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Theorem addcanprlemu 7121
Description: Lemma for addcanprg 7122. (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 6981 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnminu 6995 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  v  e.  ( 2nd `  B ) )  ->  E. r  e.  ( 2nd `  B ) r 
<Q  v )
31, 2sylan 277 . . . . . 6  |-  ( ( B  e.  P.  /\  v  e.  ( 2nd `  B ) )  ->  E. r  e.  ( 2nd `  B ) r 
<Q  v )
433ad2antl2 1104 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  v  e.  ( 2nd `  B ) )  ->  E. r  e.  ( 2nd `  B ) r  <Q  v )
54adantlr 461 . . . 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 499 . . . . . 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 6915 . . . . . 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 498 . . . . . . 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 6916 . . . . . . 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 6981 . . . . . . . . . . . . . 14  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
13 prarloc2 7010 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
1412, 13sylan 277 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
1514adantrr 463 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  ->  E. u  e.  ( 1st `  A
) ( u  +Q  t )  e.  ( 2nd `  A ) )
16153ad2antl1 1103 . . . . . . . . . . 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 461 . . . . . . . . . 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 461 . . . . . . . . 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 461 . . . . . . . 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 461 . . . . . . 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 500 . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . 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 953 . . . . . . . . . . . 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 954 . . . . . . . . . . . 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 7043 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
2623, 24, 25syl2anc 403 . . . . . . . . . . 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 6981 . . . . . . . . . . 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 498 . . . . . . . . . . . . 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 6987 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
3229, 30, 31syl2anc 403 . . . . . . . . . . . 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 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 ) )  /\  ( t  e.  Q.  /\  (
t  +Q  t )  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  t  e.  Q. )
34 addclnq 6881 . . . . . . . . . . . 12  |-  ( ( u  e.  Q.  /\  t  e.  Q. )  ->  ( u  +Q  t
)  e.  Q. )
3532, 33, 34syl2anc 403 . . . . . . . . . . 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 498 . . . . . . . . . . . . 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 476 . . . . . . . . . . . 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 6988 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  r  e.  ( 2nd `  B ) )  -> 
r  e.  Q. )
4036, 38, 39syl2anc 403 . . . . . . . . . . 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 6881 . . . . . . . . . . 11  |-  ( ( ( u  +Q  t
)  e.  Q.  /\  r  e.  Q. )  ->  ( ( u  +Q  t )  +Q  r
)  e.  Q. )
4235, 40, 41syl2anc 403 . . . . . . . . . 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 6998 . . . . . . . . . 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 403 . . . . . . . . 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 6888 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  Q.  /\  t  e.  Q.  /\  r  e.  Q. )  ->  (
( u  +Q  t
)  +Q  r )  =  ( u  +Q  ( t  +Q  r
) ) )
4632, 33, 40, 45syl3anc 1172 . . . . . . . . . . . . . 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 6887 . . . . . . . . . . . . . . . 16  |-  ( ( t  e.  Q.  /\  r  e.  Q. )  ->  ( t  +Q  r
)  =  ( r  +Q  t ) )
4847oveq2d 5631 . . . . . . . . . . . . . . 15  |-  ( ( t  e.  Q.  /\  r  e.  Q. )  ->  ( u  +Q  (
t  +Q  r ) )  =  ( u  +Q  ( r  +Q  t ) ) )
4933, 40, 48syl2anc 403 . . . . . . . . . . . . . 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 2117 . . . . . . . . . . . . 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 270 . . . . . . . . . . . 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 502 . . . . . . . . . . . . 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 108 . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 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 955 . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 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 6974 . . . . . . . . . . . . . . 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 6881 . . . . . . . . . . . . . . 15  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
5957, 58genpprecll 7020 . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 2161 . . . . . . . . . . 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 5270 . . . . . . . . . . . . 13  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( 1st `  ( A  +P.  B
) )  =  ( 1st `  ( A  +P.  C ) ) )
6463eleq2d 2154 . . . . . . . . . . . 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 485 . . . . . . . . . . 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 165 . . . . . . . . . 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 7021 . . . . . . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . . . . . 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 961 . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . 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 122 . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . 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 392 . . . . . . . . . . . . 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 495 . . . . . . . . . . . 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 461 . . . . . . . . . . 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 270 . . . . . . . . . 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 300 . . . . . . . . 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 624 . . . . . . . 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 6981 . . . . . . . . . . 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 6913 . . . . . . . . . . . . . 14  |-  ( ( t  e.  Q.  /\  t  e.  Q. )  ->  t  <Q  ( t  +Q  t ) )
8233, 33, 81syl2anc 403 . . . . . . . . . . . . 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 503 . . . . . . . . . . . . 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 3846 . . . . . . . . . . . 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 6908 . . . . . . . . . . . 12  |-  ( ( t  <Q  w  /\  r  e.  Q. )  ->  ( r  +Q  t
)  <Q  ( r  +Q  w ) )
8684, 40, 85syl2anc 403 . . . . . . . . . . 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 499 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 3846 . . . . . . . . . 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 6997 . . . . . . . . . 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 403 . . . . . . . . 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 681 . . . . . . . 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 1283 . . . . . . 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 2489 . . . . . 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 2489 . . . . 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 2489 . . . 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 2489 . . 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 113 . 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 3020 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 102    <-> wb 103    \/ wo 662    /\ w3a 922    = wceq 1287    e. wcel 1436   E.wrex 2356    C_ wss 2988   <.cop 3434   class class class wbr 3822   ` cfv 4983  (class class class)co 5615   1stc1st 5868   2ndc2nd 5869   Q.cnq 6786    +Q cplq 6788    <Q cltq 6791   P.cnp 6797    +P. cpp 6799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-eprel 4092  df-id 4096  df-po 4099  df-iso 4100  df-iord 4169  df-on 4171  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-irdg 6091  df-1o 6137  df-2o 6138  df-oadd 6141  df-omul 6142  df-er 6246  df-ec 6248  df-qs 6252  df-ni 6810  df-pli 6811  df-mi 6812  df-lti 6813  df-plpq 6850  df-mpq 6851  df-enq 6853  df-nqqs 6854  df-plqqs 6855  df-mqqs 6856  df-1nqqs 6857  df-rq 6858  df-ltnqqs 6859  df-enq0 6930  df-nq0 6931  df-0nq0 6932  df-plq0 6933  df-mq0 6934  df-inp 6972  df-iplp 6974
This theorem is referenced by:  addcanprg  7122
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