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Theorem adderpqlem 8765
Description: Lemma for adderpq 8767. (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 6316 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
213ad2ant1 978 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  A )  e.  N. )
3 xp2nd 6317 . . . . . 6  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
433ad2ant3 980 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  C )  e.  N. )
5 mulclpi 8704 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
62, 4, 5syl2anc 643 . . . 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 6316 . . . . . 6  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
873ad2ant3 980 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  C )  e.  N. )
9 xp2nd 6317 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
1093ad2ant1 978 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  A )  e.  N. )
11 mulclpi 8704 . . . . 5  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  A ) )  e. 
N. )
128, 10, 11syl2anc 643 . . . 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 8703 . . . 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 643 . . 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 8704 . . . 4  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
1610, 4, 15syl2anc 643 . . 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 6316 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
18173ad2ant2 979 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  B )  e.  N. )
19 mulclpi 8704 . . . . 5  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
2018, 4, 19syl2anc 643 . . . 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 6317 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
22213ad2ant2 979 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  B )  e.  N. )
23 mulclpi 8704 . . . . 5  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
248, 22, 23syl2anc 643 . . . 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 8703 . . . 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 643 . . 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 8704 . . . 4  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
2822, 4, 27syl2anc 643 . . 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 8730 . . 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 1185 . 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 8747 . . . 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 976 . . 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 8747 . . . 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 975 . . 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 4167 . 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 8731 . . . 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 977 . . 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 8704 . . . . 5  |-  ( ( ( 2nd `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
394, 4, 38syl2anc 643 . . . 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 8704 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
412, 22, 40syl2anc 643 . . . 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 8711 . . . 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 643 . . 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 8707 . . . . . . . 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 5683 . . . . . . . . 9  |-  ( 1st `  A )  e.  _V
46 fvex 5683 . . . . . . . . 9  |-  ( 2nd `  B )  e.  _V
47 fvex 5683 . . . . . . . . 9  |-  ( 2nd `  C )  e.  _V
48 mulcompi 8707 . . . . . . . . 9  |-  ( x  .N  y )  =  ( y  .N  x
)
49 mulasspi 8708 . . . . . . . . 9  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
5045, 46, 47, 48, 49, 47caov4 6218 . . . . . . . 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 2408 . . . . . . 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 5683 . . . . . . . . 9  |-  ( 2nd `  A )  e.  _V
53 fvex 5683 . . . . . . . . 9  |-  ( 1st `  C )  e.  _V
5452, 47, 53, 48, 49, 46caov4 6218 . . . . . . . 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 8707 . . . . . . . . 9  |-  ( ( 2nd `  A )  .N  ( 1st `  C
) )  =  ( ( 1st `  C
)  .N  ( 2nd `  A ) )
56 mulcompi 8707 . . . . . . . . 9  |-  ( ( 2nd `  C )  .N  ( 2nd `  B
) )  =  ( ( 2nd `  B
)  .N  ( 2nd `  C ) )
5755, 56oveq12i 6033 . . . . . . . 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 2408 . . . . . . 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 6033 . . . . . 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 8705 . . . . . 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 6046 . . . . . . 7  |-  ( ( 1st `  A )  .N  ( 2nd `  C
) )  e.  _V
62 ovex 6046 . . . . . . 7  |-  ( ( 1st `  C )  .N  ( 2nd `  A
) )  e.  _V
63 ovex 6046 . . . . . . 7  |-  ( ( 2nd `  B )  .N  ( 2nd `  C
) )  e.  _V
64 distrpi 8709 . . . . . . 7  |-  ( x  .N  ( y  +N  z ) )  =  ( ( x  .N  y )  +N  (
x  .N  z ) )
6561, 62, 63, 48, 64caovdir 6221 . . . . . 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 2418 . . . . 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 8705 . . . . . 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 8708 . . . . . . . 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 8707 . . . . . . . . . 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 8708 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( 1st `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  C )  .N  ( 1st `  B
) ) )
71 mulcompi 8707 . . . . . . . . . . . 12  |-  ( ( 2nd `  A )  .N  ( ( 2nd `  C )  .N  ( 1st `  B ) ) )  =  ( ( ( 2nd `  C
)  .N  ( 1st `  B ) )  .N  ( 2nd `  A
) )
72 mulasspi 8708 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  C
)  .N  ( 1st `  B ) )  .N  ( 2nd `  A
) )  =  ( ( 2nd `  C
)  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )
7370, 71, 723eqtrri 2413 . . . . . . . . . . 11  |-  ( ( 2nd `  C )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( 1st `  B
) )
7473oveq1i 6031 . . . . . . . . . 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 2408 . . . . . . . . 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 8708 . . . . . . . . 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 2408 . . . . . . . 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 2408 . . . . . . 7  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) )
7978oveq2i 6032 . . . . . 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 8709 . . . . . 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 2418 . . . . 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 2401 . . . 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 8704 . . . . . 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 643 . . . . 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 8704 . . . . . 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 643 . . . . 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 8710 . . . . 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 643 . . . 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 252 . . 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 273 . 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 278 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 177    /\ w3a 936    = wceq 1649    e. wcel 1717   <.cop 3761   class class class wbr 4154    X. cxp 4817   ` cfv 5395  (class class class)co 6021   1stc1st 6287   2ndc2nd 6288   N.cnpi 8653    +N cpli 8654    .N cmi 8655    +pQ cplpq 8657    ~Q ceq 8660
This theorem is referenced by:  adderpq  8767
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-recs 6570  df-rdg 6605  df-oadd 6665  df-omul 6666  df-ni 8683  df-pli 8684  df-mi 8685  df-plpq 8719  df-enq 8722
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