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Theorem addassnq 8837
Description: Addition of positive fractions is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
addassnq  |-  ( ( A  +Q  B )  +Q  C )  =  ( A  +Q  ( B  +Q  C ) )

Proof of Theorem addassnq
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addasspi 8774 . . . . . . . 8  |-  ( ( ( ( 1st `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  +N  ( ( ( 1st `  B )  .N  ( 2nd `  A ) )  .N  ( 2nd `  C
) ) )  +N  ( ( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) )  =  ( ( ( 1st `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  +N  ( ( ( ( 1st `  B
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) ) )
2 ovex 6108 . . . . . . . . . . 11  |-  ( ( 1st `  A )  .N  ( 2nd `  B
) )  e.  _V
3 ovex 6108 . . . . . . . . . . 11  |-  ( ( 1st `  B )  .N  ( 2nd `  A
) )  e.  _V
4 fvex 5744 . . . . . . . . . . 11  |-  ( 2nd `  C )  e.  _V
5 mulcompi 8775 . . . . . . . . . . 11  |-  ( x  .N  y )  =  ( y  .N  x
)
6 distrpi 8777 . . . . . . . . . . 11  |-  ( x  .N  ( y  +N  z ) )  =  ( ( x  .N  y )  +N  (
x  .N  z ) )
72, 3, 4, 5, 6caovdir 6283 . . . . . . . . . 10  |-  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  .N  ( 2nd `  C
) )  =  ( ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  .N  ( 2nd `  C
) )  +N  (
( ( 1st `  B
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  C
) ) )
8 mulasspi 8776 . . . . . . . . . . 11  |-  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  C
) )  =  ( ( 1st `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )
98oveq1i 6093 . . . . . . . . . 10  |-  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  C
) )  +N  (
( ( 1st `  B
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  C
) ) )  =  ( ( ( 1st `  A )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) )  +N  ( ( ( 1st `  B )  .N  ( 2nd `  A
) )  .N  ( 2nd `  C ) ) )
107, 9eqtri 2458 . . . . . . . . 9  |-  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  .N  ( 2nd `  C
) )  =  ( ( ( 1st `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  +N  ( ( ( 1st `  B )  .N  ( 2nd `  A ) )  .N  ( 2nd `  C
) ) )
1110oveq1i 6093 . . . . . . . 8  |-  ( ( ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) )  =  ( ( ( ( 1st `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  +N  ( ( ( 1st `  B )  .N  ( 2nd `  A ) )  .N  ( 2nd `  C
) ) )  +N  ( ( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) )
12 ovex 6108 . . . . . . . . . . 11  |-  ( ( 1st `  B )  .N  ( 2nd `  C
) )  e.  _V
13 ovex 6108 . . . . . . . . . . 11  |-  ( ( 1st `  C )  .N  ( 2nd `  B
) )  e.  _V
14 fvex 5744 . . . . . . . . . . 11  |-  ( 2nd `  A )  e.  _V
1512, 13, 14, 5, 6caovdir 6283 . . . . . . . . . 10  |-  ( ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  .N  ( 2nd `  A
) )  =  ( ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  .N  ( 2nd `  A
) )  +N  (
( ( 1st `  C
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  A
) ) )
16 fvex 5744 . . . . . . . . . . . 12  |-  ( 1st `  B )  e.  _V
17 mulasspi 8776 . . . . . . . . . . . 12  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
1816, 4, 14, 5, 17caov32 6276 . . . . . . . . . . 11  |-  ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  .N  ( 2nd `  A
) )  =  ( ( ( 1st `  B
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  C
) )
19 mulasspi 8776 . . . . . . . . . . . 12  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  A
) )  =  ( ( 1st `  C
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  A
) ) )
20 mulcompi 8775 . . . . . . . . . . . . 13  |-  ( ( 2nd `  B )  .N  ( 2nd `  A
) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  B ) )
2120oveq2i 6094 . . . . . . . . . . . 12  |-  ( ( 1st `  C )  .N  ( ( 2nd `  B )  .N  ( 2nd `  A ) ) )  =  ( ( 1st `  C )  .N  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) )
2219, 21eqtri 2458 . . . . . . . . . . 11  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  A
) )  =  ( ( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) )
2318, 22oveq12i 6095 . . . . . . . . . 10  |-  ( ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  .N  ( 2nd `  A
) )  +N  (
( ( 1st `  C
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  A
) ) )  =  ( ( ( ( 1st `  B )  .N  ( 2nd `  A
) )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  (
( 2nd `  A
)  .N  ( 2nd `  B ) ) ) )
2415, 23eqtri 2458 . . . . . . . . 9  |-  ( ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  .N  ( 2nd `  A
) )  =  ( ( ( ( 1st `  B )  .N  ( 2nd `  A ) )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) )
2524oveq2i 6094 . . . . . . . 8  |-  ( ( ( 1st `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  +N  ( ( ( ( 1st `  B )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) )  .N  ( 2nd `  A
) ) )  =  ( ( ( 1st `  A )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) )  +N  ( ( ( ( 1st `  B
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) ) )
261, 11, 253eqtr4i 2468 . . . . . . 7  |-  ( ( ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) )  =  ( ( ( 1st `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  +N  ( ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  .N  ( 2nd `  A
) ) )
27 mulasspi 8776 . . . . . . 7  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  C
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )
2826, 27opeq12i 3991 . . . . . 6  |-  <. (
( ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  +N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  B
) )  .N  ( 2nd `  C ) )
>.  =  <. ( ( ( 1st `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  +N  ( ( ( ( 1st `  B )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) )  .N  ( 2nd `  A
) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) ) >.
