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Theorem mulerpqlem 9385
Description: Lemma for mulerpq 9387. (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 6828 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
213ad2ant1 1030 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  A )  e.  N. )
3 xp1st 6828 . . . . 5  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
433ad2ant3 1032 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  C )  e.  N. )
5 mulclpi 9323 . . . 4  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  C ) )  e. 
N. )
62, 4, 5syl2anc 667 . . 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 6829 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
873ad2ant1 1030 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  A )  e.  N. )
9 xp2nd 6829 . . . . 5  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
1093ad2ant3 1032 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  C )  e.  N. )
11 mulclpi 9323 . . . 4  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  C ) )  e. 
N. )
128, 10, 11syl2anc 667 . . 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 6828 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
14133ad2ant2 1031 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 1st `  B )  e.  N. )
15 mulclpi 9323 . . . 4  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N. )
1614, 4, 15syl2anc 667 . . 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 6829 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
18173ad2ant2 1031 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. )
)  ->  ( 2nd `  B )  e.  N. )
19 mulclpi 9323 . . . 4  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
2018, 10, 19syl2anc 667 . . 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 9349 . . 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 1270 . 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 9369 . . . 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 1028 . . 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 9369 . . . 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 1027 . . 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 4418 . 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 9350 . . . 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 1029 . . 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 9323 . . . . 5  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
314, 10, 30syl2anc 667 . . . 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 9323 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
332, 18, 32syl2anc 667 . . . 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 9330 . . . 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 667 . . 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 9326 . . . . . 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 5880 . . . . . . 7  |-  ( 1st `  A )  e.  _V
38 fvex 5880 . . . . . . 7  |-  ( 2nd `  B )  e.  _V
39 fvex 5880 . . . . . . 7  |-  ( 1st `  C )  e.  _V
40 mulcompi 9326 . . . . . . 7  |-  ( x  .N  y )  =  ( y  .N  x
)
41 mulasspi 9327 . . . . . . 7  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
42 fvex 5880 . . . . . . 7  |-  ( 2nd `  C )  e.  _V
4337, 38, 39, 40, 41, 42caov4 6505 . . . . . 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 2475 . . . . 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 9326 . . . . . 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 5880 . . . . . . 7  |-  ( 1st `  B )  e.  _V
47 fvex 5880 . . . . . . 7  |-  ( 2nd `  A )  e.  _V
4846, 47, 39, 40, 41, 42caov4 6505 . . . . . 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 9326 . . . . . 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 2479 . . . . 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 2467 . . . 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 285 . 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 290 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 188    /\ w3a 986    = wceq 1446    e. wcel 1889   <.cop 3976   class class class wbr 4405    X. cxp 4835   ` cfv 5585  (class class class)co 6295   1stc1st 6796   2ndc2nd 6797   N.cnpi 9274    .N cmi 9276    .pQ cmpq 9279    ~Q ceq 9281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-oadd 7191  df-omul 7192  df-ni 9302  df-mi 9304  df-mpq 9339  df-enq 9341
This theorem is referenced by:  mulerpq  9387
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