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Theorem addcanprlemu 7391
Description: Lemma for addcanprg 7392. (Contributed by Jim Kingdon, 25-Dec-2019.)
Assertion
Ref Expression
addcanprlemu  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 2nd `  B
)  C_  ( 2nd `  C ) )

Proof of Theorem addcanprlemu
Dummy variables  f  g  h  q  r  s  t  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7251 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnminu 7265 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  v  e.  ( 2nd `  B ) )  ->  E. r  e.  ( 2nd `  B ) r 
<Q  v )
31, 2sylan 281 . . . . . 6  |-  ( ( B  e.  P.  /\  v  e.  ( 2nd `  B ) )  ->  E. r  e.  ( 2nd `  B ) r 
<Q  v )
433ad2antl2 1129 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  v  e.  ( 2nd `  B ) )  ->  E. r  e.  ( 2nd `  B ) r  <Q  v )
54adantlr 468 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  ->  E. r  e.  ( 2nd `  B
) r  <Q  v
)
6 simprr 506 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  -> 
r  <Q  v )
7 ltexnqi 7185 . . . . . 6  |-  ( r 
<Q  v  ->  E. w  e.  Q.  ( r  +Q  w )  =  v )
86, 7syl 14 . . . . 5  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  ->  E. w  e.  Q.  ( r  +Q  w
)  =  v )
9 simprl 505 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  ->  w  e.  Q. )
10 halfnqq 7186 . . . . . . 7  |-  ( w  e.  Q.  ->  E. t  e.  Q.  ( t  +Q  t )  =  w )
119, 10syl 14 . . . . . 6  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  ->  E. t  e.  Q.  ( t  +Q  t )  =  w )
12 prop 7251 . . . . . . . . . . . . . 14  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
13 prarloc2 7280 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
1412, 13sylan 281 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
1514adantrr 470 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  ->  E. u  e.  ( 1st `  A
) ( u  +Q  t )  e.  ( 2nd `  A ) )
16153ad2antl1 1128 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  ->  E. u  e.  ( 1st `  A
) ( u  +Q  t )  e.  ( 2nd `  A ) )
1716adantlr 468 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  ->  E. u  e.  ( 1st `  A
) ( u  +Q  t )  e.  ( 2nd `  A ) )
1817adantlr 468 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  ->  E. u  e.  ( 1st `  A
) ( u  +Q  t )  e.  ( 2nd `  A ) )
1918adantlr 468 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  ->  E. u  e.  ( 1st `  A
) ( u  +Q  t )  e.  ( 2nd `  A ) )
2019adantlr 468 . . . . . . 7  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
21 simplll 507 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  -> 
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )
)
2221ad3antrrr 483 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. ) )
2322simp1d 978 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  A  e.  P. )
2422simp2d 979 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  B  e.  P. )
25 addclpr 7313 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
2623, 24, 25syl2anc 408 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  ( A  +P.  B )  e. 
