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Theorem adderpqlem 8668
Description: Lemma for adderpq 8670. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
adderpqlem  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( A  ~Q  B  <->  ( A  +pQ  C )  ~Q  ( B 
+pQ  C ) ) )

Proof of Theorem adderpqlem
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 6236 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
213ad2ant1 976 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  A )  e.  N. )
3 xp2nd 6237 . . . . . 6  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
433ad2ant3 978 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  C )  e.  N. )
5 mulclpi 8607 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
62, 4, 5syl2anc 642 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  A )  .N  ( 2nd `  C
) )  e.  N. )
7 xp1st 6236 . . . . . 6  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
873ad2ant3 978 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  C )  e.  N. )
9 xp2nd 6237 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
1093ad2ant1 976 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  A )  e.  N. )
11 mulclpi 8607 . . . . 5  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  A ) )  e. 
N. )
128, 10, 11syl2anc 642 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  C )  .N  ( 2nd `  A
) )  e.  N. )
13 addclpi 8606 . . . 4  |-  ( ( ( ( 1st `  A
)  .N  ( 2nd `  C ) )  e. 
N.  /\  ( ( 1st `  C )  .N  ( 2nd `  A
) )  e.  N. )  ->  ( ( ( 1st `  A )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  A ) ) )  e.  N. )
146, 12, 13syl2anc 642 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( (
( 1st `  A
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  e.  N. )
15 mulclpi 8607 . . . 4  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
1610, 4, 15syl2anc 642 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 2nd `  A )  .N  ( 2nd `  C
) )  e.  N. )
17 xp1st 6236 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
18173ad2ant2 977 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  B )  e.  N. )
19 mulclpi 8607 . . . . 5  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
2018, 4, 19syl2anc 642 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  B )  .N  ( 2nd `  C
) )  e.  N. )
21 xp2nd 6237 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
22213ad2ant2 977 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  B )  e.  N. )
23 mulclpi 8607 . . . . 5  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
248, 22, 23syl2anc 642 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  C )  .N  ( 2nd `  B
) )  e.  N. )
25 addclpi 8606 . . . 4  |-  ( ( ( ( 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. )
2620, 24, 25syl2anc 642 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( (
( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  e.  N. )
27 mulclpi 8607 . . . 4  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
2822, 4, 27syl2anc 642 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 2nd `  B )  .N  ( 2nd `  C
) )  e.  N. )
29 enqbreq 8633 . . 3  |-  ( ( ( ( ( ( 1st `  A )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  A ) ) )  e.  N.  /\  (
( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )  /\  (
( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) )  e.  N.  /\  ( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. ) )  -> 
( <. ( ( ( 1st `  A )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  C ) )
>.  ~Q  <. ( ( ( 1st `  B )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C ) )
>. 
<->  ( ( ( ( 1st `  A )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  A ) ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) ) ) ) )
3014, 16, 26, 28, 29syl22anc 1183 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( <. ( ( ( 1st `  A
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  C ) )
>.  ~Q  <. ( ( ( 1st `  B )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C ) )
>. 
