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Theorem distrnq 9126
Description: Multiplication of positive fractions is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
distrnq  |-  ( A  .Q  ( B  +Q  C ) )  =  ( ( A  .Q  B )  +Q  ( A  .Q  C ) )

Proof of Theorem distrnq
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulcompi 9061 . . . . . . . . . . . . 13  |-  ( ( 1st `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 1st `  A ) )
21oveq1i 6100 . . . . . . . . . . . 12  |-  ( ( ( 1st `  A
)  .N  ( 1st `  B ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 1st `  B )  .N  ( 1st `  A
) )  .N  (
( 2nd `  A
)  .N  ( 2nd `  C ) ) )
3 fvex 5698 . . . . . . . . . . . . 13  |-  ( 1st `  B )  e.  _V
4 fvex 5698 . . . . . . . . . . . . 13  |-  ( 1st `  A )  e.  _V
5 fvex 5698 . . . . . . . . . . . . 13  |-  ( 2nd `  A )  e.  _V
6 mulcompi 9061 . . . . . . . . . . . . 13  |-  ( x  .N  y )  =  ( y  .N  x
)
7 mulasspi 9062 . . . . . . . . . . . . 13  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
8 fvex 5698 . . . . . . . . . . . . 13  |-  ( 2nd `  C )  e.  _V
93, 4, 5, 6, 7, 8caov411 6294 . . . . . . . . . . . 12  |-  ( ( ( 1st `  B
)  .N  ( 1st `  A ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A )  .N  ( 1st `  A
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) )
102, 9eqtri 2461 . . . . . . . . . . 11  |-  ( ( ( 1st `  A
)  .N  ( 1st `  B ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A )  .N  ( 1st `  A
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) )
11 mulcompi 9061 . . . . . . . . . . . . 13  |-  ( ( 1st `  A )  .N  ( 1st `  C
) )  =  ( ( 1st `  C
)  .N  ( 1st `  A ) )
1211oveq1i 6100 . . . . . . . . . . . 12  |-  ( ( ( 1st `  A
)  .N  ( 1st `  C ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  C )  .N  ( 1st `  A
) )  .N  (
( 2nd `  A
)  .N  ( 2nd `  B ) ) )
13 fvex 5698 . . . . . . . . . . . . 13  |-  ( 1st `  C )  e.  _V
14 fvex 5698 . . . . . . . . . . . . 13  |-  ( 2nd `  B )  e.  _V
1513, 4, 5, 6, 7, 14caov411 6294 . . . . . . . . . . . 12  |-  ( ( ( 1st `  C
)  .N  ( 1st `  A ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  A )  .N  ( 1st `  A
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) )
1612, 15eqtri 2461 . . . . . . . . . . 11  |-  ( ( ( 1st `  A
)  .N  ( 1st `  C ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  A )  .N  ( 1st `  A
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) )
1710, 16oveq12i 6102 . . . . . . . . . 10  |-  ( ( ( ( 1st `  A
)  .N  ( 1st `  B ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )  +N  ( ( ( 1st `  A )  .N  ( 1st `  C
) )  .N  (
( 2nd `  A
)  .N  ( 2nd `  B ) ) ) )  =  ( ( ( ( 2nd `  A
)  .N  ( 1st `  A ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  C ) ) )  +N  ( ( ( 2nd `  A )  .N  ( 1st `  A
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) ) )
18 distrpi 9063 . . . . . . . . . 10  |-  ( ( ( 2nd `  A
)  .N  ( 1st `  A ) )  .N  ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) ) )  =  ( ( ( ( 2nd `  A )  .N  ( 1st `  A ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  C ) ) )  +N  ( ( ( 2nd `  A
)  .N  ( 1st `  A ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) )
19 mulasspi 9062 . . . . . . . . . 10  |-  ( ( ( 2nd `  A
)  .N  ( 1st `  A ) )  .N  ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) ) )  =  ( ( 2nd `  A
)  .N  ( ( 1st `  A )  .N  ( ( ( 1st `  B )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) ) ) )
2017, 18, 193eqtr2i 2467 . . . . . . . . 9  |-  ( ( ( ( 1st `  A
)  .N  ( 1st `  B ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )  +N  ( ( ( 1st `  A )  .N  ( 1st `  C
) )  .N  (
( 2nd `  A
)  .N  ( 2nd `  B ) ) ) )  =  ( ( 2nd `  A )  .N  ( ( 1st `  A )  .N  (
( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) ) )
21 mulasspi 9062 . . . . . . . . . 10  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )  =  ( ( 2nd `  A )  .N  (
( 2nd `  B
)  .N  ( ( 2nd `  A )  .N  ( 2nd `  C
) ) ) )