P. )
27 prop 7251 . . . . . . . . . . 11  |-  ( ( A  +P.  B )  e.  P.  ->  <. ( 1st `  ( A  +P.  B ) ) ,  ( 2nd `  ( A  +P.  B ) )
>.  e.  P. )
2826, 27syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  <. ( 1st `  ( A  +P.  B ) ) ,  ( 2nd `  ( A  +P.  B ) )
>.  e.  P. )
2923, 12syl 14 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
30 simprl 505 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  u  e.  ( 1st `  A
) )
31 elprnql 7257 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
3229, 30, 31syl2anc 408 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  u  e.  Q. )
33 simplrl 509 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  t  e.  Q. )
34 addclnq 7151 . . . . . . . . . . . 12  |-  ( ( u  e.  Q.  /\  t  e.  Q. )  ->  ( u  +Q  t
)  e.  Q. )
3532, 33, 34syl2anc 408 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
u  +Q  t )  e.  Q. )
3624, 1syl 14 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
37 simprl 505 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  -> 
r  e.  ( 2nd `  B ) )
3837ad3antrrr 483 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  r  e.  ( 2nd `  B
) )
39 elprnqu 7258 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  r  e.  ( 2nd `  B ) )  -> 
r  e.  Q. )
4036, 38, 39syl2anc 408 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  r  e.  Q. )
41 addclnq 7151 . . . . . . . . . . 11  |-  ( ( ( u  +Q  t
)  e.  Q.  /\  r  e.  Q. )  ->  ( ( u  +Q  t )  +Q  r
)  e.  Q. )
4235, 40, 41syl2anc 408 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
( u  +Q  t
)  +Q  r )  e.  Q. )
43 prdisj 7268 . . . . . . . . . 10  |-  ( (
<. ( 1st `  ( A  +P.  B ) ) ,  ( 2nd `  ( A  +P.  B ) )
>.  e.  P.  /\  (
( u  +Q  t
)  +Q  r )  e.  Q. )  ->  -.  ( ( ( u  +Q  t )  +Q  r )  e.  ( 1st `  ( A  +P.  B ) )  /\  ( ( u  +Q  t )  +Q  r )  e.  ( 2nd `  ( A  +P.  B ) ) ) )
4428, 42, 43syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  -.  ( ( ( u  +Q  t )  +Q  r )  e.  ( 1st `  ( A  +P.  B ) )  /\  ( ( u  +Q  t )  +Q  r )  e.  ( 2nd `  ( A  +P.  B ) ) ) )
45 addassnqg 7158 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  Q.  /\  t  e.  Q.  /\  r  e.  Q. )  ->  (
( u  +Q  t
)  +Q  r )  =  ( u  +Q  ( t  +Q  r
) ) )
4632, 33, 40, 45syl3anc 1201 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
( u  +Q  t
)  +Q  r )  =  ( u  +Q  ( t  +Q  r
) ) )
47 addcomnqg 7157 . . . . . . . . . . . . . . . 16  |-  ( ( t  e.  Q.  /\  r  e.  Q. )  ->  ( t  +Q  r
)  =  ( r  +Q  t ) )
4847oveq2d 5758 . . . . . . . . . . . . . . 15  |-  ( ( t  e.  Q.  /\  r  e.  Q. )  ->  ( u  +Q  (
t  +Q  r ) )  =  ( u  +Q  ( r  +Q  t ) ) )
4933, 40, 48syl2anc 408 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
u  +Q  ( t  +Q  r ) )  =  ( u  +Q  ( r  +Q  t
) ) )
5046, 49eqtrd 2150 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
( u  +Q  t
)  +Q  r )  =  ( u  +Q  ( r  +Q  t
) ) )
5150adantr 274 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
r  +Q  t )  e.  ( 1st `  C
) )  ->  (
( u  +Q  t
)  +Q  r )  =  ( u  +Q  ( r  +Q  t
) ) )
52 simplrl 509 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
r  +Q  t )  e.  ( 1st `  C
) )  ->  u  e.  ( 1st `  A
) )
53 simpr 109 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
r  +Q  t )  e.  ( 1st `  C
) )  ->  (
r  +Q  t )  e.  ( 1st `  C
) )
5423adantr 274 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
r  +Q  t )  e.  ( 1st `  C
) )  ->  A  e.  P. )
5522simp3d 980 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  C  e.  P. )
