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Theorem mulerpqlem 9122
Description: Lemma for mulerpq 9124. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulerpqlem  |-  ( ( 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 mulerpqlem
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 6604 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
213ad2ant1 1009 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  A )  e.  N. )
3 xp1st 6604 . . . . 5  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
433ad2ant3 1011 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  C )  e.  N. )
5 mulclpi 9060 . . . 4  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  C ) )  e. 
N. )
62, 4, 5syl2anc 661 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  A )  .N  ( 1st `  C
) )  e.  N. )
7 xp2nd 6605 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
873ad2ant1 1009 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  A )  e.  N. )
9 xp2nd 6605 . . . . 5  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
1093ad2ant3 1011 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  C )  e.  N. )
11 mulclpi 9060 . . . 4  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
128, 10, 11syl2anc 661 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 2nd `  A )  .N  ( 2nd `  C
) )  e.  N. )
13 xp1st 6604 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
14133ad2ant2 1010 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  B )  e.  N. )
15 mulclpi 9060 . . . 4  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N. )
1614, 4, 15syl2anc 661 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  B )  .N  ( 1st `  C
) )  e.  N. )
17 xp2nd 6605 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
18173ad2ant2 1010 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  B )  e.  N. )
19 mulclpi 9060 . . . 4  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
2018, 10, 19syl2anc 661 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 2nd `  B )  .N  ( 2nd `  C
) )  e.  N. )
21 enqbreq 9086 . . 3  |-  ( ( ( ( ( 1st `  A )  .N  ( 1st `  C ) )  e.  N.  /\  (
( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )  /\  (
( ( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N.  /\  ( ( 2nd `  B )  .N  ( 2nd `  C
) )  e.  N. ) )  ->  ( <. ( ( 1st `  A
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) >.  ~Q  <. ( ( 1st `  B )  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C ) )
>. 
<->  ( ( ( 1st `  A )  .N  ( 1st `  C ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 1st `  C ) ) ) ) )
226, 12, 16, 20, 21syl22anc 1219 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( <. ( ( 1st `  A
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) >.  ~Q  <. ( ( 1st `  B )  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C ) )
>. 
<->  ( ( ( 1st `  A )  .N  ( 1st `  C ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 1st `  C ) ) ) ) )
23 mulpipq2 9106 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  C )  = 
<. ( ( 1st `  A
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) >.
)
24233adant2 1007 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( A  .pQ  C )  =  <. ( ( 1st `  A
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) >.
)
25 mulpipq2 9106 . . . 4  |-  ( ( B  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( B  .pQ  C )  = 
<. ( ( 1st `  B
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B
)  .N  ( 2nd `  C ) ) >.
)
26253adant1 1006 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( B  .pQ  C )  =  <. ( ( 1st `  B
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B
)  .N  ( 2nd `  C ) ) >.
)
2724, 26breq12d 4303 . 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  ( 1st `  C ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) >.  ~Q  <. ( ( 1st `  B )  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B )  .N  ( 2nd `  C ) )
>. ) )
28 enqbreq2 9087 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  ~Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
29283adant3 1008 . . 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 ) ) ) )
30 mulclpi 9060 . . . . 5  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
314, 10, 30syl2anc 661 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  C )  .N  ( 2nd `  C
) )  e.  N. )
32 mulclpi 9060 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
332, 18, 32syl2anc 661 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  e.  N. )
34 mulcanpi 9067 . . . 4  |-  ( ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N.  /\  ( ( 1st `  A )  .N  ( 2nd `  B
) )  e.  N. )  ->  ( ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
3531, 33, 34syl2anc 661 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( (
( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
36 mulcompi 9063 . . . . . 6  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  C ) ) )
37 fvex 5699 . . . . . . 7  |-  ( 1st `  A )  e.  _V
38 fvex 5699 . . . . . . 7  |-  ( 2nd `  B )  e.  _V
39 fvex 5699 . . . . . . 7  |-  ( 1st `  C )  e.  _V
40 mulcompi 9063 . . . . . . 7  |-  ( x  .N  y )  =  ( y  .N  x
)
41 mulasspi 9064 . . . . . . 7  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
42 fvex 5699 . . . . . . 7  |-  ( 2nd `  C )  e.  _V
4337, 38, 39, 40, 41, 42caov4 6292 . . . . . 6  |-  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 1st `  A )  .N  ( 1st `  C
) )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) )
4436, 43eqtri 2461 . . . . 5  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  A )  .N  ( 1st `  C
) )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) )
45 mulcompi 9063 . . . . . 6  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 1st `  B )  .N  ( 2nd `  A
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  C ) ) )
46 fvex 5699 . . . . . . 7  |-  ( 1st `  B )  e.  _V
47 fvex 5699 . . . . . . 7  |-  ( 2nd `  A )  e.  _V
4846, 47, 39, 40, 41, 42caov4 6292 . . . . . 6  |-  ( ( ( 1st `  B
)  .N  ( 2nd `  A ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 1st `  B )  .N  ( 1st `  C
) )  .N  (
( 2nd `  A
)  .N  ( 2nd `  C ) ) )
49 mulcompi 9063 . . . . . 6  |-  ( ( ( 1st `  B
)  .N  ( 1st `  C ) )  .N  ( ( 2nd `  A
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 1st `  C ) ) )
5045, 48, 493eqtri 2465 . . . . 5  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 1st `  C ) ) )
5144, 50eqeq12i 2454 . . . 4  |-  ( ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( ( 1st `  A )  .N  ( 1st `  C ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 1st `  C ) ) ) )
5251a1i 11 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( (
( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( ( 1st `  A )  .N  ( 1st `  C ) )  .N  ( ( 2nd `  B )  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 1st `  C ) ) ) ) )
5329, 35, 523bitr2d 281 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( A  ~Q  B  <->  ( ( ( 1st `  A )  .N  ( 1st `  C
) )  .N  (
( 2nd `  B
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  A )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 1st `  C ) ) ) ) )
5422, 27, 533bitr4rd 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 965    = wceq 1369    e. wcel 1756   <.cop 3881   class class class wbr 4290    X. cxp 4836   ` cfv 5416  (class class class)co 6089   1stc1st 6573   2ndc2nd 6574   N.cnpi 9009    .N cmi 9011    .pQ cmpq 9014    ~Q ceq 9016
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-oadd 6922  df-omul 6923  df-ni 9039  df-mi 9041  df-mpq 9076  df-enq 9078
This theorem is referenced by:  mulerpq  9124
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