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Theorem addlocpr 7308
Description: Locatedness of addition on positive reals. Lemma 11.16 in [BauerTaylor], p. 53. The proof in BauerTaylor relies on signed rationals, so we replace it with another proof which applies prarloc 7275 to both  A and  B, and uses nqtri3or 7168 rather than prloc 7263 to decide whether  q is too big to be in the lower cut of  A  +P.  B (and deduce that if it is, then  r must be in the upper cut). What the two proofs have in common is that they take the difference between  q and  r to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.)
Assertion
Ref Expression
addlocpr  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. q  e.  Q.  A. r  e.  Q.  (
q  <Q  r  ->  (
q  e.  ( 1st `  ( A  +P.  B
) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
Distinct variable groups:    A, q, r    B, q, r

Proof of Theorem addlocpr
Dummy variables  d  e  h  p  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqq 7180 . . . . . 6  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( q  <Q  r  <->  E. p  e.  Q.  (
q  +Q  p )  =  r ) )
21biimpa 292 . . . . 5  |-  ( ( ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  ->  E. p  e.  Q.  ( q  +Q  p
)  =  r )
323adant1 982 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  ->  E. p  e.  Q.  ( q  +Q  p
)  =  r )
4 halfnqq 7182 . . . . . 6  |-  ( p  e.  Q.  ->  E. h  e.  Q.  ( h  +Q  h )  =  p )
54ad2antrl 479 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  ->  E. h  e.  Q.  ( h  +Q  h )  =  p )
6 prop 7247 . . . . . . . . . 10  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
7 prarloc 7275 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  h  e.  Q. )  ->  E. d  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( d  +Q  h ) )
86, 7sylan 279 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  h  e.  Q. )  ->  E. d  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( d  +Q  h ) )
98adantlr 466 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  h  e.  Q. )  ->  E. d  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( d  +Q  h ) )
1093ad2antl1 1126 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  h  e.  Q. )  ->  E. d  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( d  +Q  h ) )
1110ad2ant2r 498 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  ->  E. d  e.  ( 1st `  A
) E. u  e.  ( 2nd `  A
) u  <Q  (
d  +Q  h ) )
12 prop 7247 . . . . . . . . . . . . . 14  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
13 prarloc 7275 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  h  e.  Q. )  ->  E. e  e.  ( 1st `  B ) E. t  e.  ( 2nd `  B ) t  <Q  ( e  +Q  h ) )
1412, 13sylan 279 . . . . . . . . . . . . 13  |-  ( ( B  e.  P.  /\  h  e.  Q. )  ->  E. e  e.  ( 1st `  B ) E. t  e.  ( 2nd `  B ) t  <Q  ( e  +Q  h ) )
1514adantll 465 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  h  e.  Q. )  ->  E. e  e.  ( 1st `  B ) E. t  e.  ( 2nd `  B ) t  <Q  ( e  +Q  h ) )
16153ad2antl1 1126 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  h  e.  Q. )  ->  E. e  e.  ( 1st `  B ) E. t  e.  ( 2nd `  B ) t  <Q  ( e  +Q  h ) )
1716ad2ant2r 498 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  ->  E. e  e.  ( 1st `  B
) E. t  e.  ( 2nd `  B
) t  <Q  (
e  +Q  h ) )
1817adantr 272 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  ->  E. e  e.  ( 1st `  B ) E. t  e.  ( 2nd `  B ) t  <Q 
( e  +Q  h
) )
19 simpll1 1003 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  ->  ( A  e.  P.  /\  B  e. 