29 elpqn 8804 . . . . . . . . . 10  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
30293ad2ant1 979 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
31 elpqn 8804 . . . . . . . . . 10  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
32313ad2ant2 980 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
33 addpipq2 8815 . . . . . . . . 9  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  +pQ  B )  = 
<. ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
3430, 32, 33syl2anc 644 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  +pQ  B )  = 
<. ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >. )
35 relxp 4985 . . . . . . . . 9  |-  Rel  ( N.  X.  N. )
36 elpqn 8804 . . . . . . . . . 10  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
37363ad2ant3 981 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
38 1st2nd 6395 . . . . . . . . 9  |-  ( ( Rel  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
3935, 37, 38sylancr 646 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
4034, 39oveq12d 6101 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  +pQ  B
)  +pQ  C )  =  ( <. (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>.  +pQ  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
)
41 xp1st 6378 . . . . . . . . . . 11  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
4230, 41syl 16 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  A )  e. 
N. )
43 xp2nd 6379 . . . . . . . . . . 11  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
4432, 43syl 16 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  B )  e. 
N. )
45 mulclpi 8772 . . . . . . . . . 10  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
4642, 44, 45syl2anc 644 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
47 xp1st 6378 . . . . . . . . . . 11  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
4832, 47syl 16 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  B )  e. 
N. )
49 xp2nd 6379 . . . . . . . . . . 11  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
5030, 49syl 16 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  A )  e. 
N. )
51 mulclpi 8772 . . . . . . . . . 10  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  e. 
N. )
5248, 50, 51syl2anc 644 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  B
)  .N  ( 2nd `  A ) )  e. 
N. )
53 addclpi 8771 . . . . . . . . 9  |-  ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N.  /\  ( ( 1st `  B )  .N  ( 2nd `  A
) )  e.  N. )  ->  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  +N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  e.  N. )
5446, 52, 53syl2anc 644 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  e.  N. )
55 mulclpi 8772 . . . . . . . . 9  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
5650, 44, 55syl2anc 644 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
57 xp1st 6378 . . . . . . . . 9  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
5837, 57syl 16 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  C )  e. 
N. )
59 xp2nd 6379 . . . . . . . . 9  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
6037, 59syl 16 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  C )  e. 
N. )
61 addpipq 8816 . . . . . . . 8  |-  ( ( ( ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  +N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  e.  N.  /\  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )  /\  (
( 1st `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. ) )  ->  ( <. (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>.  +pQ  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )  =  <. ( ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  B
) )  .N  ( 2nd `  C ) )
>. )
6254, 56, 58, 60, 61syl22anc 1186 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( <. ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) >.  +pQ  <. ( 1st `  C ) ,  ( 2nd `  C
) >. )  =  <. ( ( ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  +N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  B
) )  .N  ( 2nd `  C ) )
>. )
6340, 62eqtrd 2470 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  +pQ  B
)  +pQ  C )  =  <. ( ( ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  B
) )  .N  ( 2nd `  C ) )
>. )
64 1st2nd 6395 . . . . . . . . 9  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
6535, 30, 64sylancr 646 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
66 addpipq2 8815 . . . . . . . . 9  |-  ( ( B  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( B  +pQ  C )  = 
<. ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) >. )
6732, 37, 66syl2anc 644 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +pQ  C )  = 
<. ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) >. )
6865, 67oveq12d 6101 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  +pQ  ( B  +pQ  C ) )  =  (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  +pQ  <. (
( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C ) )
>. ) )
69 mulclpi 8772 . . . . . . . . . 10  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
7048, 60, 69syl2anc 644 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
71 mulclpi 8772 . . . . . . . . . 10  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
7258, 44, 71syl2anc 644 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
73 addclpi 8771 . . . . . . . . 9  |-  ( ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N.  /\  ( ( 1st `  C )  .N  ( 2nd `  B
) )  e.  N. )  ->  ( ( ( 1st `  B )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) )  e.  N. )
7470, 72, 73syl2anc 644 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  e.  N. )
75 mulclpi 8772 . . . . . . . . 9  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
7644, 60, 75syl2anc 644 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
77 addpipq 8816 . . . . . . . 8  |-  ( ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  /\  ( ( ( ( 1st `  B )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) )  e.  N.  /\  (
( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. ) )  -> 
( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  +pQ  <. (
( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C ) )
>. )  =  <. ( ( ( 1st `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  +N  ( ( ( ( 1st `  B )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) )  .N  ( 2nd `  A
) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) ) >.
