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Theorem distrnq 9130
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 9065 . . . . . . . . . . . . 13  |-  ( ( 1st `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 1st `  A ) )
21oveq1i 6101 . . . . . . . . . . . 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 9065 . . . . . . . . . . . . 13  |-  ( x  .N  y )  =  ( y  .N  x
)
7 mulasspi 9066 . . . . . . . . . . . . 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 6295 . . . . . . . . . . . 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 2463 . . . . . . . . . . 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 9065 . . . . . . . . . . . . 13  |-  ( ( 1st `  A )  .N  ( 1st `  C
) )  =  ( ( 1st `  C
)  .N  ( 1st `  A ) )
1211oveq1i 6101 . . . . . . . . . . . 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 6295 . . . . . . . . . . . 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 2463 . . . . . . . . . . 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 6103 . . . . . . . . . 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 9067 . . . . . . . . . 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 9066 . . . . . . . . . 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 2469 . . . . . . . . 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 9066 . . . . . . . . . 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 6291 . . . . . . . . . . 11  |-  ( ( 2nd `  B )  .N  ( ( 2nd `  A )  .N  ( 2nd `  C ) ) )  =  ( ( 2nd `  A )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )
2322oveq2i 6102 . . . . . . . . . 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 2463 . . . . . . . . 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 4064 . . . . . . . 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 9094 . . . . . . . . . . 11  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
27263ad2ant1 1009 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
28 xp2nd 6607 . . . . . . . . . 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 6606 . . . . . . . . . . 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 9094 . . . . . . . . . . . . . 14  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
33323ad2ant2 1010 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
34 xp1st 6606 . . . . . . . . . . . . 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 9094 . . . . . . . . . . . . . 14  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
37363ad2ant3 1011 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
38 xp2nd 6607 . . . . . . . . . . . . 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 9062 . . . . . . . . . . . 12  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
4135, 39, 40syl2anc 661 . . . . . . . . . . 11  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
42 xp1st 6606 . . . . . . . . . . . . 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 6607 . . . . . . . . . . . . 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 9062 . . . . . . . . . . . 12  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
4743, 45, 46syl2anc 661 . . . . . . . . . . 11  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
48 addclpi 9061 . . . . . . . . . . 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 661 . . . . . . . . . 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 9062 . . . . . . . . . 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 661 . . . . . . . . 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 9062 . . . . . . . . . . 11  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
5345, 39, 52syl2anc 661 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
54 mulclpi 9062 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )  e. 
N. )
56 mulcanenq 9129 . . . . . . . . 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 1218 . . . . . . . 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 4325 . . . . . . 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 9108 . . . . . . . . . 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 661 . . . . . . . . 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 9108 . . . . . . . . . 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 661 . . . . . . . . 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 6109 . . . . . . . 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 9062 . . . . . . . . . 10  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
6531, 35, 64syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
66 mulclpi 9062 . . . . . . . . . 10  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
6729, 45, 66syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
68 mulclpi 9062 . . . . . . . . . 10  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  C ) )  e. 
N. )
6931, 43, 68syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( 1st `  C ) )  e. 
