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Theorem enqbreq2 9310
Description: Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
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 6832 . . 3  |-  ( A  e.  ( N.  X.  N. )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 1st2nd2 6832 . . 3  |-  ( B  e.  ( N.  X.  N. )  ->  B  = 
<. ( 1st `  B
) ,  ( 2nd `  B ) >. )
31, 2breqan12d 4468 . 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 6825 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
5 xp2nd 6826 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
64, 5jca 532 . . 3  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  A )  e.  N.  /\  ( 2nd `  A )  e. 
N. ) )
7 xp1st 6825 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
8 xp2nd 6826 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
97, 8jca 532 . . 3  |-  ( B  e.  ( N.  X.  N. )  ->  ( ( 1st `  B )  e.  N.  /\  ( 2nd `  B )  e. 
N. ) )
10 enqbreq 9309 . . 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 477 . 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 mulcompi 9286 . . . 4  |-  ( ( 2nd `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )
1312eqeq2i 2485 . . 3  |-  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  =  ( ( 2nd `  A
)  .N  ( 1st `  B ) )  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )
1413a1i 11 . 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 279 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4039   class class class wbr 4453    X. cxp 5003   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   N.cnpi 9234    .N cmi 9236    ~Q ceq 9241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-oadd 7146  df-omul 7147  df-ni 9262  df-mi 9264  df-enq 9301
This theorem is referenced by:  adderpqlem  9344  mulerpqlem  9345  ltsonq  9359  lterpq  9360
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