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Theorem addcanprleml 7152
Description: Lemma for addcanprg 7154. (Contributed by Jim Kingdon, 25-Dec-2019.)
Assertion
Ref Expression
addcanprleml  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 1st `  B
)  C_  ( 1st `  C ) )

Proof of Theorem addcanprleml
Dummy variables  f  g  h  r  s  t  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7013 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnmaddl 7028 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  v  e.  ( 1st `  B ) )  ->  E. w  e.  Q.  ( v  +Q  w
)  e.  ( 1st `  B ) )
31, 2sylan 277 . . . . . 6  |-  ( ( B  e.  P.  /\  v  e.  ( 1st `  B ) )  ->  E. w  e.  Q.  ( v  +Q  w
)  e.  ( 1st `  B ) )
433ad2antl2 1106 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  v  e.  ( 1st `  B ) )  ->  E. w  e.  Q.  ( v  +Q  w
)  e.  ( 1st `  B ) )
54adantlr 461 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  ->  E. w  e.  Q.  ( v  +Q  w )  e.  ( 1st `  B ) )
6 simprl 498 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  ->  w  e.  Q. )
7 halfnqq 6948 . . . . . 6  |-  ( w  e.  Q.  ->  E. t  e.  Q.  ( t  +Q  t )  =  w )
86, 7syl 14 . . . . 5  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  ->  E. t  e.  Q.  ( t  +Q  t
)  =  w )
9 simplll 500 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  ->  ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. ) )
109adantr 270 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  ->  ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. ) )
1110simp1d 955 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  ->  A  e.  P. )
12 prop 7013 . . . . . . . 8  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
1311, 12syl 14 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
14 simprl 498 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  ->  t  e.  Q. )
15 prarloc2 7042 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
1613, 14, 15syl2anc 403 . . . . . 6  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  ->  E. u  e.  ( 1st `  A
) ( u  +Q  t )  e.  ( 2nd `  A ) )
179ad2antrr 472 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( 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. ) )
1817simp1d 955 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  A  e.  P. )
1917simp2d 956 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  B  e.  P. )
20 addclpr 7075 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
2118, 19, 20syl2anc 403 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  ( A  +P.  B )  e. 
P. )
22 prop 7013 . . . . . . . . . 10  |-  ( ( A  +P.  B )  e.  P.  ->  <. ( 1st `  ( A  +P.  B ) ) ,  ( 2nd `  ( A  +P.  B ) )
>.  e.  P. )
2321, 22syl 14 . . . . . . . . 9  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( 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. )
2418, 12syl 14 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
25 simprl 498 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  u  e.  ( 1st `  A
) )
26 elprnql 7019 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
2724, 25, 26syl2anc 403 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  u  e.  Q. )
2819, 1syl 14 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
29 simplr 497 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  ->  v  e.  ( 1st `  B ) )
3029ad2antrr 472 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  v  e.  ( 1st `  B
) )
31 elprnql 7019 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  v  e.  ( 1st `  B ) )  -> 
v  e.  Q. )
3228, 30, 31syl2anc 403 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  v  e.  Q. )
33 simplrl 502 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  ->  w  e.  Q. )
3433adantr 270 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  w  e.  Q. )
35 addclnq 6913 . . . . . . . . . . 11  |-  ( ( v  e.  Q.  /\  w  e.  Q. )  ->  ( v  +Q  w
)  e.  Q. )
3632, 34, 35syl2anc 403 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
v  +Q  w )  e.  Q. )
37 addclnq 6913 . . . . . . . . . 10  |-  ( ( u  e.  Q.  /\  ( v  +Q  w
)  e.  Q. )  ->  ( u  +Q  (
v  +Q  w ) )  e.  Q. )
3827, 36, 37syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
u  +Q  ( v  +Q  w ) )  e.  Q. )
39 prdisj 7030 . . . . . . . . 9  |-  ( (
<. ( 1st `  ( A  +P.  B ) ) ,  ( 2nd `  ( A  +P.  B ) )
>.  e.  P.  /\  (
