ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  addcanprlemu Unicode version

Theorem addcanprlemu 7435
Description: Lemma for addcanprg 7436. (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 7295 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnminu 7309 . . . . . . 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 1144 . . . . 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 521 . . . . . 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 7229 . . . . . 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 520 . . . . . . 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 7230 . . . . . . 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 7295 . . . . . . . . . . . . . 14  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
13 prarloc2 7324 . . . . . . . . . . . . . 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 1143 . . . . . . . . . . 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 522 . . . . . . . . . . . . . 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 993 . . . . . . . . . . . 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 994 . . . . . . . . . . . 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 7357 . . . . . . . . . . . 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 7295 . . . . . . . . . . 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 520 . . . . . . . . . . . . 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 7301 . . . . . . . . . . . . 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 524 . . . . . . . . . . . 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 7195 . . . . . . . . . . . 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 520 . . . . . . . . . . . . 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 7302 . . . . . . . . . . . 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 7195 . . . . . . . . . . 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 7312 . . . . . . . . . 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 7202 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  Q.  /\  t  e.  Q.  /\  r  e.  Q. )  ->  (
( u  +Q  t
)  +Q  r )  =  ( u  +Q  ( t  +Q  r
) ) )
4632, 33, 40, 45syl3anc 1216 . . . . . . . . . . . . . 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 7201 . . . . . . . . . . . . . . . 16  |-  ( ( t  e.  Q.  /\  r  e.  Q. )  ->  ( t  +Q  r
)  =  ( r  +Q  t ) )
4847oveq2d 5790 . . . . . . . . . . . . . . 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 2172 . . . . . . . . . . . . 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 524 . . . . . . . . . . . . 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 995 . . . . . . . . . . . . . . 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 7288 . . . . . . . . . . . . . . 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 7195 . . . . . . . . . . . . . . 15  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
5957, 58genpprecll 7334 . . . . . . . . . . . . . 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 2216 . . . . . . . . . . 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 5421 . . . . . . . . . . . . 13  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( 1st `  ( A  +P.  B
) )  =  ( 1st `  ( A  +P.  C ) ) )
6463eleq2d 2209 . . . . . . . . . . . 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 7335 . . . . . . . . . . . . . . . . . . 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 1001 . . . . . . . . . . . . . . . . 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 654 . . . . . . . 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 7295 . . . . . . . . . . 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 7227 . . . . . . . . . . . . . 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 525 . . . . . . . . . . . . 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 3954 . . . . . . . . . . . 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 7222 . . . . . . . . . . . 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 521 . . . . . . . . . . . 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 3954 . . . . . . . . . 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 7311 . . . . . . . . . 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 718 . . . . . . . 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 1327 . . . . . . 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 2554 . . . . . 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 2554 . . . . 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 2554 . . . 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 2554 . . 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 3103 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 697    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2417    C_ wss 3071   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7100    +Q cplq 7102    <Q cltq 7105   P.cnp 7111    +P. cpp 7113
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-iplp 7288
This theorem is referenced by:  addcanprg  7436
  Copyright terms: Public domain W3C validator