2214, 5, 8, 6, 7caov12 6290 . . . . . . . . . . 11  |-  ( ( 2nd `  B )  .N  ( ( 2nd `  A )  .N  ( 2nd `  C ) ) )  =  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )
2322oveq2i 6101 . . . . . . . . . 10  |-  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  (
( 2nd `  A
)  .N  ( 2nd `  C ) ) ) )  =  ( ( 2nd `  A )  .N  ( ( 2nd `  A )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) ) )
2421, 23eqtri 2461 . . . . . . . . 9  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )  =  ( ( 2nd `  A )  .N  (
( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) ) )
2520, 24opeq12i 4061 . . . . . . . 8  |-  <. (
( ( ( 1st `  A )  .N  ( 1st `  B ) )  .N  ( ( 2nd `  A )  .N  ( 2nd `  C ) ) )  +N  ( ( ( 1st `  A
)  .N  ( 1st `  C ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) ) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )
>.  =  <. ( ( 2nd `  A )  .N  ( ( 1st `  A )  .N  (
( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) ) ) >.
26 elpqn 9090 . . . . . . . . . . 11  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
27263ad2ant1 1004 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
28 xp2nd 6606 . . . . . . . . . 10  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
2927, 28syl 16 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  A )  e. 
N. )
30 xp1st 6605 . . . . . . . . . . 11  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
3127, 30syl 16 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  A )  e. 
N. )
32 elpqn 9090 . . . . . . . . . . . . . 14  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
33323ad2ant2 1005 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
34 xp1st 6605 . . . . . . . . . . . . 13  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
3533, 34syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  B )  e. 
N. )
36 elpqn 9090 . . . . . . . . . . . . . 14  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
37363ad2ant3 1006 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
38 xp2nd 6606 . . . . . . . . . . . . 13  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
3937, 38syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  C )  e. 
N. )
40 mulclpi 9058 . . . . . . . . . . . 12  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
4135, 39, 40syl2anc 656 . . . . . . . . . . 11  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
42 xp1st 6605 . . . . . . . . . . . . 13  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
4337, 42syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  C )  e. 
N. )
44 xp2nd 6606 . . . . . . . . . . . . 13  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
4533, 44syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  B )  e. 
N. )
46 mulclpi 9058 . . . . . . . . . . . 12  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
4743, 45, 46syl2anc 656 . . . . . . . . . . 11  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
48 addclpi 9057 . . . . . . . . . . 11  |-  ( ( ( ( 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. )
4941, 47, 48syl2anc 656 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  e.  N. )
50 mulclpi 9058 . . . . . . . . . 10  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) )  e.  N. )  ->  ( ( 1st `  A
)  .N  ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) )  e.  N. )
5131, 49, 50syl2anc 656 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) )  e.  N. )
52 mulclpi 9058 . . . . . . . . . . 11  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
5345, 39, 52syl2anc 656 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
54 mulclpi 9058 . . . . . . . . . 10  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )  ->  (
( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  e. 
N. )
5529, 53, 54syl2anc 656 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  e. 
N. )
56 mulcanenq 9125 . . . . . . . . 9  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( ( 1st `  A
)  .N  ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) )  e.  N.  /\  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  e. 
