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Theorem addcanprleml 7444
Description: Lemma for addcanprg 7446. (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 7305 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnmaddl 7320 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  v  e.  ( 1st `  B ) )  ->  E. w  e.  Q.  ( v  +Q  w
)  e.  ( 1st `  B ) )
31, 2sylan 281 . . . . . 6  |-  ( ( B  e.  P.  /\  v  e.  ( 1st `  B ) )  ->  E. w  e.  Q.  ( v  +Q  w
)  e.  ( 1st `  B ) )
433ad2antl2 1145 . . . . 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 469 . . . 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 521 . . . . . 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 7240 . . . . . 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 523 . . . . . . . . . 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 274 . . . . . . . . 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 994 . . . . . . . 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 7305 . . . . . . . 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 521 . . . . . . 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 7334 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
1613, 14, 15syl2anc 409 . . . . . 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 480 . . . . . . . . . . . 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 994 . . . . . . . . . . 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 995 . . . . . . . . . . 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 7367 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
2118, 19, 20syl2anc 409 . . . . . . . . . 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 7305 . . . . . . . . . 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 521 . . . . . . . . . . 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 7311 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
2724, 25, 26syl2anc 409 . . . . . . . . . 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 520 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 7311 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  v  e.  ( 1st `  B ) )  -> 
v  e.  Q. )
3228, 30, 31syl2anc 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  e.  Q. )
33 simplrl 525 . . . . . . . . . . . 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 274 . . . . . . . . . . 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 7205 . . . . . . . . . . 11  |-  ( ( v  e.  Q.  /\  w  e.  Q. )  ->  ( v  +Q  w
)  e.  Q. )
3632, 34, 35syl2anc 409 . . . . . . . . . 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 7205 . . . . . . . . . 10  |-  ( ( u  e.  Q.  /\  ( v  +Q  w
)  e.  Q. )  ->  ( u  +Q  (
v  +Q  w ) )  e.  Q. )
3827, 36, 37syl2anc 409 . . . . . . . . 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 7322 . . . . . . . . 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 409 . . . . . . . 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 274 . . . . . . . . . 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 274 . . . . . . . . . 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 525 . . . . . . . . . 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 526 . . . . . . . . . . 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 480 . . . . . . . . . 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 7298 . . . . . . . . . . . 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 7205 . . . . . . . . . . . 12  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
4846, 47genpprecll 7344 . . . . . . . . . . 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 123 . . . . . . . . . 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 1218 . . . . . . . . 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 274 . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 274 . . . . . . . . . . . . 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 7211 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
5554adantl 275 . . . . . . . . . . . . 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 7212 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
5756adantl 275 . . . . . . . . . . . . 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 7205 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  e.  Q. )
5958adantl 275 . . . . . . . . . . . . 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 5961 . . . . . . . . . . . 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 522 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 5796 . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 7212 . . . . . . . . . . . . 13  |-  ( ( u  e.  Q.  /\  v  e.  Q.  /\  w  e.  Q. )  ->  (
( u  +Q  v
)  +Q  w )  =  ( u  +Q  ( v  +Q  w
) ) )
6651, 53, 64, 65syl3anc 1217 . . . . . . . . . . . 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 2177 . . . . . . . . . . 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 526 . . . . . . . . . . . 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 109 . . . . . . . . . . . 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 996 . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . 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 7345 . . . . . . . . . . . . 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 409 . . . . . . . . . . . 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 430 . . . . . . . . . . 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 2218 . . . . . . . . . 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 109 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( A  +P.  B )  =  ( A  +P.  C ) )
7776ad3antrrr 484 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 5427 . . . . . . . . . . . 12  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( 2nd `  ( A  +P.  B
) )  =  ( 2nd `  ( A  +P.  C ) ) )
8079eleq2d 2210 . . . . . . . . . . 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 166 . . . . . . . . 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 304 . . . . . . . 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 655 . . . . . . 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 7305 . . . . . . . . 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 525 . . . . . . . . 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 7237 . . . . . . . . 9  |-  ( ( v  e.  Q.  /\  t  e.  Q. )  ->  v  <Q  ( v  +Q  t ) )
8932, 87, 88syl2anc 409 . . . . . . . 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 7321 . . . . . . . 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 409 . . . . . . 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 1328 . . . . . 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 2557 . . . . 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 2557 . . . 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 2557 . . 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 114 . 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 3106 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 103    <-> wb 104    \/ wo 698    /\ w3a 963    = wceq 1332    e. wcel 1481   E.wrex 2418    C_ wss 3074   <.cop 3533   class class class wbr 3935   ` cfv 5129  (class class class)co 5780   1stc1st 6042   2ndc2nd 6043   Q.cnq 7110    +Q cplq 7112    <Q cltq 7115   P.cnp 7121    +P. cpp 7123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4049  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-iinf 4508
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-tr 4033  df-eprel 4217  df-id 4221  df-po 4224  df-iso 4225  df-iord 4294  df-on 4296  df-suc 4299  df-iom 4511  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-ov 5783  df-oprab 5784  df-mpo 5785  df-1st 6044  df-2nd 6045  df-recs 6208  df-irdg 6273  df-1o 6319  df-2o 6320  df-oadd 6323  df-omul 6324  df-er 6435  df-ec 6437  df-qs 6441  df-ni 7134  df-pli 7135  df-mi 7136  df-lti 7137  df-plpq 7174  df-mpq 7175  df-enq 7177  df-nqqs 7178  df-plqqs 7179  df-mqqs 7180  df-1nqqs 7181  df-rq 7182  df-ltnqqs 7183  df-enq0 7254  df-nq0 7255  df-0nq0 7256  df-plq0 7257  df-mq0 7258  df-inp 7296  df-iplp 7298
This theorem is referenced by:  addcanprg  7446
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