u  +Q  ( v  +Q  w ) )  e.  Q. )  ->  -.  ( ( u  +Q  ( v  +Q  w
) )  e.  ( 1st `  ( A  +P.  B ) )  /\  ( u  +Q  ( v  +Q  w
) )  e.  ( 2nd `  ( A  +P.  B ) ) ) )
4023, 38, 39syl2anc 403 . . . . . . . 8  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  -.  ( ( u  +Q  ( v  +Q  w
) )  e.  ( 1st `  ( A  +P.  B ) )  /\  ( u  +Q  ( v  +Q  w
) )  e.  ( 2nd `  ( A  +P.  B ) ) ) )
4118adantr 270 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  A  e.  P. )
4219adantr 270 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  B  e.  P. )
43 simplrl 502 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  u  e.  ( 1st `  A
) )
44 simplrr 503 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  ->  (
v  +Q  w )  e.  ( 1st `  B
) )
4544ad2antrr 472 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
v  +Q  w )  e.  ( 1st `  B
) )
46 df-iplp 7006 . . . . . . . . . . . 12  |-  +P.  =  ( r  e.  P. ,  s  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  r )  /\  h  e.  ( 1st `  s
)  /\  f  =  ( g  +Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  r )  /\  h  e.  ( 2nd `  s
)  /\  f  =  ( g  +Q  h
) ) } >. )
47 addclnq 6913 . . . . . . . . . . . 12  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
4846, 47genpprecll 7052 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( u  e.  ( 1st `  A
)  /\  ( v  +Q  w )  e.  ( 1st `  B ) )  ->  ( u  +Q  ( v  +Q  w
) )  e.  ( 1st `  ( A  +P.  B ) ) ) )
4948imp 122 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( u  e.  ( 1st `  A )  /\  ( v  +Q  w )  e.  ( 1st `  B ) ) )  ->  (
u  +Q  ( v  +Q  w ) )  e.  ( 1st `  ( A  +P.  B ) ) )
5041, 42, 43, 45, 49syl22anc 1175 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
u  +Q  ( v  +Q  w ) )  e.  ( 1st `  ( A  +P.  B ) ) )
5127adantr 270 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  u  e.  Q. )
5214ad2antrr 472 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  t  e.  Q. )
5332adantr 270 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  v  e.  Q. )
54 addcomnqg 6919 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
5554adantl 271 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  /\  (
u  e.  ( 1st `  A )  /\  (
u  +Q  t )  e.  ( 2nd `  A
) ) )  /\  ( v  +Q  t
)  e.  ( 2nd `  C ) )  /\  ( f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
56 addassnqg 6920 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
5756adantl 271 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  /\  (
u  e.  ( 1st `  A )  /\  (
u  +Q  t )  e.  ( 2nd `  A
) ) )  /\  ( v  +Q  t
)  e.  ( 2nd `  C ) )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( (
f  +Q  g )  +Q  h )  =  ( f  +Q  (
g  +Q  h ) ) )
58 addclnq 6913 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  e.  Q. )
5958adantl 271 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  /\  (
u  e.  ( 1st `  A )  /\  (
u  +Q  t )  e.  ( 2nd `  A
) ) )  /\  ( v  +Q  t
)  e.  ( 2nd `  C ) )  /\  ( f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  e.  Q. )
6051, 52, 53, 55, 57, 52, 59caov4d 5811 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
( u  +Q  t
)  +Q  ( v  +Q  t ) )  =  ( ( u  +Q  v )  +Q  ( t  +Q  t
) ) )
61 simprr 499 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  ->  (
t  +Q  t )  =  w )
6261ad2antrr 472 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
t  +Q  t )  =  w )
6362oveq2d 5650 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
( u  +Q  v
)  +Q  ( t  +Q  t ) )  =  ( ( u  +Q  v )  +Q  w ) )
6433ad2antrr 472 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  w  e.  Q. )
65 addassnqg 6920 . . . . . . . . . . . . 13  |-  ( ( u  e.  Q.  /\  v  e.  Q.  /\  w  e.  Q. )  ->  (
( u  +Q  v
)  +Q  w )  =  ( u  +Q  ( v  +Q  w
) ) )
6651, 53, 64, 65syl3anc 1174 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
( u  +Q  v
)  +Q  w )  =  ( u  +Q  ( v  +Q  w
) ) )
6760, 63, 663eqtrd 2124 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
( u  +Q  t
)  +Q  ( v  +Q  t ) )  =  ( u  +Q  ( v  +Q  w
) ) )
68 simplrr 503 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
u  +Q  t )  e.  ( 2nd `  A
) )
69 simpr 108 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
v  +Q  t )  e.  ( 2nd `  C
) )
7017simp3d 957 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  C  e.  P. )
7170adantr 270 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  C  e.  P. )