N. )  ->  <. (
( 2nd `  A
)  .N  ( ( 1st `  A )  .N  ( ( ( 1st `  B )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) ) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) ) ) >.  ~Q  <. ( ( 1st `  A
)  .N  ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) ) ,  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) ) >. )
5729, 51, 55, 56syl3anc 1213 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  <. (
( 2nd `  A
)  .N  ( ( 1st `  A )  .N  ( ( ( 1st `  B )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) ) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) ) ) >.  ~Q  <. ( ( 1st `  A
)  .N  ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) ) ,  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) ) >. )
5825, 57syl5eqbr 4322 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  <. (
( ( ( 1st `  A )  .N  ( 1st `  B ) )  .N  ( ( 2nd `  A )  .N  ( 2nd `  C ) ) )  +N  ( ( ( 1st `  A
)  .N  ( 1st `  C ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) ) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )
>.  ~Q  <. ( ( 1st `  A )  .N  (
( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) ) ,  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) ) >. )
59 mulpipq2 9104 . . . . . . . . . 10  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  = 
<. ( ( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)
6027, 33, 59syl2anc 656 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  B )  = 
<. ( ( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)
61 mulpipq2 9104 . . . . . . . . . 10  |-  ( ( A  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  C )  = 
<. ( ( 1st `  A
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) >.
)
6227, 37, 61syl2anc 656 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  C )  = 
<. ( ( 1st `  A
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) >.
)
6360, 62oveq12d 6108 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .pQ  B
)  +pQ  ( A  .pQ  C ) )  =  ( <. ( ( 1st `  A )  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
>.  +pQ  <. ( ( 1st `  A )  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  C ) )
>. ) )
64 mulclpi 9058 . . . . . . . . . 10  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
6531, 35, 64syl2anc 656 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
66 mulclpi 9058 . . . . . . . . . 10  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
6729, 45, 66syl2anc 656 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
68 mulclpi 9058 . . . . . . . . . 10  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  C ) )  e. 
N. )
6931, 43, 68syl2anc 656 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( 1st `  C ) )  e. 
N. )
70 mulclpi 9058 . . . . . . . . . 10  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
7129, 39, 70syl2anc 656 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
72 addpipq 9102 . . . . . . . . 9  |-  ( ( ( ( ( 1st `  A )  .N  ( 1st `  B ) )  e.  N.  /\  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )  /\  (
( ( 1st `  A
)  .N  ( 1st `  C ) )  e. 
N.  /\  ( ( 2nd `  A )  .N  ( 2nd `  C
) )  e.  N. ) )  ->  ( <. ( ( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.  +pQ  <. ( ( 1st `  A )  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  C ) )
>. )  =  <. ( ( ( ( 1st `  A )  .N  ( 1st `  B ) )  .N  ( ( 2nd `  A )  .N  ( 2nd `  C ) ) )  +N  ( ( ( 1st `  A
)  .N  ( 1st `  C ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) ) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )
>. )
7365, 67, 69, 71, 72syl22anc 1214 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( <. ( ( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.  +pQ  <. ( ( 1st `  A )  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  C ) )
