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Theorem addlocpr 6692
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 6659 to both  A and  B, and uses nqtri3or 6552 rather than prloc 6647 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 6564 . . . . . 6  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( q  <Q  r  <->  E. p  e.  Q.  (
q  +Q  p )  =  r ) )
21biimpa 284 . . . . 5  |-  ( ( ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  ->  E. p  e.  Q.  ( q  +Q  p
)  =  r )
323adant1 933 . . . 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 6566 . . . . . 6  |-  ( p  e.  Q.  ->  E. h  e.  Q.  ( h  +Q  h )  =  p )
54ad2antrl 467 . . . . 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 6631 . . . . . . . . . 10  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
7 prarloc 6659 . . . . . . . . . 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 271 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  h  e.  Q. )  ->  E. d  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( d  +Q  h ) )
98adantlr 454 . . . . . . . 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 1077 . . . . . . 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 486 . . . . . 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 6631 . . . . . . . . . . . . . 14  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
13 prarloc 6659 . . . . . . . . . . . . . 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 271 . . . . . . . . . . . . 13  |-  ( ( B  e.  P.  /\  h  e.  Q. )  ->  E. e  e.  ( 1st `  B ) E. t  e.  ( 2nd `  B ) t  <Q  ( e  +Q  h ) )
1514adantll 453 . . . . . . . . . . . 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 1077 . . . . . . . . . . 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 486 . . . . . . . . . 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 265 . . . . . . . . 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 954 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 109 . . . . . . . . . . . 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 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
) ) )  ->  B  e.  P. )
23 simpll3 956 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 495 . . . . . . . . . . . . 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 265 . . . . . . . . . . . 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 496 . . . . . . . . . . . . . 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 5548 . . . . . . . . . . . . . . . 16  |-  ( ( h  +Q  h )  =  p  ->  (
q  +Q  ( h  +Q  h ) )  =  ( q  +Q  p ) )
2928eqeq1d 2064 . . . . . . . . . . . . . . 15  |-  ( ( h  +Q  h )  =  p  ->  (
( q  +Q  (
h  +Q  h ) )  =  r  <->  ( q  +Q  p )  =  r ) )
3029ad2antll 468 . . . . . . . . . . . . . 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 160 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 497 . . . . . . . . . . . . 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 265 . . . . . . . . . . . 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 498 . . . . . . . . . . . . 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 265 . . . . . . . . . . . 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 496 . . . . . . . . . . . 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 497 . . . . . . . . . . . 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 498 . . . . . . . . . . . 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 492 . . . . . . . . . . . 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 6691 . . . . . . . . . . 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 361 . . . . . . . . . 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 2457 . . . . . . . . 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 361 . . . . . . 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 2457 . . . . . 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 2454 . . . 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 2454 . . 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 1117 . 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 2418 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 101    <-> wb 102    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   A.wral 2323   E.wrex 2324   <.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. cpp 6449
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-2o 6033  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-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624
This theorem is referenced by:  addclpr  6693
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