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Theorem enqbreq2 6645
Description: Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)
Assertion
Ref Expression
enqbreq2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  ~Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) )

Proof of Theorem enqbreq2
StepHypRef Expression
1 1st2nd2 5853 . . 3  |-  ( A  e.  ( N.  X.  N. )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 1st2nd2 5853 . . 3  |-  ( B  e.  ( N.  X.  N. )  ->  B  = 
<. ( 1st `  B
) ,  ( 2nd `  B ) >. )
31, 2breqan12d 3821 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  ~Q  B  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  ~Q  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
)
4 xp1st 5844 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
5 xp2nd 5845 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
64, 5jca 300 . . 3  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  A )  e.  N.  /\  ( 2nd `  A )  e. 
N. ) )
7 xp1st 5844 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
8 xp2nd 5845 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
97, 8jca 300 . . 3  |-  ( B  e.  ( N.  X.  N. )  ->  ( ( 1st `  B )  e.  N.  /\  ( 2nd `  B )  e. 
N. ) )
10 enqbreq 6644 . . 3  |-  ( ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  /\  ( ( 1st `  B
)  e.  N.  /\  ( 2nd `  B )  e.  N. ) )  ->  ( <. ( 1st `  A ) ,  ( 2nd `  A
) >.  ~Q  <. ( 1st `  B ) ,  ( 2nd `  B )
>. 
<->  ( ( 1st `  A
)  .N  ( 2nd `  B ) )  =  ( ( 2nd `  A
)  .N  ( 1st `  B ) ) ) )
116, 9, 10syl2an 283 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  ~Q  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  .N  ( 2nd `  B ) )  =  ( ( 2nd `  A
)  .N  ( 1st `  B ) ) ) )
12 mulcompig 6619 . . . 4  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 1st `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 1st `  B ) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )
135, 7, 12syl2an 283 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( 2nd `  A
)  .N  ( 1st `  B ) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )
1413eqeq2d 2094 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  =  ( ( 2nd `  A
)  .N  ( 1st `  B ) )  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
153, 11, 143bitrd 212 1  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  ~Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   <.cop 3420   class class class wbr 3806    X. cxp 4390   ` cfv 4953  (class class class)co 5564   1stc1st 5817   2ndc2nd 5818   N.cnpi 6560    .N cmi 6562    ~Q ceq 6567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-oadd 6090  df-omul 6091  df-ni 6592  df-mi 6594  df-enq 6635
This theorem is referenced by: (None)
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