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Theorem distrnq 8838
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 8773 . . . . . . . . . . . . 13  |-  ( ( 1st `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 1st `  A ) )
21oveq1i 6091 . . . . . . . . . . . 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 5742 . . . . . . . . . . . . 13  |-  ( 1st `  B )  e.  _V
4 fvex 5742 . . . . . . . . . . . . 13  |-  ( 1st `  A )  e.  _V
5 fvex 5742 . . . . . . . . . . . . 13  |-  ( 2nd `  A )  e.  _V
6 mulcompi 8773 . . . . . . . . . . . . 13  |-  ( x  .N  y )  =  ( y  .N  x
)
7 mulasspi 8774 . . . . . . . . . . . . 13  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
8 fvex 5742 . . . . . . . . . . . . 13  |-  ( 2nd `  C )  e.  _V
93, 4, 5, 6, 7, 8caov411 6279 . . . . . . . . . . . 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 2456 . . . . . . . . . . 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 8773 . . . . . . . . . . . . 13  |-  ( ( 1st `  A )  .N  ( 1st `  C
) )  =  ( ( 1st `  C
)  .N  ( 1st `  A ) )
1211oveq1i 6091 . . . . . . . . . . . 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 5742 . . . . . . . . . . . . 13  |-  ( 1st `  C )  e.  _V
14 fvex 5742 . . . . . . . . . . . . 13  |-  ( 2nd `  B )  e.  _V
1513, 4, 5, 6, 7, 14caov411 6279 . . . . . . . . . . . 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 2456 . . . . . . . . . . 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 6093 . . . . . . . . . 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 8775 . . . . . . . . . 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 8774 . . . . . . . . . 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 2462 . . . . . . . . 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 8774 . . . . . . . . . 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 6275 . . . . . . . . . . 11  |-  ( ( 2nd `  B )  .N  ( ( 2nd `  A )  .N  ( 2nd `  C ) ) )  =  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )
2322oveq2i 6092 . . . . . . . . . 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 2456 . . . . . . . . 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 3989 . . . . . . . 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 8802 . . . . . . . . . . 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 6377 . . . . . . . . . 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 6376 . . . . . . . . . . 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 8802 . . . . . . . . . . . . . 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 6376 . . . . . . . . . . . . 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 8802 . . . . . . . . . . . . . 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 6377 . . . . . . . . . . . . 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 8770 . . . . . . . . . . . 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 6376 . . . . . . . . . . . . 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 6377 . . . . . . . . . . . . 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 8770 . . . . . . . . . . . 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 8769 . . . . . . . . . . 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 8770 . . . . . . . . . 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 8770 . . . . . . . . . . 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 8770 . . . . . . . . . 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 8837 . . . . . . . . 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 4245 . . . . . . 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 8816 . . . . . . . . . 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 8816 . . . . . . . . . 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 6099 . . . . . . . 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 8770 . . . . . . . . . 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 8770 . . . . . . . . . 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 8770 . . . . . . . . . 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 8770 . . . . . . . . . 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 8814 . . . . . . . . 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 2468 . . . . . . 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 4983 . . . . . . . . . 10  |-  Rel  ( N.  X.  N. )
76 1st2nd 6393 . . . . . . . . . 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 8813 . . . . . . . . . 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 6099 . . . . . . . 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 8817 . . . . . . . . 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 2468 . . . . . . 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 4242 . . . . . 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 8823 . . . . . . . . . 10  |-  .pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
8685fovcl 6175 . . . . . . . . 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 6175 . . . . . . . . 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 8821 . . . . . . . . 9  |-  +pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
9190fovcl 6175 . . . . . . . 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 6175 . . . . . . . . 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 6175 . . . . . . . 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 8812 . . . . . . 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 2441 . . . 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 8834 . . . 4  |-  ( ( /Q `  A )  .Q  ( /Q `  ( B  +pQ  C ) ) )  =  ( /Q `  ( A 
.pQ  ( B  +pQ  C ) ) )
102 adderpq 8833 . . . 4  |-  ( ( /Q `  ( A 
.pQ  B ) )  +Q  ( /Q `  ( A  .pQ  C ) ) )  =  ( /Q `  ( ( A  .pQ  B ) 
+pQ  ( A  .pQ  C ) ) )
103100, 101, 1023eqtr4g 2493 . . 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 8810 . . . . . 6  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
105104eqcomd 2441 . . . . 5  |-  ( A  e.  Q.  ->  A  =  ( /Q `  A ) )
1061053ad2ant1 978 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  ( /Q `  A ) )
107 addpqnq 8815 . . . . 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 6099 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  ( B  +Q  C ) )  =  ( ( /Q `  A )  .Q  ( /Q `  ( B  +pQ  C ) ) ) )
110 mulpqnq 8818 . . . . 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 8818 . . . . 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 6099 . . 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 2478 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  ( B  +Q  C ) )  =  ( ( A  .Q  B )  +Q  ( A  .Q  C ) ) )
116 addnqf 8825 . . . 4  |-  +Q  :
( Q.  X.  Q. )
--> Q.
117116fdmi 5596 . . 3  |-  dom  +Q  =  ( Q.  X.  Q. )
118 0nnq 8801 . . 3  |-  -.  (/)  e.  Q.
119 mulnqf 8826 . . . 4  |-  .Q  :
( Q.  X.  Q. )
--> Q.
120119fdmi 5596 . . 3  |-  dom  .Q  =  ( Q.  X.  Q. )
121117, 118, 120ndmovdistr 6236 . 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 1652    e. wcel 1725   <.cop 3817   class class class wbr 4212    X. cxp 4876   Rel wrel 4883   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   N.cnpi 8719    +N cpli 8720    .N cmi 8721    +pQ cplpq 8723    .pQ cmpq 8724    ~Q ceq 8726   Q.cnq 8727   /Qcerq 8729    +Q cplq 8730    .Q cmq 8731
This theorem is referenced by:  ltaddnq  8851  halfnq  8853  addclprlem2  8894  distrlem1pr  8902  distrlem4pr  8903  prlem934  8910  prlem936  8924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-ni 8749  df-pli 8750  df-mi 8751  df-lti 8752  df-plpq 8785  df-mpq 8786  df-enq 8788  df-nq 8789  df-erq 8790  df-plq 8791  df-mq 8792  df-1nq 8793
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