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Theorem addcanprlemu 7077
Description: Lemma for addcanprg 7078. (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 6937 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnminu 6951 . . . . . . 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 1102 . . . . 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 6871 . . . . . 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 6872 . . . . . . 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 6937 . . . . . . . . . . . . . 14  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
13 prarloc2 6966 . . . . . . . . . . . . . 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 1101 . . . . . . . . . . 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 951 . . . . . . . . . . . 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 952 . . . . . . . . . . . 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 6999 . . . . . . . . . . . 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 6937 . . . . . . . . . . 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 6943 . . . . . . . . . . . . 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 6837 . . . . . . . . . . . 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 6944 . . . . . . . . . . . 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 6837 . . . . . . . . . . 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 6954 . . . . . . . . . 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 6844 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  Q.  /\  t  e.  Q.  /\  r  e.  Q. )  ->  (
( u  +Q  t
)  +Q  r )  =  ( u  +Q  ( t  +Q  r
) ) )
4632, 33, 40, 45syl3anc 1170 . . . . . . . . . . . . . 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 6843 . . . . . . . . . . . . . . . 16  |-  ( ( t  e.  Q.  /\  r  e.  Q. )  ->  ( t  +Q  r
)  =  ( r  +Q  t ) )
4847oveq2d 5607 . . . . . . . . . . . . . . 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 2115 . . . . . . . . . . . . 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 953 . . . . . . . . . . . . . . 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 6930 . . . . . . . . . . . . . . 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 6837 . . . . . . . . . . . . . . 15  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
5957, 58genpprecll 6976 . . . . . . . . . . . . . 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 2159 . . . . . . . . . . 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 5253 . . . . . . . . . . . . 13  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( 1st `  ( A  +P.  B
) )  =  ( 1st `  ( A  +P.  C ) ) )
6463eleq2d 2152 . . . . . . . . . . . 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 6977 . . . . . . . . . . . . . . . . . . 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 959 . . . . . . . . . . . . . . . . 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 6937 . . . . . . . . . . 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 6869 . . . . . . . . . . . . . 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 3835 . . . . . . . . . . . 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 6864 . . . . . . . . . . . 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 3835 . . . . . . . . . 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 6953 . . . . . . . . . 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 1281 . . . . . . 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 2487 . . . . . 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 2487 . . . . 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 2487 . . . 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 2487 . . 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 3016 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 920    = wceq 1285    e. wcel 1434   E.wrex 2354    C_ wss 2984   <.cop 3425   class class class wbr 3811   ` cfv 4969  (class class class)co 5591   1stc1st 5844   2ndc2nd 5845   Q.cnq 6742    +Q cplq 6744    <Q cltq 6747   P.cnp 6753    +P. cpp 6755
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928  df-iplp 6930
This theorem is referenced by:  addcanprg  7078
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