 2216 . . . . . . . . . . . . . 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 304 . . . . . . . . . . . . 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 308 . . . . . . . . . . . 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 372 . . . . . . . . . . 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 2534 . . . . . . . . . 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 2554 . . . . . . . 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 1579 . . . . . . 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 134 . . . . . 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 2709 . . . . . 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 133 . . . . 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 1878 . . . . . . 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 1878 . . . . . . . 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 452 . . . . . . 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 183 . . . . . 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 2442 . . . . 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 121 . . . 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 7426 . . . . . 6  |-  ( q  e.  ( 2nd `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )
8281anbi2i 452 . . . . 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 2442 . . . 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 133 . . 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 402 . 2  |-  ( ( A  <P  B  /\  ( r  e.  Q.  /\  r  e.  ( 2nd `  C ) ) )  ->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) )
86853impb 1177 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 103    <-> wb 104    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3932   ` cfv 5126  (class class class)co 5777   1stc1st 6039   2ndc2nd 6040   Q.cnq 7107    +Q cplq 7109    <Q cltq 7112   P.cnp 7118    <P cltp 7122
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-iinf 4505
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-eprel 4214  df-id 4218  df-po 4221  df-iso 4222  df-iord 4291  df-on 4293  df-suc 4296  df-iom 4508  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-recs 6205  df-irdg 6270  df-1o 6316  df-oadd 6320  df-omul 6321  df-er 6432  df-ec 6434  df-qs 6438  df-ni 7131  df-pli 7132  df-mi 7133  df-lti 7134  df-plpq 7171  df-mpq 7172  df-enq 7174  df-nqqs 7175  df-plqqs 7176  df-mqqs 7177  df-1nqqs 7178  df-ltnqqs 7180  df-inp 7293  df-iltp 7297
This theorem is referenced by:  ltexprlemrnd  7432
  Copyright terms: Public domain W3C validator