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Theorem adderpqlem 8594
Description: Lemma for adderpq 8596. (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 6165 . . . . . 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 6166 . . . . . 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 8533 . . . . 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 6165 . . . . . 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 6166 . . . . . 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 8533 . . . . 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 8532 . . . 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 8533 . . . 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 6165 . . . . . 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 8533 . . . . 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 6166 . . . . . 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 8533 . . . . 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 8532 . . . 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 8533 . . . 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 8559 . . 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 8576 . . . 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 8576 . . . 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 4052 . 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 8560 . . . 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 8533 . . . . 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 8533 . . . . 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 8540 . . . 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 8536 . . . . . . . 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 5555 . . . . . . . . 9  |-  ( 1st `  A )  e.  _V
46 fvex 5555 . . . . . . . . 9  |-  ( 2nd `  B )  e.  _V
47 fvex 5555 . . . . . . . . 9  |-  ( 2nd `  C )  e.  _V
48 mulcompi 8536 . . . . . . . . 9  |-  ( x  .N  y )  =  ( y  .N  x
)
49 mulasspi 8537 . . . . . . . . 9  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
5045, 46, 47, 48, 49, 47caov4 6067 . . . . . . . 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 2316 . . . . . . 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 5555 . . . . . . . . 9  |-  ( 2nd `  A )  e.  _V
53 fvex 5555 . . . . . . . . 9  |-  ( 1st `  C )  e.  _V
5452, 47, 53, 48, 49, 46caov4 6067 . . . . . . . 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 8536 . . . . . . . . 9  |-  ( ( 2nd `  A )  .N  ( 1st `  C
) )  =  ( ( 1st `  C
)  .N  ( 2nd `  A ) )
56 mulcompi 8536 . . . . . . . . 9  |-  ( ( 2nd `  C )  .N  ( 2nd `  B
) )  =  ( ( 2nd `  B
)  .N  ( 2nd `  C ) )
5755, 56oveq12i 5886 . . . . . . . 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 2316 . . . . . . 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 5886 . . . . . 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 8534 . . . . . 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 5899 . . . . . . 7  |-  ( ( 1st `  A )  .N  ( 2nd `  C
) )  e.  _V
62 ovex 5899 . . . . . . 7  |-  ( ( 1st `  C )  .N  ( 2nd `  A
) )  e.  _V
63 ovex 5899 . . . . . . 7  |-  ( ( 2nd `  B )  .N  ( 2nd `  C
) )  e.  _V
64 distrpi 8538 . . . . . . 7  |-  ( x  .N  ( y  +N  z ) )  =  ( ( x  .N  y )  +N  (
x  .N  z ) )
6561, 62, 63, 48, 64caovdir 6070 . . . . . 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 2326 . . . . 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 8534 . . . . . 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 8537 . . . . . . . 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 8536 . . . . . . . . . 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 8537 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( 1st `  B
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  C )  .N  ( 1st `  B
) ) )
71 mulcompi 8536 . . . . . . . . . . . 12  |-  ( ( 2nd `  A )  .N  ( ( 2nd `  C )  .N  ( 1st `  B ) ) )  =  ( ( ( 2nd `  C
)  .N  ( 1st `  B ) )  .N  ( 2nd `  A
) )
72 mulasspi 8537 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  C
)  .N  ( 1st `  B ) )  .N  ( 2nd `  A
) )  =  ( ( 2nd `  C
)  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )
7370, 71, 723eqtrri 2321 . . . . . . . . . . 11  |-  ( ( 2nd `  C )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( 1st `  B
) )
7473oveq1i 5884 . . . . . . . . . 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 2316 . . . . . . . . 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 8537 . . . . . . . . 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 2316 . . . . . . . 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 2316 . . . . . . 7  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) )
7978oveq2i 5885 . . . . . 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 8538 . . . . . 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 2326 . . . . 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 2309 . . . 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 8533 . . . . . 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 8533 . . . . . 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 8539 . . . . 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 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039    X. cxp 4703   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   N.cnpi 8482    +N cpli 8483    .N cmi 8484    +pQ cplpq 8486    ~Q ceq 8489
This theorem is referenced by:  adderpq  8596
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-oadd 6499  df-omul 6500  df-ni 8512  df-pli 8513  df-mi 8514  df-plpq 8548  df-enq 8551
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