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Theorem addcanprleml 6769
Description: Lemma for addcanprg 6771. (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 6630 . . . . . . 7  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnmaddl 6645 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  v  e.  ( 1st `  B ) )  ->  E. w  e.  Q.  ( v  +Q  w
)  e.  ( 1st `  B ) )
31, 2sylan 271 . . . . . 6  |-  ( ( B  e.  P.  /\  v  e.  ( 1st `  B ) )  ->  E. w  e.  Q.  ( v  +Q  w
)  e.  ( 1st `  B ) )
433ad2antl2 1078 . . . . 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 454 . . . 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 491 . . . . . 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 6565 . . . . . 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 493 . . . . . . . . . 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 265 . . . . . . . . 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 927 . . . . . . . 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 6630 . . . . . . . 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 491 . . . . . . 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 6659 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
1613, 14, 15syl2anc 397 . . . . . 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 465 . . . . . . . . . . . 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 927 . . . . . . . . . . 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 928 . . . . . . . . . . 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 6692 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
2118, 19, 20syl2anc 397 . . . . . . . . . 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 6630 . . . . . . . . . 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 491 . . . . . . . . . . 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 6636 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
2724, 25, 26syl2anc 397 . . . . . . . . . 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 490 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 6636 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  v  e.  ( 1st `  B ) )  -> 
v  e.  Q. )
3228, 30, 31syl2anc 397 . . . . . . . . . . 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 495 . . . . . . . . . . . 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 265 . . . . . . . . . . 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 6530 . . . . . . . . . . 11  |-  ( ( v  e.  Q.  /\  w  e.  Q. )  ->  ( v  +Q  w
)  e.  Q. )
3632, 34, 35syl2anc 397 . . . . . . . . . 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 6530 . . . . . . . . . 10  |-  ( ( u  e.  Q.  /\  ( v  +Q  w
)  e.  Q. )  ->  ( u  +Q  (
v  +Q  w ) )  e.  Q. )
3827, 36, 37syl2anc 397 . . . . . . . . 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 6647 . . . . . . . . 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 397 . . . . . . . 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 265 . . . . . . . . . 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 265 . . . . . . . . . 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 495 . . . . . . . . . 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 496 . . . . . . . . . . 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 465 . . . . . . . . . 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 6623 . . . . . . . . . . . 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 6530 . . . . . . . . . . . 12  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
4846, 47genpprecll 6669 . . . . . . . . . . 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 119 . . . . . . . . . 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 1147 . . . . . . . . 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 265 . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 265 . . . . . . . . . . . . 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 6536 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
5554adantl 266 . . . . . . . . . . . . 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 6537 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
5756adantl 266 . . . . . . . . . . . . 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 6530 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  e.  Q. )
5958adantl 266 . . . . . . . . . . . . 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 5712 . . . . . . . . . . . 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 492 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 5555 . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 6537 . . . . . . . . . . . . 13  |-  ( ( u  e.  Q.  /\  v  e.  Q.  /\  w  e.  Q. )  ->  (
( u  +Q  v
)  +Q  w )  =  ( u  +Q  ( v  +Q  w
) ) )
6651, 53, 64, 65syl3anc 1146 . . . . . . . . . . . 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 2092 . . . . . . . . . . 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 496 . . . . . . . . . . . 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 107 . . . . . . . . . . . 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 929 . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . 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 6670 . . . . . . . . . . . . 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 397 . . . . . . . . . . . 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 417 . . . . . . . . . . 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 2131 . . . . . . . . . 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 107 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( A  +P.  B )  =  ( A  +P.  C ) )
7776ad3antrrr 469 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 5205 . . . . . . . . . . . 12  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( 2nd `  ( A  +P.  B
) )  =  ( 2nd `  ( A  +P.  C ) ) )
8079eleq2d 2123 . . . . . . . . . . 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 160 . . . . . . . . 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 294 . . . . . . . 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 601 . . . . . . 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 6630 . . . . . . . . 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 495 . . . . . . . . 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 6562 . . . . . . . . 9  |-  ( ( v  e.  Q.  /\  t  e.  Q. )  ->  v  <Q  ( v  +Q  t ) )
8932, 87, 88syl2anc 397 . . . . . . . 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 6646 . . . . . . . 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 397 . . . . . . 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 1255 . . . . . 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 2454 . . . . 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 2454 . . . 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 2454 . . 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 112 . 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 2978 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 101    <-> wb 102    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324    C_ wss 2944   <.cop 3405   class class class wbr 3791   ` cfv 4929  (class class class)co 5539   1stc1st 5792   2ndc2nd 5793   Q.cnq 6435    +Q cplq 6437    <Q cltq 6440   P.cnp 6446    +P. cpp 6448
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-iplp 6623
This theorem is referenced by:  addcanprg  6771
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