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Theorem addcanprleml 6712
Description: Lemma for addcanprg 6714. (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 6573 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnmaddl 6588 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  v  e.  ( 1st `  B ) )  ->  E. w  e.  Q.  ( v  +Q  w
)  e.  ( 1st `  B ) )
31, 2sylan 267 . . . . . 6  |-  ( ( B  e.  P.  /\  v  e.  ( 1st `  B ) )  ->  E. w  e.  Q.  ( v  +Q  w
)  e.  ( 1st `  B ) )
433ad2antl2 1067 . . . . 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 446 . . . 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 483 . . . . . 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 6508 . . . . . 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 485 . . . . . . . . . 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 261 . . . . . . . . 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 916 . . . . . . . 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 6573 . . . . . . . 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 483 . . . . . . 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 6602 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
1613, 14, 15syl2anc 391 . . . . . 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 457 . . . . . . . . . . . 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 916 . . . . . . . . . . 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 917 . . . . . . . . . . 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 6635 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
2118, 19, 20syl2anc 391 . . . . . . . . . 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 6573 . . . . . . . . . 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 483 . . . . . . . . . . 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 6579 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
2724, 25, 26syl2anc 391 . . . . . . . . . 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 482 . . . . . . . . . . . . 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 457 . . . . . . . . . . . 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 6579 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  v  e.  ( 1st `  B ) )  -> 
v  e.  Q. )
3228, 30, 31syl2anc 391 . . . . . . . . . . 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 487 . . . . . . . . . . . 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 261 . . . . . . . . . . 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 6473 . . . . . . . . . . 11  |-  ( ( v  e.  Q.  /\  w  e.  Q. )  ->  ( v  +Q  w
)  e.  Q. )
3632, 34, 35syl2anc 391 . . . . . . . . . 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 6473 . . . . . . . . . 10  |-  ( ( u  e.  Q.  /\  ( v  +Q  w
)  e.  Q. )  ->  ( u  +Q  (
v  +Q  w ) )  e.  Q. )
3827, 36, 37syl2anc 391 . . . . . . . . 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 6590 . . . . . . . . 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 391 . . . . . . . 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 261 . . . . . . . . . 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 261 . . . . . . . . . 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 487 . . . . . . . . . 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 488 . . . . . . . . . . 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 457 . . . . . . . . . 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 6566 . . . . . . . . . . . 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 6473 . . . . . . . . . . . 12  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
4846, 47genpprecll 6612 . . . . . . . . . . 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 115 . . . . . . . . . 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 1136 . . . . . . . . 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 261 . . . . . . . . . . . . 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 457 . . . . . . . . . . . . 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 261 . . . . . . . . . . . . 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 6479 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
5554adantl 262 . . . . . . . . . . . . 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 6480 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
5756adantl 262 . . . . . . . . . . . . 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 6473 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  e.  Q. )
5958adantl 262 . . . . . . . . . . . . 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 5685 . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . 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 5528 . . . . . . . . . . . 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 457 . . . . . . . . . . . . 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 6480 . . . . . . . . . . . . 13  |-  ( ( u  e.  Q.  /\  v  e.  Q.  /\  w  e.  Q. )  ->  (
( u  +Q  v
)  +Q  w )  =  ( u  +Q  ( v  +Q  w
) ) )
6651, 53, 64, 65syl3anc 1135 . . . . . . . . . . . 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 2076 . . . . . . . . . . 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 488 . . . . . . . . . . . 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 103 . . . . . . . . . . . 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 918 . . . . . . . . . . . . . 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 261 . . . . . . . . . . . . 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 6613 . . . . . . . . . . . . 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 391 . . . . . . . . . . . 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 409 . . . . . . . . . . 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 2115 . . . . . . . . . 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 103 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( A  +P.  B )  =  ( A  +P.  C ) )
7776ad3antrrr 461 . . . . . . . . . . . 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 457 . . . . . . . . . . 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 5178 . . . . . . . . . . . 12  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( 2nd `  ( A  +P.  B
) )  =  ( 2nd `  ( A  +P.  C ) ) )
8079eleq2d 2107 . . . . . . . . . . 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 156 . . . . . . . . 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 290 . . . . . . . 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 591 . . . . . . 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 6573 . . . . . . . . 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 487 . . . . . . . . 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 6505 . . . . . . . . 9  |-  ( ( v  e.  Q.  /\  t  e.  Q. )  ->  v  <Q  ( v  +Q  t ) )
8932, 87, 88syl2anc 391 . . . . . . . 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 6589 . . . . . . . 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 391 . . . . . . 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 1239 . . . . . 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 2437 . . . . 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 2437 . . . 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 2437 . . 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 108 . 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 2951 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 97    <-> wb 98    \/ wo 629    /\ w3a 885    = wceq 1243    e. wcel 1393   E.wrex 2307    C_ wss 2917   <.cop 3378   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378    +Q cplq 6380    <Q cltq 6383   P.cnp 6389    +P. cpp 6391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566
This theorem is referenced by:  addcanprg  6714
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