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Theorem adderpqlem 9344
Description: Lemma for adderpq 9346. (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 6825 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
213ad2ant1 1017 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  A )  e.  N. )
3 xp2nd 6826 . . . . . 6  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
433ad2ant3 1019 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  C )  e.  N. )
5 mulclpi 9283 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
62, 4, 5syl2anc 661 . . . 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 6825 . . . . . 6  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
873ad2ant3 1019 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  C )  e.  N. )
9 xp2nd 6826 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
1093ad2ant1 1017 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  A )  e.  N. )
11 mulclpi 9283 . . . . 5  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  A ) )  e. 
N. )
128, 10, 11syl2anc 661 . . . 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 9282 . . . 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 661 . . 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 9283 . . . 4  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
1610, 4, 15syl2anc 661 . . 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 6825 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
18173ad2ant2 1018 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  B )  e.  N. )
19 mulclpi 9283 . . . . 5  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
2018, 4, 19syl2anc 661 . . . 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 6826 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
22213ad2ant2 1018 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  B )  e.  N. )
23 mulclpi 9283 . . . . 5  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
248, 22, 23syl2anc 661 . . . 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 9282 . . . 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 661 . . 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 9283 . . . 4  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
2822, 4, 27syl2anc 661 . . 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 9309 . . 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 1229 . 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 9326 . . . 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 1015 . . 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 9326 . . . 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 1014 . . 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 4466 . 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 9310 . . . 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 1016 . . 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 9283 . . . . 5  |-  ( ( ( 2nd `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
394, 4, 38syl2anc 661 . . . 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 9283 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
412, 22, 40syl2anc 661 . . . 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 9290 . . . 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 661 . . 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 9286 . . . . . . . 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 5882 . . . . . . . . 9  |-  ( 1st `  A )  e.  _V
46 fvex 5882 . . . . . . . . 9  |-  ( 2nd `  B )  e.  _V
47 fvex 5882 . . . . . . . . 9  |-  ( 2nd `  C )  e.  _V
48 mulcompi 9286 . . . . . . . . 9  |-  ( x  .N  y )  =  ( y  .N  x
)
49 mulasspi 9287 . . . . . . . . 9  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
5045, 46, 47, 48, 49, 47caov4 6501 . . . . . . . 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 2496 . . . . . . 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 5882 . . . . . . . . 9  |-  ( 2nd `  A )  e.  _V
53 fvex 5882 . . . . . . . . 9  |-  ( 1st `  C )  e.  _V
5452, 47, 53, 48, 49, 46caov4 6501 . . . . . . . 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 9286 . . . . . . . . 9  |-  ( ( 2nd `  A )  .N  ( 1st `  C
) )  =  ( ( 1st `  C
)  .N  ( 2nd `  A ) )
56 mulcompi 9286 . . . . . . . . 9  |-  ( ( 2nd `  C )  .N  ( 2nd `  B
) )  =  ( ( 2nd `  B
)  .N  ( 2nd `  C ) )
5755, 56oveq12i 6307 . . . . . . . 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 2496 . . . . . . 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 6307 . . . . . 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 9284 . . . . . 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 6320 . . . . . . 7  |-  ( ( 1st `  A )  .N  ( 2nd `  C
) )  e.  _V
62 ovex 6320 . . . . . . 7  |-  ( ( 1st `  C )  .N  ( 2nd `  A
) )  e.  _V
63 ovex 6320 . . . . . . 7  |-  ( ( 2nd `  B )  .N  ( 2nd `  C
) )  e.  _V
64 distrpi 9288 . . . . . . 7  |-  ( x  .N  ( y  +N  z ) )  =  ( ( x  .N  y )  +N  (
x  .N  z ) )
6561, 62, 63, 48, 64caovdir 6504 . . . . . 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 2506 . . . . 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 9284 . . . . . 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 9287 . . . . . . . 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 9286 . . . . . . . . . 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 9287 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( 1st `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  C )  .N  ( 1st `  B
) ) )
71 mulcompi 9286 . . . . . . . . . . . 12  |-  ( ( 2nd `  A )  .N  ( ( 2nd `  C )  .N  ( 1st `  B ) ) )  =  ( ( ( 2nd `  C
)  .N  ( 1st `  B ) )  .N  ( 2nd `  A
) )
72 mulasspi 9287 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  C
)  .N  ( 1st `  B ) )  .N  ( 2nd `  A
) )  =  ( ( 2nd `  C
)  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )
7370, 71, 723eqtrri 2501 . . . . . . . . . . 11  |-  ( ( 2nd `  C )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( 1st `  B
) )
7473oveq1i 6305 . . . . . . . . . 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 2496 . . . . . . . . 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 9287 . . . . . . . . 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 2496 . . . . . . . 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 2496 . . . . . . 7  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) )
7978oveq2i 6306 . . . . . 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 9288 . . . . . 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 2506 . . . . 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 2487 . . . 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 9283 . . . . . 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 661 . . . . 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 9283 . . . . . 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 661 . . . . 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 9289 . . . . 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 661 . . . 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 260 . . 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 281 . 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 286 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767   <.cop 4039   class class class wbr 4453    X. cxp 5003   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   N.cnpi 9234    +N cpli 9235    .N cmi 9236    +pQ cplpq 9238    ~Q ceq 9241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-oadd 7146  df-omul 7147  df-ni 9262  df-pli 9263  df-mi 9264  df-plpq 9298  df-enq 9301
This theorem is referenced by:  adderpq  9346
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