<->  ( ( ( ( 1st `  A )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  A ) ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) ) ) ) )
31 addpipq2 8650 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( A  +pQ  C )  = 
<. ( ( ( 1st `  A )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  C
) ) >. )
32313adant2 974 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( A  +pQ  C )  =  <. ( ( ( 1st `  A
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  C ) )
>. )
33 addpipq2 8650 . . . 4  |-  ( ( 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
) ) >. )
34333adant1 973 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  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 ) )
>. )
3532, 34breq12d 4117 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( A  +pQ  C )  ~Q  ( B  +pQ  C )  <->  <. ( ( ( 1st `  A )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  C
) ) >.  ~Q  <. ( ( ( 1st `  B
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C ) )
>. ) )
36 enqbreq2 8634 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  ~Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
37363adant3 975 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( A  ~Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B ) )  =  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) )
38 mulclpi 8607 . . . . 5  |-  ( ( ( 2nd `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
394, 4, 38syl2anc 642 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 2nd `  C )  .N  ( 2nd `  C
) )  e.  N. )
40 mulclpi 8607 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
412, 22, 40syl2anc 642 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  e.  N. )
42 mulcanpi 8614 . . . 4  |-  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N.  /\  ( ( 1st `  A )  .N  ( 2nd `  B
) )  e.  N. )  ->  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
4339, 41, 42syl2anc 642 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( (
( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
44 mulcompi 8610 . . . . . . . 8  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  .N  (
( 2nd `  C
)  .N  ( 2nd `  C ) ) )
45 fvex 5622 . . . . . . . . 9  |-  ( 1st `  A )  e.  _V
46 fvex 5622 . . . . . . . . 9  |-  ( 2nd `  B )  e.  _V
47 fvex 5622 . . . . . . . . 9  |-  ( 2nd `  C )  e.  _V
48 mulcompi 8610 . . . . . . . . 9  |-  ( x  .N  y )  =  ( y  .N  x
)
49 mulasspi 8611 . . . . . . . . 9  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
5045, 46, 47, 48, 49, 47caov4 6138 . . . . . . . 8  |-  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 2nd `  C
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  C
) )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) )
5144, 50eqtri 2378 . . . . . . 7  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  C
) )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) )
52 fvex 5622 . . . . . . . . 9  |-  ( 2nd `  A )  e.  _V
53 fvex 5622 . . . . . . . . 9  |-  ( 1st `  C )  e.  _V
5452, 47, 53, 48, 49, 46caov4 6138 . . . . . . . 8  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  A )  .N  ( 1st `  C
) )  .N  (
( 2nd `  C
)  .N  ( 2nd `  B ) ) )
55 mulcompi 8610 . . . . . . . . 9  |-  ( ( 2nd `  A )  .N  ( 1st `  C
) )  =  ( ( 1st `  C
)  .N  ( 2nd `  A ) )
56 mulcompi 8610 . . . . . . . . 9  |-  ( ( 2nd `  C )  .N  ( 2nd `  B
) )  =  ( ( 2nd `  B
)  .N  ( 2nd `  C ) )
5755, 56oveq12i 5957 . . . . . . . 8  |-  ( ( ( 2nd `  A
)  .N  ( 1st `  C ) )  .N  ( ( 2nd `  C
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  C )  .N  ( 2nd `  A
) )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) )
5854, 57eqtri 2378 . . . . . . 7  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  C )  .N  ( 2nd `  A
) )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) )
5951, 58oveq12i 5957 . . . . . 6  |-  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) ) )  =  ( ( ( ( 1st `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 2nd `  B
)  .N  ( 2nd `  C ) ) )  +N  ( ( ( 1st `  C )  .N  ( 2nd `  A
) )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) ) )
60 addcompi 8608 . . . . . 6  |-  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  B ) ) ) )  =  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) ) )
61 ovex 5970 . . . . . . 7  |-  ( ( 1st `  A )  .N  ( 2nd `  C
) )  e.  _V
62 ovex 5970 . . . . . . 7  |-  ( ( 1st `  C )  .N  ( 2nd `  A
) )  e.  _V
63 ovex 5970 . . . . . . 7  |-  ( ( 2nd `  B )  .N  ( 2nd `  C
) )  e.  _V
64 distrpi 8612 . . . . . . 7  |-  ( x  .N  ( y  +N  z ) )  =  ( ( x  .N  y )  +N  (
x  .N  z ) )
6561, 62, 63, 48, 64caovdir 6141 . . . . . 6  |-  ( ( ( ( 1st `  A
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( ( 1st `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 2nd `  B
)  .N  ( 2nd `  C ) ) )  +N  ( ( ( 1st `  C )  .N  ( 2nd `  A
) )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) ) )
6659, 60, 653eqtr4i 2388 . . . . 5  |-  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  B ) ) ) )  =  ( ( ( ( 1st `  A
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )
67 addcompi 8608 . . . . . 6  |-  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) ) )  =  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  C ) ) )  +N  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) ) )
