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Theorem distrnq 8794
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 8729 . . . . . . . . . . . . 13  |-  ( ( 1st `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 1st `  A ) )
21oveq1i 6050 . . . . . . . . . . . 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 5701 . . . . . . . . . . . . 13  |-  ( 1st `  B )  e.  _V
4 fvex 5701 . . . . . . . . . . . . 13  |-  ( 1st `  A )  e.  _V
5 fvex 5701 . . . . . . . . . . . . 13  |-  ( 2nd `  A )  e.  _V
6 mulcompi 8729 . . . . . . . . . . . . 13  |-  ( x  .N  y )  =  ( y  .N  x
)
7 mulasspi 8730 . . . . . . . . . . . . 13  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
8 fvex 5701 . . . . . . . . . . . . 13  |-  ( 2nd `  C )  e.  _V
93, 4, 5, 6, 7, 8caov411 6238 . . . . . . . . . . . 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 2424 . . . . . . . . . . 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 8729 . . . . . . . . . . . . 13  |-  ( ( 1st `  A )  .N  ( 1st `  C
) )  =  ( ( 1st `  C
)  .N  ( 1st `  A ) )
1211oveq1i 6050 . . . . . . . . . . . 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 5701 . . . . . . . . . . . . 13  |-  ( 1st `  C )  e.  _V
14 fvex 5701 . . . . . . . . . . . . 13  |-  ( 2nd `  B )  e.  _V
1513, 4, 5, 6, 7, 14caov411 6238 . . . . . . . . . . . 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 2424 . . . . . . . . . . 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 6052 . . . . . . . . . 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 8731 . . . . . . . . . 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 8730 . . . . . . . . . 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 2430 . . . . . . . . 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 8730 . . . . . . . . . 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 6234 . . . . . . . . . . 11  |-  ( ( 2nd `  B )  .N  ( ( 2nd `  A )  .N  ( 2nd `  C ) ) )  =  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )
2322oveq2i 6051 . . . . . . . . . 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 2424 . . . . . . . . 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 3949 . . . . . . . 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 8758 . . . . . . . . . . 11  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
27263ad2ant1 978 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
28 xp2nd 6336 . . . . . . . . . 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 6335 . . . . . . . . . . 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 8758 . . . . . . . . . . . . . 14  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
33323ad2ant2 979 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
34 xp1st 6335 . . . . . . . . . . . . 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 8758 . . . . . . . . . . . . . 14  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
37363ad2ant3 980 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
38 xp2nd 6336 . . . . . . . . . . . . 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 8726 . . . . . . . . . . . 12  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
4135, 39, 40syl2anc 643 . . . . . . . . . . 11  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
42 xp1st 6335 . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . 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 8726 . . . . . . . . . . . 12  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
4743, 45, 46syl2anc 643 . . . . . . . . . . 11  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
48 addclpi 8725 . . . . . . . . . . 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 643 . . . . . . . . . 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 8726 . . . . . . . . . 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 643 . . . . . . . . 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 8726 . . . . . . . . . . 11  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
5345, 39, 52syl2anc 643 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
54 mulclpi 8726 . . . . . . . . . 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 643 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  e. 
N. )
56 mulcanenq 8793 . . . . . . . . 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 1184 . . . . . . . 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 4205 . . . . . . 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 8772 . . . . . . . . . 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 643 . . . . . . . . 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 8772 . . . . . . . . . 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 643 . . . . . . . . 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 6058 . . . . . . . 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 8726 . . . . . . . . . 10  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
6531, 35, 64syl2anc 643 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
66 mulclpi 8726 . . . . . . . . . 10  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
6729, 45, 66syl2anc 643 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
68 mulclpi 8726 . . . . . . . . . 10  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  C ) )  e. 
N. )
6931, 43, 68syl2anc 643 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( 1st `  C ) )  e. 