7246, 47genppreclu 7053 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( ( ( u  +Q  t )  e.  ( 2nd `  A
)  /\  ( v  +Q  t )  e.  ( 2nd `  C ) )  ->  ( (
u  +Q  t )  +Q  ( v  +Q  t ) )  e.  ( 2nd `  ( A  +P.  C ) ) ) )
7341, 71, 72syl2anc 403 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
( ( u  +Q  t )  e.  ( 2nd `  A )  /\  ( v  +Q  t )  e.  ( 2nd `  C ) )  ->  ( (
u  +Q  t )  +Q  ( v  +Q  t ) )  e.  ( 2nd `  ( A  +P.  C ) ) ) )
7468, 69, 73mp2and 424 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
( u  +Q  t
)  +Q  ( v  +Q  t ) )  e.  ( 2nd `  ( A  +P.  C ) ) )
7567, 74eqeltrrd 2165 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
u  +Q  ( v  +Q  w ) )  e.  ( 2nd `  ( A  +P.  C ) ) )
76 simpr 108 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( A  +P.  B )  =  ( A  +P.  C ) )
7776ad3antrrr 476 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  ->  ( A  +P.  B )  =  ( A  +P.  C
) )
7877ad2antrr 472 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  ( A  +P.  B )  =  ( A  +P.  C
) )
79 fveq2 5289 . . . . . . . . . . . 12  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( 2nd `  ( A  +P.  B
) )  =  ( 2nd `  ( A  +P.  C ) ) )
8079eleq2d 2157 . . . . . . . . . . 11  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( (
u  +Q  ( v  +Q  w ) )  e.  ( 2nd `  ( A  +P.  B ) )  <-> 
( u  +Q  (
v  +Q  w ) )  e.  ( 2nd `  ( A  +P.  C
) ) ) )
8178, 80syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
( u  +Q  (
v  +Q  w ) )  e.  ( 2nd `  ( A  +P.  B
) )  <->  ( u  +Q  ( v  +Q  w
) )  e.  ( 2nd `  ( A  +P.  C ) ) ) )
8275, 81mpbird 165 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
u  +Q  ( v  +Q  w ) )  e.  ( 2nd `  ( A  +P.  B ) ) )
8350, 82jca 300 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  (
v  +Q  t )  e.  ( 2nd `  C
) )  ->  (
( u  +Q  (
v  +Q  w ) )  e.  ( 1st `  ( A  +P.  B
) )  /\  (
u  +Q  ( v  +Q  w ) )  e.  ( 2nd `  ( A  +P.  B ) ) ) )
8440, 83mtand 626 . . . . . . 7  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  -.  ( v  +Q  t
)  e.  ( 2nd `  C ) )
85 prop 7013 . . . . . . . . 9  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
8670, 85syl 14 . . . . . . . 8  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
87 simplrl 502 . . . . . . . . 9  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  t  e.  Q. )
88 ltaddnq 6945 . . . . . . . . 9  |-  ( ( v  e.  Q.  /\  t  e.  Q. )  ->  v  <Q  ( v  +Q  t ) )
8932, 87, 88syl2anc 403 . . . . . . . 8  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  v  <Q  ( v  +Q  t
) )
90 prloc 7029 . . . . . . . 8  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  v  <Q  ( v  +Q  t ) )  -> 
( v  e.  ( 1st `  C )  \/  ( v  +Q  t )  e.  ( 2nd `  C ) ) )
9186, 89, 90syl2anc 403 . . . . . . 7  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
v  e.  ( 1st `  C )  \/  (
v  +Q  t )  e.  ( 2nd `  C
) ) )
9284, 91ecased 1285 . . . . . 6  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  /\  ( w  e.  Q.  /\  (
v  +Q  w )  e.  ( 1st `  B
) ) )  /\  ( t  e.  Q.  /\  ( t  +Q  t
)  =  w ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  v  e.  ( 1st `  C
) )
9316, 92rexlimddv 2493 . . . . 5  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  /\  ( t  e. 
Q.  /\  ( t  +Q  t )  =  w ) )  ->  v  e.  ( 1st `  C
) )
948, 93rexlimddv 2493 . . . 4  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( A  +P.  B )  =  ( A  +P.  C
) )  /\  v  e.  ( 1st `  B
) )  /\  (
w  e.  Q.  /\  ( v  +Q  w
)  e.  ( 1st `  B ) ) )  ->  v  e.  ( 1st `  C ) )
955, 94rexlimddv 2493 . . 3  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B )  =  ( A  +P.  C ) )  /\  v  e.  ( 1st `  B ) )  ->  v  e.  ( 1st `  C ) )
9695ex 113 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( v  e.  ( 1st `  B
)  ->  v  e.  ( 1st `  C ) ) )
9796ssrdv 3029 1  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 1st `  B
)  C_  ( 1st `  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360    C_ wss 2997   <.cop 3444   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   1stc1st 5891   2ndc2nd 5892   Q.cnq 6818    +Q cplq 6820    <Q cltq 6823   P.cnp 6829    +P. cpp 6831
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006
This theorem is referenced by:  addcanprg  7154
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