68 mulasspi 8611 . . . . . . . 8  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( 2nd `  C )  .N  (
( 2nd `  C
)  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) ) )
69 mulcompi 8610 . . . . . . . . . 10  |-  ( ( 2nd `  C )  .N  ( ( 2nd `  C )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )  =  ( ( ( 2nd `  C
)  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )  .N  ( 2nd `  C
) )
70 mulasspi 8611 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( 1st `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  C )  .N  ( 1st `  B
) ) )
71 mulcompi 8610 . . . . . . . . . . . 12  |-  ( ( 2nd `  A )  .N  ( ( 2nd `  C )  .N  ( 1st `  B ) ) )  =  ( ( ( 2nd `  C
)  .N  ( 1st `  B ) )  .N  ( 2nd `  A
) )
72 mulasspi 8611 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  C
)  .N  ( 1st `  B ) )  .N  ( 2nd `  A
) )  =  ( ( 2nd `  C
)  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )
7370, 71, 723eqtrri 2383 . . . . . . . . . . 11  |-  ( ( 2nd `  C )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( 1st `  B
) )
7473oveq1i 5955 . . . . . . . . . 10  |-  ( ( ( 2nd `  C
)  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )  .N  ( 2nd `  C
) )  =  ( ( ( ( 2nd `  A )  .N  ( 2nd `  C ) )  .N  ( 1st `  B
) )  .N  ( 2nd `  C ) )
7569, 74eqtri 2378 . . . . . . . . 9  |-  ( ( 2nd `  C )  .N  ( ( 2nd `  C )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )  =  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( 1st `  B
) )  .N  ( 2nd `  C ) )
76 mulasspi 8611 . . . . . . . . 9  |-  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( 1st `  B
) )  .N  ( 2nd `  C ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) )
7775, 76eqtri 2378 . . . . . . . 8  |-  ( ( 2nd `  C )  .N  ( ( 2nd `  C )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  C ) ) )
7868, 77eqtri 2378 . . . . . . 7  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) )
7978oveq2i 5956 . . . . . 6  |-  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )  =  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) ) )
80 distrpi 8612 . . . . . 6  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) ) )  =  ( ( ( ( 2nd `  A )  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  C ) ) )  +N  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) ) )
8167, 79, 803eqtr4i 2388 . . . . 5  |-  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) ) )
8266, 81eqeq12i 2371 . . . 4  |-  ( ( ( ( ( 2nd `  A )  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) ) )  =  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )  <->  ( ( ( ( 1st `  A
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) ) ) )
83 mulclpi 8607 . . . . . 6  |-  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N.  /\  ( ( 1st `  C )  .N  ( 2nd `  B
) )  e.  N. )  ->  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) )  e.  N. )
8416, 24, 83syl2anc 642 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( (
( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  e.  N. )
85 mulclpi 8607 . . . . . 6  |-  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N.  /\  ( ( 1st `  A )  .N  ( 2nd `  B
) )  e.  N. )  ->  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  B ) ) )  e.  N. )
8639, 41, 85syl2anc 642 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( (
( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  e.  N. )
87 addcanpi 8613 . . . . 5  |-  ( ( ( ( ( 2nd `  A )  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) )  e.  N.  /\  ( ( ( 2nd `  C )  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A )  .N  ( 2nd `  B ) ) )  e.  N. )  ->  ( ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  B ) ) ) )  =  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )  <->  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) ) )
8884, 86, 87syl2anc 642 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( (
( ( ( 2nd `  A )  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) ) )  =  ( ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )  <->  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) ) )
8982, 88syl5rbbr 251 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( (
( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( ( ( 1st `  A )  .N  ( 2nd `  C
) )  +N  (
( 1st `  C
)  .N  ( 2nd `  A ) ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) ) ) ) )
9037, 43, 893bitr2d 272 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( A  ~Q  B  <->  ( ( ( ( 1st `  A
)  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( ( 1st `  B )  .N  ( 2nd `  C ) )  +N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) ) ) ) )
9130, 35, 903bitr4rd 277 1  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( A  ~Q  B  <->  ( A  +pQ  C )  ~Q  ( B 
+pQ  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1642    e. wcel 1710   <.cop 3719   class class class wbr 4104    X. cxp 4769   ` cfv 5337  (class class class)co 5945   1stc1st 6207   2ndc2nd 6208   N.cnpi 8556    +N cpli 8557    .N cmi 8558    +pQ cplpq 8560    ~Q ceq 8563
This theorem is referenced by:  adderpq  8670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-recs 6475  df-rdg 6510  df-oadd 6570  df-omul 6571  df-ni 8586  df-pli 8587  df-mi 8588  df-plpq 8622  df-enq 8625
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