P. ) )
2019ad2antrr 477 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  -> 
( A  e.  P.  /\  B  e.  P. )
)
2120simpld 111 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  ->  A  e.  P. )
2220simprd 113 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  ->  B  e.  P. )
23 simpll3 1005 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  ->  q  <Q  r )
2423ad2antrr 477 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  -> 
q  <Q  r )
25 simplrl 507 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  ->  h  e.  Q. )
2625adantr 272 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  ->  h  e.  Q. )
27 simplrr 508 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  ->  ( q  +Q  p )  =  r )
28 oveq2 5748 . . . . . . . . . . . . . . . 16  |-  ( ( h  +Q  h )  =  p  ->  (
q  +Q  ( h  +Q  h ) )  =  ( q  +Q  p ) )
2928eqeq1d 2124 . . . . . . . . . . . . . . 15  |-  ( ( h  +Q  h )  =  p  ->  (
( q  +Q  (
h  +Q  h ) )  =  r  <->  ( q  +Q  p )  =  r ) )
3029ad2antll 480 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  ->  ( (
q  +Q  ( h  +Q  h ) )  =  r  <->  ( q  +Q  p )  =  r ) )
3127, 30mpbird 166 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  ->  ( q  +Q  ( h  +Q  h
) )  =  r )
3231ad2antrr 477 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  -> 
( q  +Q  (
h  +Q  h ) )  =  r )
33 simprll 509 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  -> 
d  e.  ( 1st `  A ) )
3433adantr 272 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  -> 
d  e.  ( 1st `  A ) )
35 simprlr 510 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  ->  u  e.  ( 2nd `  A ) )
3635adantr 272 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  ->  u  e.  ( 2nd `  A ) )
37 simplrr 508 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  ->  u  <Q  ( d  +Q  h ) )
38 simprll 509 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  -> 
e  e.  ( 1st `  B ) )
39 simprlr 510 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  -> 
t  e.  ( 2nd `  B ) )
40 simprr 504 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  -> 
t  <Q  ( e  +Q  h ) )
4121, 22, 24, 26, 32, 34, 36, 37, 38, 39, 40addlocprlem 7307 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( ( e  e.  ( 1st `  B
)  /\  t  e.  ( 2nd `  B ) )  /\  t  <Q 
( e  +Q  h
) ) )  -> 
( q  e.  ( 1st `  ( A  +P.  B ) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) )
4241expr 370 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  /\  ( e  e.  ( 1st `  B )  /\  t  e.  ( 2nd `  B ) ) )  ->  (
t  <Q  ( e  +Q  h )  ->  (
q  e.  ( 1st `  ( A  +P.  B
) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
4342rexlimdvva 2532 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  -> 
( E. e  e.  ( 1st `  B
) E. t  e.  ( 2nd `  B
) t  <Q  (
e  +Q  h )  ->  ( q  e.  ( 1st `  ( A  +P.  B ) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
4418, 43mpd 13 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( ( d  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) )  /\  u  <Q  ( d  +Q  h
) ) )  -> 
( q  e.  ( 1st `  ( A  +P.  B ) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) )
4544expr 370 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  /\  ( d  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  ->  (
u  <Q  ( d  +Q  h )  ->  (
q  e.  ( 1st `  ( A  +P.  B
) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
4645rexlimdvva 2532 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  ->  ( E. d  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A
) u  <Q  (
d  +Q  h )  ->  ( q  e.  ( 1st `  ( A  +P.  B ) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
4711, 46mpd 13 . . . . 5  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  /\  (
h  e.  Q.  /\  ( h  +Q  h
)  =  p ) )  ->  ( q  e.  ( 1st `  ( A  +P.  B ) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) )
485, 47rexlimddv 2529 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  /\  ( p  e. 
Q.  /\  ( q  +Q  p )  =  r ) )  ->  (
q  e.  ( 1st `  ( A  +P.  B
) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) )
493, 48rexlimddv 2529 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  ->  ( q  e.  ( 1st `  ( A  +P.  B ) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) )
50493expia 1166 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  ->  ( q  <Q  r  ->  ( q  e.  ( 1st `  ( A  +P.  B ) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
5150ralrimivva 2489 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. q  e.  Q.  A. r  e.  Q.  (
q  <Q  r  ->  (
q  e.  ( 1st `  ( A  +P.  B
) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463   A.wral 2391   E.wrex 2392   <.cop 3498   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052    +Q cplq 7054    <Q cltq 7057   P.cnp 7063    +P. cpp 7065
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240
This theorem is referenced by:  addclpr  7309
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