>. )  =  <. ( ( ( ( 1st `  A )  .N  ( 1st `  B ) )  .N  ( ( 2nd `  A )  .N  ( 2nd `  C ) ) )  +N  ( ( ( 1st `  A
)  .N  ( 1st `  C ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) ) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )
>. )
7463, 73eqtrd 2473 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .pQ  B
)  +pQ  ( A  .pQ  C ) )  = 
<. ( ( ( ( 1st `  A )  .N  ( 1st `  B
) )  .N  (
( 2nd `  A
)  .N  ( 2nd `  C ) ) )  +N  ( ( ( 1st `  A )  .N  ( 1st `  C
) )  .N  (
( 2nd `  A
)  .N  ( 2nd `  B ) ) ) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )
>. )
75 relxp 4943 . . . . . . . . . 10  |-  Rel  ( N.  X.  N. )
76 1st2nd 6619 . . . . . . . . . 10  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
7775, 27, 76sylancr 658 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
78 addpipq2 9101 . . . . . . . . . 10  |-  ( ( 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
) ) >. )
7933, 37, 78syl2anc 656 . . . . . . . . 9  |-  ( ( 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
) ) >. )
8077, 79oveq12d 6108 . . . . . . . 8  |-  ( ( 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 ) )
>. ) )
81 mulpipq 9105 . . . . . . . . 9  |-  ( ( ( ( 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  ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) ) ,  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) ) >. )
8231, 29, 49, 53, 81syl22anc 1214 . . . . . . . 8  |-  ( ( 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  ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) ) ,  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) ) >. )
8380, 82eqtrd 2473 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  ( B  +pQ  C ) )  =  <. ( ( 1st `  A
)  .N  ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) ) ,  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) ) >. )
8458, 74, 833brtr4d 4319 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .pQ  B
)  +pQ  ( A  .pQ  C ) )  ~Q  ( A  .pQ  ( B 
+pQ  C ) ) )
85 mulpqf 9111 . . . . . . . . . 10  |-  .pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
8685fovcl 6194 . . . . . . . . 9  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  e.  ( N.  X.  N. ) )
8727, 33, 86syl2anc 656 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  B )  e.  ( N.  X.  N. ) )
8885fovcl 6194 . . . . . . . . 9  |-  ( ( A  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  C )  e.  ( N.  X.  N. ) )
8927, 37, 88syl2anc 656 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  C )  e.  ( N.  X.  N. ) )
90 addpqf 9109 . . . . . . . . 9  |-  +pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
9190fovcl 6194 . . . . . . . 8  |-  ( ( ( A  .pQ  B
)  e.  ( N. 
X.  N. )  /\  ( A  .pQ  C )  e.  ( N.  X.  N. ) )  ->  (
( A  .pQ  B
)  +pQ  ( A  .pQ  C ) )  e.  ( N.  X.  N. ) )
9287, 89, 91syl2anc 656 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .pQ  B
)  +pQ  ( A  .pQ  C ) )  e.  ( N.  X.  N. ) )
9390fovcl 6194 . . . . . . . . 9  |-  ( ( B  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( B  +pQ  C )  e.  ( N.  X.  N. ) )
9433, 37, 93syl2anc 656 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +pQ  C )  e.  ( N.  X.  N. ) )
9585fovcl 6194 . . . . . . . 8  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  +pQ  C )  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  ( B  +pQ  C ) )  e.  ( N.  X.  N. )
)
9627, 94, 95syl2anc 656 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  ( B  +pQ  C ) )  e.  ( N.  X.  N. )
)
97 nqereq 9100 . . . . . . 7  |-  ( ( ( ( A  .pQ  B )  +pQ  ( A 
.pQ  C ) )  e.  ( N.  X.  N. )  /\  ( A  .pQ  ( B  +pQ  C ) )  e.  ( N.  X.  N. )
)  ->  ( (
( A  .pQ  B
)  +pQ  ( A  .pQ  C ) )  ~Q  ( A  .pQ  ( B 
+pQ  C ) )  <-> 
( /Q `  (
( A  .pQ  B