)
7842, 50, 74, 76, 77syl22anc 1186 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  +pQ  <. (
( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C ) )
>. )  =  <. ( ( ( 1st `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  +N  ( ( ( ( 1st `  B )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) )  .N  ( 2nd `  A
) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) ) >.
)
7968, 78eqtrd 2470 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  +pQ  ( B  +pQ  C ) )  =  <. ( ( ( 1st `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  +N  ( ( ( ( 1st `  B )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) )  .N  ( 2nd `  A
) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) ) >.
)
8028, 63, 793eqtr4a 2496 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  +pQ  B
)  +pQ  C )  =  ( A  +pQ  ( B  +pQ  C ) ) )
8180fveq2d 5734 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( /Q `  ( ( A 
+pQ  B )  +pQ  C ) )  =  ( /Q `  ( A 
+pQ  ( B  +pQ  C ) ) ) )
82 adderpq 8835 . . . 4  |-  ( ( /Q `  ( A 
+pQ  B ) )  +Q  ( /Q `  C ) )  =  ( /Q `  (
( A  +pQ  B
)  +pQ  C )
)
83 adderpq 8835 . . . 4  |-  ( ( /Q `  A )  +Q  ( /Q `  ( B  +pQ  C ) ) )  =  ( /Q `  ( A 
+pQ  ( B  +pQ  C ) ) )
8481, 82, 833eqtr4g 2495 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( /Q `  ( A  +pQ  B ) )  +Q  ( /Q `  C ) )  =  ( ( /Q `  A )  +Q  ( /Q `  ( B  +pQ  C ) ) ) )
85 addpqnq 8817 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( /Q
`  ( A  +pQ  B ) ) )
86853adant3 978 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  +Q  B )  =  ( /Q `  ( A  +pQ  B ) ) )
87 nqerid 8812 . . . . . 6  |-  ( C  e.  Q.  ->  ( /Q `  C )  =  C )
8887eqcomd 2443 . . . . 5  |-  ( C  e.  Q.  ->  C  =  ( /Q `  C ) )
89883ad2ant3 981 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  =  ( /Q `  C ) )
9086, 89oveq12d 6101 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  +Q  B
)  +Q  C )  =  ( ( /Q
`  ( A  +pQ  B ) )  +Q  ( /Q `  C ) ) )
91 nqerid 8812 . . . . . 6  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
9291eqcomd 2443 . . . . 5  |-  ( A  e.  Q.  ->  A  =  ( /Q `  A ) )
93923ad2ant1 979 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  ( /Q `  A ) )
94 addpqnq 8817 . . . . 5  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +Q  C
)  =  ( /Q
`  ( B  +pQ  C ) ) )
95943adant1 976 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +Q  C )  =  ( /Q `  ( B  +pQ  C ) ) )
9693, 95oveq12d 6101 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  +Q  ( B  +Q  C ) )  =  ( ( /Q `  A )  +Q  ( /Q `  ( B  +pQ  C ) ) ) )
9784, 90, 963eqtr4d 2480 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  +Q  B
)  +Q  C )  =  ( A  +Q  ( B  +Q  C
) ) )
98 addnqf 8827 . . . 4  |-  +Q  :
( Q.  X.  Q. )
--> Q.
9998fdmi 5598 . . 3  |-  dom  +Q  =  ( Q.  X.  Q. )
100 0nnq 8803 . . 3  |-  -.  (/)  e.  Q.
10199, 100ndmovass 6237 . 2  |-  ( -.  ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( ( A  +Q  B )  +Q  C
)  =  ( A  +Q  ( B  +Q  C ) ) )
10297, 101pm2.61i 159 1  |-  ( ( A  +Q  B )  +Q  C )  =  ( A  +Q  ( B  +Q  C ) )
Colors of variables: wff set class
Syntax hints:    /\ w3a 937    = wceq 1653    e. wcel 1726   <.cop 3819    X. cxp 4878   Rel wrel 4885   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350   N.cnpi 8721    +N cpli 8722    .N cmi 8723    +pQ cplpq 8725   Q.cnq 8729   /Qcerq 8731    +Q cplq 8732
This theorem is referenced by:  ltaddnq  8853  addasspr  8901  prlem934  8912  ltexprlem7  8921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-omul 6731  df-er 6907  df-ni 8751  df-pli 8752  df-mi 8753  df-lti 8754  df-plpq 8787  df-enq 8790  df-nq 8791  df-erq 8792  df-plq 8793  df-1nq 8795
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