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Theorem mulerpqlem 8837
Description: Lemma for mulerpq 8839. (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 6379 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
213ad2ant1 979 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  A )  e.  N. )
3 xp1st 6379 . . . . 5  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
433ad2ant3 981 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  C )  e.  N. )
5 mulclpi 8775 . . . 4  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  C ) )  e. 
N. )
62, 4, 5syl2anc 644 . . 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 6380 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
873ad2ant1 979 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  A )  e.  N. )
9 xp2nd 6380 . . . . 5  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
1093ad2ant3 981 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  C )  e.  N. )
11 mulclpi 8775 . . . 4  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
128, 10, 11syl2anc 644 . . 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 6379 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
14133ad2ant2 980 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  B )  e.  N. )
15 mulclpi 8775 . . . 4  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N. )
1614, 4, 15syl2anc 644 . . 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 6380 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
18173ad2ant2 980 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  B )  e.  N. )
19 mulclpi 8775 . . . 4  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
2018, 10, 19syl2anc 644 . . 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 8801 . . 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 1186 . 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 8821 . . . 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 977 . . 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 8821 . . . 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 976 . . 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 4228 . 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 8802 . . . 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 978 . . 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 8775 . . . . 5  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
314, 10, 30syl2anc 644 . . . 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 8775 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
332, 18, 32syl2anc 644 . . . 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 8782 . . . 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 644 . . 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 8778 . . . . . 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 5745 . . . . . . 7  |-  ( 1st `  A )  e.  _V
38 fvex 5745 . . . . . . 7  |-  ( 2nd `  B )  e.  _V
39 fvex 5745 . . . . . . 7  |-  ( 1st `  C )  e.  _V
40 mulcompi 8778 . . . . . . 7  |-  ( x  .N  y )  =  ( y  .N  x
)
41 mulasspi 8779 . . . . . . 7  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
42 fvex 5745 . . . . . . 7  |-  ( 2nd `  C )  e.  _V
4337, 38, 39, 40, 41, 42caov4 6281 . . . . . 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 2458 . . . . 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 8778 . . . . . 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 5745 . . . . . . 7  |-  ( 1st `  B )  e.  _V
47 fvex 5745 . . . . . . 7  |-  ( 2nd `  A )  e.  _V
4846, 47, 39, 40, 41, 42caov4 6281 . . . . . 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 8778 . . . . . 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 2462 . . . . 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 2451 . . . 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 274 . 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 279 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 178    /\ w3a 937    = wceq 1653    e. wcel 1726   <.cop 3819   class class class wbr 4215    X. cxp 4879   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   N.cnpi 8724    .N cmi 8726    .pQ cmpq 8729    ~Q ceq 8731
This theorem is referenced by:  mulerpq  8839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-oadd 6731  df-omul 6732  df-ni 8754  df-mi 8756  df-mpq 8791  df-enq 8793
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