N. )
70 mulclpi 9062 . . . . . . . . . 10  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
7129, 39, 70syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
72 addpipq 9106 . . . . . . . . 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 1219 . . . . . . . 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 2475 . . . . . . 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 4947 . . . . . . . . . 10  |-  Rel  ( N.  X.  N. )
76 1st2nd 6620 . . . . . . . . . 10  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
7775, 27, 76sylancr 663 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
78 addpipq2 9105 . . . . . . . . . 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 661 . . . . . . . . 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 6109 . . . . . . . 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 9109 . . . . . . . . 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 1219 . . . . . . . 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 2475 . . . . . . 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 4322 . . . . . 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 9115 . . . . . . . . . 10  |-  .pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
8685fovcl 6195 . . . . . . . . 9  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  e.  ( N.  X.  N. ) )
8727, 33, 86syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  B )  e.  ( N.  X.  N. ) )
8885fovcl 6195 . . . . . . . . 9  |-  ( ( A  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  C )  e.  ( N.  X.  N. ) )
8927, 37, 88syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  C )  e.  ( N.  X.  N. ) )
90 addpqf 9113 . . . . . . . . 9  |-  +pQ  :
( ( N.  X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
9190fovcl 6195 . . . . . . . 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 661 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .pQ  B
)  +pQ  ( A  .pQ  C ) )  e.  ( N.  X.  N. ) )
9390fovcl 6195 . . . . . . . . 9  |-  ( ( B  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( B  +pQ  C )  e.  ( N.  X.  N. ) )
9433, 37, 93syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +pQ  C )  e.  ( N.  X.  N. ) )
9585fovcl 6195 . . . . . . . 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 661 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  ( B  +pQ  C ) )  e.  ( N.  X.  N. )
)
97 nqereq 9104 . . . . . . 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 661 . . . . . 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 2448 . . . 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 9126 . . . 4  |-  ( ( /Q `  A )  .Q  ( /Q `  ( B  +pQ  C ) ) )  =  ( /Q `  ( A 
.pQ  ( B  +pQ  C ) ) )
102 adderpq 9125 . . . 4  |-  ( ( /Q `  ( A 
.pQ  B ) )  +Q  ( /Q `  ( A  .pQ  C ) ) )  =  ( /Q `  ( ( A  .pQ  B ) 
+pQ  ( A  .pQ  C ) ) )
103100, 101, 1023eqtr4g 2500 . . 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 9102 . . . . . 6  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
105104eqcomd 2448 . . . . 5  |-  ( A  e.  Q.  ->  A  =  ( /Q `  A ) )
1061053ad2ant1 1009 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  ( /Q `  A ) )
107 addpqnq 9107 . . . . 5  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +Q  C
)  =  ( /Q
`  ( B  +pQ  C ) ) )
1081073adant1 1006 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  +Q  C )  =  ( /Q `  ( B  +pQ  C ) ) )
109106, 108oveq12d 6109 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  ( B  +Q  C ) )  =  ( ( /Q `  A )  .Q  ( /Q `  ( B  +pQ  C ) ) ) )
110 mulpqnq 9110 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( /Q
`  ( A  .pQ  B ) ) )
1111103adant3 1008 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  B )  =  ( /Q `  ( A  .pQ  B ) ) )
112 mulpqnq 9110 . . . . 5  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  C
)  =  ( /Q
`  ( A  .pQ  C ) ) )
1131123adant2 1007 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  C )  =  ( /Q `  ( A  .pQ  C ) ) )
114111, 113oveq12d 6109 . . 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 2485 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  ( B  +Q  C ) )  =  ( ( A  .Q  B )  +Q  ( A  .Q  C ) ) )
116 addnqf 9117 . . . 4  |-  +Q  :
( Q.  X.  Q. )
--> Q.
117116fdmi 5564 . . 3  |-  dom  +Q  =  ( Q.  X.  Q. )
118 0nnq 9093 . . 3  |-  -.  (/)  e.  Q.
119 mulnqf 9118 . . . 4  |-  .Q  :
( Q.  X.  Q. )
--> Q.
120119fdmi 5564 . . 3  |-  dom  .Q  =  ( Q.  X.  Q. )
121117, 118, 120ndmovdistr 6252 . 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 965    = wceq 1369    e. wcel 1756   <.cop 3883   class class class wbr 4292    X. cxp 4838   Rel wrel 4845   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   N.cnpi 9011    +N cpli 9012    .N cmi 9013    +pQ cplpq 9015    .pQ cmpq 9016    ~Q ceq 9018   Q.cnq 9019   /Qcerq 9021    +Q cplq 9022    .Q cmq 9023
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 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-omul 6925  df-er 7101  df-ni 9041  df-pli 9042  df-mi 9043  df-lti 9044  df-plpq 9077  df-mpq 9078  df-enq 9080  df-nq 9081  df-erq 9082  df-plq 9083  df-mq 9084  df-1nq 9085
This theorem is referenced by:  ltaddnq  9143  halfnq  9145  addclprlem2  9186  distrlem1pr  9194  distrlem4pr  9195  prlem934  9202  prlem936  9216
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