5655adantr 274 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
r  +Q  t )  e.  ( 1st `  C
) )  ->  C  e.  P. )
57 df-iplp 7244 . . . . . . . . . . . . . . 15  |-  +P.  =  ( q  e.  P. ,  s  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  q )  /\  h  e.  ( 1st `  s
)  /\  f  =  ( g  +Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  q )  /\  h  e.  ( 2nd `  s
)  /\  f  =  ( g  +Q  h
) ) } >. )
58 addclnq 7151 . . . . . . . . . . . . . . 15  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
5957, 58genpprecll 7290 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( ( u  e.  ( 1st `  A
)  /\  ( r  +Q  t )  e.  ( 1st `  C ) )  ->  ( u  +Q  ( r  +Q  t
) )  e.  ( 1st `  ( A  +P.  C ) ) ) )
6054, 56, 59syl2anc 408 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
r  +Q  t )  e.  ( 1st `  C
) )  ->  (
( u  e.  ( 1st `  A )  /\  ( r  +Q  t )  e.  ( 1st `  C ) )  ->  ( u  +Q  ( r  +Q  t
) )  e.  ( 1st `  ( A  +P.  C ) ) ) )
6152, 53, 60mp2and 429 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
r  +Q  t )  e.  ( 1st `  C
) )  ->  (
u  +Q  ( r  +Q  t ) )  e.  ( 1st `  ( A  +P.  C ) ) )
6251, 61eqeltrd 2194 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
r  +Q  t )  e.  ( 1st `  C
) )  ->  (
( u  +Q  t
)  +Q  r )  e.  ( 1st `  ( A  +P.  C ) ) )
63 fveq2 5389 . . . . . . . . . . . . 13  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( 1st `  ( A  +P.  B
) )  =  ( 1st `  ( A  +P.  C ) ) )
6463eleq2d 2187 . . . . . . . . . . . 12  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( (
( u  +Q  t
)  +Q  r )  e.  ( 1st `  ( A  +P.  B ) )  <-> 
( ( u  +Q  t )  +Q  r
)  e.  ( 1st `  ( A  +P.  C
) ) ) )
6564ad7antlr 492 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
r  +Q  t )  e.  ( 1st `  C
) )  ->  (
( ( u  +Q  t )  +Q  r
)  e.  ( 1st `  ( A  +P.  B
) )  <->  ( (
u  +Q  t )  +Q  r )  e.  ( 1st `  ( A  +P.  C ) ) ) )
6662, 65mpbird 166 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
r  +Q  t )  e.  ( 1st `  C
) )  ->  (
( u  +Q  t
)  +Q  r )  e.  ( 1st `  ( A  +P.  B ) ) )
6757, 58genppreclu 7291 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( u  +Q  t )  e.  ( 2nd `  A
)  /\  r  e.  ( 2nd `  B ) )  ->  ( (
u  +Q  t )  +Q  r )  e.  ( 2nd `  ( A  +P.  B ) ) ) )
6867ancomsd 267 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( r  e.  ( 2nd `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) )  ->  ( (
u  +Q  t )  +Q  r )  e.  ( 2nd `  ( A  +P.  B ) ) ) )
69683adant3 986 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( r  e.  ( 2nd `  B )  /\  ( u  +Q  t )  e.  ( 2nd `  A ) )  ->  ( (
u  +Q  t )  +Q  r )  e.  ( 2nd `  ( A  +P.  B ) ) ) )
7069ad2antrr 479 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  ->  ( (
r  e.  ( 2nd `  B )  /\  (
u  +Q  t )  e.  ( 2nd `  A
) )  ->  (
( u  +Q  t
)  +Q  r )  e.  ( 2nd `  ( A  +P.  B ) ) ) )
7170imp 123 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  (
u  +Q  t )  e.  ( 2nd `  A
) ) )  -> 
( ( u  +Q  t )  +Q  r
)  e.  ( 2nd `  ( A  +P.  B
) ) )
7271adantrlr 476 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
( r  e.  ( 2nd `  B )  /\  r  <Q  v
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
( u  +Q  t
)  +Q  r )  e.  ( 2nd `  ( A  +P.  B ) ) )
7372anassrs 397 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( u  +Q  t
)  e.  ( 2nd `  A ) )  -> 
( ( u  +Q  t )  +Q  r
)  e.  ( 2nd `  ( A  +P.  B
) ) )
7473ad2ant2rl 502 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
( u  +Q  t
)  +Q  r )  e.  ( 2nd `  ( A  +P.  B ) ) )
