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Theorem enqbreq2 9204
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 6726 . . 3  |-  ( A  e.  ( N.  X.  N. )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 1st2nd2 6726 . . 3  |-  ( B  e.  ( N.  X.  N. )  ->  B  = 
<. ( 1st `  B
) ,  ( 2nd `  B ) >. )
31, 2breqan12d 4418 . 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 6719 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
5 xp2nd 6720 . . . 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 6719 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
8 xp2nd 6720 . . . 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 9203 . . 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 9180 . . . 4  |-  ( ( 2nd `  A )  .N  ( 1st `  B
) )  =  ( ( 1st `  B
)  .N  ( 2nd `  A ) )
1312eqeq2i 2472 . . 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 1370    e. wcel 1758   <.cop 3994   class class class wbr 4403    X. cxp 4949   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689   N.cnpi 9126    .N cmi 9128    ~Q ceq 9133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-oadd 7037  df-omul 7038  df-ni 9156  df-mi 9158  df-enq 9195
This theorem is referenced by:  adderpqlem  9238  mulerpqlem  9239  ltsonq  9253  lterpq  9254
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