N. )
70 mulclpi 8726 . . . . . . . . . 10  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
7129, 39, 70syl2anc 643 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
72 addpipq 8770 . . . . . . . . 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 1185 . . . . . . . 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 2436 . . . . . . 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 4942 . . . . . . . . . 10  |-  Rel  ( N.  X.  N. )
76 1st2nd 6352 . . . . . . . . . 10  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
7775, 27, 76sylancr 645 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
78 addpipq2 8769 . . . . . . . . . 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 643 . . . . . . . . 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 6058 . . . . . . . 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 8773 . . . . . . . . 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 1185 . . . . . . . 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 2436 . . . . . . 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 4202 . . . . . 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 8779 . . . . . . . . . 10  |-  .pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
8685fovcl 6134 . . . . . . . . 9  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  e.  ( N.  X.  N. ) )
8727, 33, 86syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  B )  e.  ( N.  X.  N. ) )
8885fovcl 6134 . . . . . . . . 9  |-  ( ( A  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  C )  e.  ( N.  X.  N. ) )
8927, 37, 88syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  C )  e.  ( N.  X.  N. ) )
90 addpqf 8777 . . . . . . . . 9  |-  +pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
9190fovcl 6134 . . . . . . . 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 643 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .pQ  B
)  +pQ  ( A  .pQ  C ) )  e.  ( N.  X.  N. ) )
9390fovcl 6134 . . . . . . . . 9  |-  ( ( B  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( B  +pQ  C )  e.  ( N.  X.  N. ) )
9433, 37, 93syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +pQ  C )  e.  ( N.  X.  N. ) )
9585fovcl 6134 . . . . . . . 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 643 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  ( B  +pQ  C ) )  e.  ( N.  X.  N. )
)
97 nqereq 8768 . . . . . . 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 643 . . . . . 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 202 . . . . 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 2409 . . . 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 8790 . . . 4  |-  ( ( /Q `  A )  .Q  ( /Q `  ( B  +pQ  C ) ) )  =  ( /Q `  ( A 
.pQ  ( B  +pQ  C ) ) )
102 adderpq 8789 . . . 4  |-  ( ( /Q `  ( A 
.pQ  B ) )  +Q  ( /Q `  ( A  .pQ  C ) ) )  =  ( /Q `  ( ( A  .pQ  B ) 
+pQ  ( A  .pQ  C ) ) )
103100, 101, 1023eqtr4g 2461 . . 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 8766 . . . . . 6  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
105104eqcomd 2409 . . . . 5  |-  ( A  e.  Q.  ->  A  =  ( /Q `  A ) )
1061053ad2ant1 978 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  ( /Q `  A ) )
107 addpqnq 8771 . . . . 5  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +Q  C
)  =  ( /Q
`  ( B  +pQ  C ) ) )
1081073adant1 975 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +Q  C )  =  ( /Q `  ( B  +pQ  C ) ) )
109106, 108oveq12d 6058 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  ( B  +Q  C ) )  =  ( ( /Q `  A )  .Q  ( /Q `  ( B  +pQ  C ) ) ) )
110 mulpqnq 8774 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( /Q
`  ( A  .pQ  B ) ) )
1111103adant3 977 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  B )  =  ( /Q `  ( A  .pQ  B ) ) )
112 mulpqnq 8774 . . . . 5  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  C
)  =  ( /Q
`  ( A  .pQ  C ) ) )
1131123adant2 976 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  C )  =  ( /Q `  ( A  .pQ  C ) ) )
114111, 113oveq12d 6058 . . 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 2446 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  ( B  +Q  C ) )  =  ( ( A  .Q  B )  +Q  ( A  .Q  C ) ) )
116 addnqf 8781 . . . 4  |-  +Q  :
( Q.  X.  Q. )
--> Q.
117116fdmi 5555 . . 3  |-  dom  +Q  =  ( Q.  X.  Q. )
118 0nnq 8757 . . 3  |-  -.  (/)  e.  Q.
119 mulnqf 8782 . . . 4  |-  .Q  :
( Q.  X.  Q. )
--> Q.
120119fdmi 5555 . . 3  |-  dom  .Q  =  ( Q.  X.  Q. )
121117, 118, 120ndmovdistr 6195 . 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 158 1  |-  ( A  .Q  ( B  +Q  C ) )  =  ( ( A  .Q  B )  +Q  ( A  .Q  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1721   <.cop 3777   class class class wbr 4172    X. cxp 4835   Rel wrel 4842   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   N.cnpi 8675    +N cpli 8676    .N cmi 8677    +pQ cplpq 8679    .pQ cmpq 8680    ~Q ceq 8682   Q.cnq 8683   /Qcerq 8685    +Q cplq 8686    .Q cmq 8687
This theorem is referenced by:  ltaddnq  8807  halfnq  8809  addclprlem2  8850  distrlem1pr  8858  distrlem4pr  8859  prlem934  8866  prlem936  8880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-omul 6688  df-er 6864  df-ni 8705  df-pli 8706  df-mi 8707  df-lti 8708  df-plpq 8741  df-mpq 8742  df-enq 8744  df-nq 8745  df-erq 8746  df-plq 8747  df-mq 8748  df-1nq 8749
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