7574adantlr 468 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
( u  +Q  t
)  +Q  r )  e.  ( 2nd `  ( A  +P.  B ) ) )
7675adantr 274 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
r  +Q  t )  e.  ( 1st `  C
) )  ->  (
( u  +Q  t
)  +Q  r )  e.  ( 2nd `  ( A  +P.  B ) ) )
7766, 76jca 304 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
r  +Q  t )  e.  ( 1st `  C
) )  ->  (
( ( u  +Q  t )  +Q  r
)  e.  ( 1st `  ( A  +P.  B
) )  /\  (
( u  +Q  t
)  +Q  r )  e.  ( 2nd `  ( A  +P.  B ) ) ) )
7844, 77mtand 639 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  -.  ( r  +Q  t
)  e.  ( 1st `  C ) )
79 prop 7251 . . . . . . . . . . 11  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
8055, 79syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
81 ltaddnq 7183 . . . . . . . . . . . . . 14  |-  ( ( t  e.  Q.  /\  t  e.  Q. )  ->  t  <Q  ( t  +Q  t ) )
8233, 33, 81syl2anc 408 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  t  <Q  ( t  +Q  t
) )
83 simplrr 510 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
t  +Q  t )  =  w )
8482, 83breqtrd 3924 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  t  <Q  w )
85 ltanqi 7178 . . . . . . . . . . . 12  |-  ( ( t  <Q  w  /\  r  e.  Q. )  ->  ( r  +Q  t
)  <Q  ( r  +Q  w ) )
8684, 40, 85syl2anc 408 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
r  +Q  t ) 
<Q  ( r  +Q  w
) )
87 simprr 506 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  ->  ( r  +Q  w )  =  v )
8887ad2antrr 479 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
r  +Q  w )  =  v )
8986, 88breqtrd 3924 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
r  +Q  t ) 
<Q  v )
90 prloc 7267 . . . . . . . . . 10  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  ( r  +Q  t
)  <Q  v )  -> 
( ( r  +Q  t )  e.  ( 1st `  C )  \/  v  e.  ( 2nd `  C ) ) )
9180, 89, 90syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
( r  +Q  t
)  e.  ( 1st `  C )  \/  v  e.  ( 2nd `  C
) ) )
9291orcomd 703 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
v  e.  ( 2nd `  C )  \/  (
r  +Q  t )  e.  ( 1st `  C
) ) )
9378, 92ecased 1312 . . . . . . 7  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  v  e.  ( 2nd `  C
) )
9420, 93rexlimddv 2531 . . . . . 6  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  /\  ( r  e.  ( 2nd `  B
)  /\  r  <Q  v ) )  /\  (
w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  ->  v  e.  ( 2nd `  C ) )
9511, 94rexlimddv 2531 . . . . 5  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  /\  ( w  e.  Q.  /\  ( r  +Q  w
)  =  v ) )  ->  v  e.  ( 2nd `  C ) )
968, 95rexlimddv 2531 . . . 4  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 2nd `  B
) )  /\  (
r  e.  ( 2nd `  B )  /\  r  <Q  v ) )  -> 
v  e.  ( 2nd `  C ) )
975, 96rexlimddv 2531 . . 3  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 2nd `  B ) )  ->  v  e.  ( 2nd `  C ) )
9897ex 114 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( v  e.  ( 2nd `  B
)  ->  v  e.  ( 2nd `  C ) ) )
9998ssrdv 3073 1  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 2nd `  B
)  C_  ( 2nd `  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 682    /\ w3a 947    = wceq 1316    e. wcel 1465   E.wrex 2394    C_ wss 3041   <.cop 3500   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056    +Q cplq 7058    <Q cltq 7061   P.cnp 7067    +P. cpp 7069
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244
This theorem is referenced by:  addcanprg  7392
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