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Theorem addcanprleml 6943
Description: Lemma for addcanprg 6945. (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 6804 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnmaddl 6819 . . . . . . 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 1102 . . . . 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 6739 . . . . . 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 951 . . . . . . . 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 6804 . . . . . . . 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 6833 . . . . . . 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 951 . . . . . . . . . . 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 952 . . . . . . . . . . 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 6866 . . . . . . . . . . 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 6804 . . . . . . . . . 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 6810 . . . . . . . . . . 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 6810 . . . . . . . . . . . 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 6704 . . . . . . . . . . 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 6704 . . . . . . . . . 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 6821 . . . . . . . . 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 6797 . . . . . . . . . . . 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 6704 . . . . . . . . . . . 12  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
4846, 47genpprecll 6843 . . . . . . . . . . 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 1171 . . . . . . . . 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 6710 . . . . . . . . . . . . . 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 6711 . . . . . . . . . . . . . 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 6704 . . . . . . . . . . . . . 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 5738 . . . . . . . . . . . 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 5581 . . . . . . . . . . . 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 6711 . . . . . . . . . . . . 13  |-  ( ( u  e.  Q.  /\  v  e.  Q.  /\  w  e.  Q. )  ->  (
( u  +Q  v
)  +Q  w )  =  ( u  +Q  ( v  +Q  w
) ) )
6651, 53, 64, 65syl3anc 1170 . . . . . . . . . . . 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 2119 . . . . . . . . . . 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 953 . . . . . . . . . . . . . 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 6844 . . . . . . . . . . . . 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 2160 . . . . . . . . . 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 5231 . . . . . . . . . . . 12  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( 2nd `  ( A  +P.  B
) )  =  ( 2nd `  ( A  +P.  C ) ) )
8079eleq2d 2152 . . . . . . . . . . 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 624 . . . . . . 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 6804 . . . . . . . . 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 6736 . . . . . . . . 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 6820 . . . . . . . 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 1281 . . . . . 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 2487 . . . . 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 2487 . . . 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 2487 . . 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 3015 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 662    /\ w3a 920    = wceq 1285    e. wcel 1434   E.wrex 2354    C_ wss 2983   <.cop 3420   class class class wbr 3806   ` cfv 4953  (class class class)co 5565   1stc1st 5818   2ndc2nd 5819   Q.cnq 6609    +Q cplq 6611    <Q cltq 6614   P.cnp 6620    +P. cpp 6622
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 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
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 2613  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-eprel 4073  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-1st 5820  df-2nd 5821  df-recs 5976  df-irdg 6041  df-1o 6087  df-2o 6088  df-oadd 6091  df-omul 6092  df-er 6195  df-ec 6197  df-qs 6201  df-ni 6633  df-pli 6634  df-mi 6635  df-lti 6636  df-plpq 6673  df-mpq 6674  df-enq 6676  df-nqqs 6677  df-plqqs 6678  df-mqqs 6679  df-1nqqs 6680  df-rq 6681  df-ltnqqs 6682  df-enq0 6753  df-nq0 6754  df-0nq0 6755  df-plq0 6756  df-mq0 6757  df-inp 6795  df-iplp 6797
This theorem is referenced by:  addcanprg  6945
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