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Theorem addassnq 9142
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 9079 . . . . . . . 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 6131 . . . . . . . . . . 11  |-  ( ( 1st `  A )  .N  ( 2nd `  B
) )  e.  _V
3 ovex 6131 . . . . . . . . . . 11  |-  ( ( 1st `  B )  .N  ( 2nd `  A
) )  e.  _V
4 fvex 5716 . . . . . . . . . . 11  |-  ( 2nd `  C )  e.  _V
5 mulcompi 9080 . . . . . . . . . . 11  |-  ( x  .N  y )  =  ( y  .N  x
)
6 distrpi 9082 . . . . . . . . . . 11  |-  ( x  .N  ( y  +N  z ) )  =  ( ( x  .N  y )  +N  (
x  .N  z ) )
72, 3, 4, 5, 6caovdir 6312 . . . . . . . . . 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 9081 . . . . . . . . . . 11  |-  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  C
) )  =  ( ( 1st `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )
98oveq1i 6116 . . . . . . . . . 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 2463 . . . . . . . . 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 6116 . . . . . . . 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 6131 . . . . . . . . . . 11  |-  ( ( 1st `  B )  .N  ( 2nd `  C
) )  e.  _V
13 ovex 6131 . . . . . . . . . . 11  |-  ( ( 1st `  C )  .N  ( 2nd `  B
) )  e.  _V
14 fvex 5716 . . . . . . . . . . 11  |-  ( 2nd `  A )  e.  _V
1512, 13, 14, 5, 6caovdir 6312 . . . . . . . . . 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 5716 . . . . . . . . . . . 12  |-  ( 1st `  B )  e.  _V
17 mulasspi 9081 . . . . . . . . . . . 12  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
1816, 4, 14, 5, 17caov32 6305 . . . . . . . . . . 11  |-  ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  .N  ( 2nd `  A
) )  =  ( ( ( 1st `  B
)  .N  ( 2nd `  A ) )  .N  ( 2nd `  C
) )
19 mulasspi 9081 . . . . . . . . . . . 12  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  A
) )  =  ( ( 1st `  C
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  A
) ) )
20 mulcompi 9080 . . . . . . . . . . . . 13  |-  ( ( 2nd `  B )  .N  ( 2nd `  A
) )  =  ( ( 2nd `  A
)  .N  ( 2nd `  B ) )
2120oveq2i 6117 . . . . . . . . . . . 12  |-  ( ( 1st `  C )  .N  ( ( 2nd `  B )  .N  ( 2nd `  A ) ) )  =  ( ( 1st `  C )  .N  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) )
2219, 21eqtri 2463 . . . . . . . . . . 11  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  A
) )  =  ( ( 1st `  C
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  B
) ) )
2318, 22oveq12i 6118 . . . . . . . . . 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 2463 . . . . . . . . 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 6117 . . . . . . . 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 2473 . . . . . . 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 9081 . . . . . . 7  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  C
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )
2826, 27opeq12i 4079 . . . . . 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 9109 . . . . . . . . . 10  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
30293ad2ant1 1009 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
31 elpqn 9109 . . . . . . . . . 10  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
32313ad2ant2 1010 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
33 addpipq2 9120 . . . . . . . . 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 661 . . . . . . . 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 4962 . . . . . . . . 9  |-  Rel  ( N.  X.  N. )
36 elpqn 9109 . . . . . . . . . 10  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
37363ad2ant3 1011 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
38 1st2nd 6635 . . . . . . . . 9  |-  ( ( Rel  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
3935, 37, 38sylancr 663 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
4034, 39oveq12d 6124 . . . . . . 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 6621 . . . . . . . . . . 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 6622 . . . . . . . . . . 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 9077 . . . . . . . . . 10  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
4642, 44, 45syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
47 xp1st 6621 . . . . . . . . . . 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 6622 . . . . . . . . . . 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 9077 . . . . . . . . . 10  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  e. 
N. )
5248, 50, 51syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  B
)  .N  ( 2nd `  A ) )  e. 
