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Theorem addlocpr 7513
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 7480 to both  A and  B, and uses nqtri3or 7373 rather than prloc 7468 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 7385 . . . . . 6  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( q  <Q  r  <->  E. p  e.  Q.  (
q  +Q  p )  =  r ) )
21biimpa 296 . . . . 5  |-  ( ( ( q  e.  Q.  /\  r  e.  Q. )  /\  q  <Q  r )  ->  E. p  e.  Q.  ( q  +Q  p
)  =  r )
323adant1 1015 . . . 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 7387 . . . . . 6  |-  ( p  e.  Q.  ->  E. h  e.  Q.  ( h  +Q  h )  =  p )
54ad2antrl 490 . . . . 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 7452 . . . . . . . . . 10  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
7 prarloc 7480 . . . . . . . . . 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 283 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  h  e.  Q. )  ->  E. d  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( d  +Q  h ) )
98adantlr 477 . . . . . . . 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 1159 . . . . . . 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 509 . . . . . 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 7452 . . . . . . . . . . . . . 14  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
13 prarloc 7480 . . . . . . . . . . . . . 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 283 . . . . . . . . . . . . 13  |-  ( ( B  e.  P.  /\  h  e.  Q. )  ->  E. e  e.  ( 1st `  B ) E. t  e.  ( 2nd `  B ) t  <Q  ( e  +Q  h ) )
1514adantll 476 . . . . . . . . . . . 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 1159 . . . . . . . . . . 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 509 . . . . . . . . . 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 276 . . . . . . . . 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 1036 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 112 . . . . . . . . . . . 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 114 . . . . . . . . . . . 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 1038 . . . . . . . . . . . . 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 488 . . . . . . . . . . . 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 535 . . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 536 . . . . . . . . . . . . . 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 5876 . . . . . . . . . . . . . . . 16  |-  ( ( h  +Q  h )  =  p  ->  (
q  +Q  ( h  +Q  h ) )  =  ( q  +Q  p ) )
2928eqeq1d 2186 . . . . . . . . . . . . . . 15  |-  ( ( h  +Q  h )  =  p  ->  (
( q  +Q  (
h  +Q  h ) )  =  r  <->  ( q  +Q  p )  =  r ) )
3029ad2antll 491 . . . . . . . . . . . . . 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 167 . . . . . . . . . . . . 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 488 . . . . . . . . . . . 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 537 . . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 538 . . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 536 . . . . . . . . . . . 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 537 . . . . . . . . . . . 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 538 . . . . . . . . . . . 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 531 . . . . . . . . . . . 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 7512 . . . . . . . . . . 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 375 . . . . . . . . . 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 2602 . . . . . . . . 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 375 . . . . . . 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 2602 . . . . . 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 2599 . . . 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 2599 . . 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 1205 . 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 2559 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 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   <.cop 3594   class class class wbr 4000   ` cfv 5211  (class class class)co 5868   1stc1st 6132   2ndc2nd 6133   Q.cnq 7257    +Q cplq 7259    <Q cltq 7262   P.cnp 7268    +P. cpp 7270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4285  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-1o 6410  df-2o 6411  df-oadd 6414  df-omul 6415  df-er 6528  df-ec 6530  df-qs 6534  df-ni 7281  df-pli 7282  df-mi 7283  df-lti 7284  df-plpq 7321  df-mpq 7322  df-enq 7324  df-nqqs 7325  df-plqqs 7326  df-mqqs 7327  df-1nqqs 7328  df-rq 7329  df-ltnqqs 7330  df-enq0 7401  df-nq0 7402  df-0nq0 7403  df-plq0 7404  df-mq0 7405  df-inp 7443  df-iplp 7445
This theorem is referenced by:  addclpr  7514
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