)  +pQ  ( A  .pQ  C ) ) )  =  ( /Q `  ( A  .pQ  ( B 
+pQ  C ) ) ) ) )
9892, 96, 97syl2anc 656 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( A  .pQ  B )  +pQ  ( A 
.pQ  C ) )  ~Q  ( A  .pQ  ( B  +pQ  C ) )  <->  ( /Q `  ( ( A  .pQ  B )  +pQ  ( A 
.pQ  C ) ) )  =  ( /Q
`  ( A  .pQ  ( B  +pQ  C ) ) ) ) )
9984, 98mpbid 210 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( /Q `  ( ( A 
.pQ  B )  +pQ  ( A  .pQ  C ) ) )  =  ( /Q `  ( A 
.pQ  ( B  +pQ  C ) ) ) )
10099eqcomd 2446 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( /Q `  ( A  .pQ  ( B  +pQ  C ) ) )  =  ( /Q `  ( ( A  .pQ  B ) 
+pQ  ( A  .pQ  C ) ) ) )
101 mulerpq 9122 . . . 4  |-  ( ( /Q `  A )  .Q  ( /Q `  ( B  +pQ  C ) ) )  =  ( /Q `  ( A 
.pQ  ( B  +pQ  C ) ) )
102 adderpq 9121 . . . 4  |-  ( ( /Q `  ( A 
.pQ  B ) )  +Q  ( /Q `  ( A  .pQ  C ) ) )  =  ( /Q `  ( ( A  .pQ  B ) 
+pQ  ( A  .pQ  C ) ) )
103100, 101, 1023eqtr4g 2498 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( /Q `  A
)  .Q  ( /Q
`  ( B  +pQ  C ) ) )  =  ( ( /Q `  ( A  .pQ  B ) )  +Q  ( /Q
`  ( A  .pQ  C ) ) ) )
104 nqerid 9098 . . . . . 6  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
105104eqcomd 2446 . . . . 5  |-  ( A  e.  Q.  ->  A  =  ( /Q `  A ) )
1061053ad2ant1 1004 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  ( /Q `  A ) )
107 addpqnq 9103 . . . . 5  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +Q  C
)  =  ( /Q
`  ( B  +pQ  C ) ) )
1081073adant1 1001 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +Q  C )  =  ( /Q `  ( B  +pQ  C ) ) )
109106, 108oveq12d 6108 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  ( B  +Q  C ) )  =  ( ( /Q `  A )  .Q  ( /Q `  ( B  +pQ  C ) ) ) )
110 mulpqnq 9106 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( /Q
`  ( A  .pQ  B ) ) )
1111103adant3 1003 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  B )  =  ( /Q `  ( A  .pQ  B ) ) )
112 mulpqnq 9106 . . . . 5  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  C
)  =  ( /Q
`  ( A  .pQ  C ) ) )
1131123adant2 1002 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  C )  =  ( /Q `  ( A  .pQ  C ) ) )
114111, 113oveq12d 6108 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .Q  B
)  +Q  ( A  .Q  C ) )  =  ( ( /Q
`  ( A  .pQ  B ) )  +Q  ( /Q `  ( A  .pQ  C ) ) ) )
115103, 109, 1143eqtr4d 2483 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  ( B  +Q  C ) )  =  ( ( A  .Q  B )  +Q  ( A  .Q  C ) ) )
116 addnqf 9113 . . . 4  |-  +Q  :
( Q.  X.  Q. )
--> Q.
117116fdmi 5561 . . 3  |-  dom  +Q  =  ( Q.  X.  Q. )
118 0nnq 9089 . . 3  |-  -.  (/)  e.  Q.
119 mulnqf 9114 . . . 4  |-  .Q  :
( Q.  X.  Q. )
--> Q.
120119fdmi 5561 . . 3  |-  dom  .Q  =  ( Q.  X.  Q. )
121117, 118, 120ndmovdistr 6251 . 2  |-  ( -.  ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  ( B  +Q  C ) )  =  ( ( A  .Q  B )  +Q  ( A  .Q  C
) ) )
122115, 121pm2.61i 164 1  |-  ( A  .Q  ( B  +Q  C ) )  =  ( ( A  .Q  B )  +Q  ( A  .Q  C ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 960    = wceq 1364    e. wcel 1761   <.cop 3880   class class class wbr 4289    X. cxp 4834   Rel wrel 4841   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   N.cnpi 9007    +N cpli 9008    .N cmi 9009    +pQ cplpq 9011    .pQ cmpq 9012    ~Q ceq 9014   Q.cnq 9015   /Qcerq 9017    +Q cplq 9018    .Q cmq 9019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-omul 6921  df-er 7097  df-ni 9037  df-pli 9038  df-mi 9039  df-lti 9040  df-plpq 9073  df-mpq 9074  df-enq 9076  df-nq 9077  df-erq 9078  df-plq 9079  df-mq 9080  df-1nq 9081
This theorem is referenced by:  ltaddnq  9139  halfnq  9141  addclprlem2  9182  distrlem1pr  9190  distrlem4pr  9191  prlem934  9198  prlem936  9212
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