N. )
53 addclpi 9076 . . . . . . . . 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 661 . . . . . . . 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 9077 . . . . . . . . 9  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
5650, 44, 55syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
57 xp1st 6621 . . . . . . . . 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 6622 . . . . . . . . 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 9121 . . . . . . . 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 1219 . . . . . . 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 2475 . . . . . 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 6635 . . . . . . . . 9  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
6535, 30, 64sylancr 663 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
66 addpipq2 9120 . . . . . . . . 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 661 . . . . . . . 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 6124 . . . . . . 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 9077 . . . . . . . . . 10  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
7048, 60, 69syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
71 mulclpi 9077 . . . . . . . . . 10  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
7258, 44, 71syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
73 addclpi 9076 . . . . . . . . 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 661 . . . . . . . 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 9077 . . . . . . . . 9  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
7644, 60, 75syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
77 addpipq 9121 . . . . . . . 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 1219 . . . . . . 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 2475 . . . . . 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 2501 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  +pQ  B
)  +pQ  C )  =  ( A  +pQ  ( B  +pQ  C ) ) )
8180fveq2d 5710 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( /Q `  ( ( A 
+pQ  B )  +pQ  C ) )  =  ( /Q `  ( A 
+pQ  ( B  +pQ  C ) ) ) )
82 adderpq 9140 . . . 4  |-  ( ( /Q `  ( A 
+pQ  B ) )  +Q  ( /Q `  C ) )  =  ( /Q `  (
( A  +pQ  B
)  +pQ  C )
)
83 adderpq 9140 . . . 4  |-  ( ( /Q `  A )  +Q  ( /Q `  ( B  +pQ  C ) ) )  =  ( /Q `  ( A 
+pQ  ( B  +pQ  C ) ) )
8481, 82, 833eqtr4g 2500 . . 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 9122 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( /Q
`  ( A  +pQ  B ) ) )
86853adant3 1008 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  +Q  B )  =  ( /Q `  ( A  +pQ  B ) ) )
87 nqerid 9117 . . . . . 6  |-  ( C  e.  Q.  ->  ( /Q `  C )  =  C )
8887eqcomd 2448 . . . . 5  |-  ( C  e.  Q.  ->  C  =  ( /Q `  C ) )
89883ad2ant3 1011 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  =  ( /Q `  C ) )
9086, 89oveq12d 6124 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  +Q  B
)  +Q  C )  =  ( ( /Q
`  ( A  +pQ  B ) )  +Q  ( /Q `  C ) ) )
91 nqerid 9117 . . . . . 6  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
9291eqcomd 2448 . . . . 5  |-  ( A  e.  Q.  ->  A  =  ( /Q `  A ) )
93923ad2ant1 1009 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  ( /Q `  A ) )
94 addpqnq 9122 . . . . 5  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +Q  C
)  =  ( /Q
`  ( B  +pQ  C ) ) )
95943adant1 1006 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +Q  C )  =  ( /Q `  ( B  +pQ  C ) ) )
9693, 95oveq12d 6124 . . 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 2485 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  +Q  B
)  +Q  C )  =  ( A  +Q  ( B  +Q  C
) ) )
98 addnqf 9132 . . . 4  |-  +Q  :
( Q.  X.  Q. )
--> Q.
9998fdmi 5579 . . 3  |-  dom  +Q  =  ( Q.  X.  Q. )
100 0nnq 9108 . . 3  |-  -.  (/)  e.  Q.
10199, 100ndmovass 6266 . 2  |-  ( -.  ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( ( A  +Q  B )  +Q  C
)  =  ( A  +Q  ( B  +Q  C ) ) )
10297, 101pm2.61i 164 1  |-  ( ( A  +Q  B )  +Q  C )  =  ( A  +Q  ( B  +Q  C ) )
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 965    = wceq 1369    e. wcel 1756   <.cop 3898    X. cxp 4853   Rel wrel 4860   ` cfv 5433  (class class class)co 6106   1stc1st 6590   2ndc2nd 6591   N.cnpi 9026    +N cpli 9027    .N cmi 9028    +pQ cplpq 9030   Q.cnq 9034   /Qcerq 9036    +Q cplq 9037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-omul 6940  df-er 7116  df-ni 9056  df-pli 9057  df-mi 9058  df-lti 9059  df-plpq 9092  df-enq 9095  df-nq 9096  df-erq 9097  df-plq 9098  df-1nq 9100
This theorem is referenced by:  ltaddnq  9158  addasspr  9206  prlem934  9217  ltexprlem7  9226
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