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Theorem enqdc1 7177
Description: The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
Assertion
Ref Expression
enqdc1  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  ( N.  X.  N. ) )  -> DECID  <. A ,  B >.  ~Q  C )

Proof of Theorem enqdc1
StepHypRef Expression
1 xp1st 6063 . . . 4  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
2 xp2nd 6064 . . . 4  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
31, 2jca 304 . . 3  |-  ( C  e.  ( N.  X.  N. )  ->  ( ( 1st `  C )  e.  N.  /\  ( 2nd `  C )  e. 
N. ) )
4 enqdc 7176 . . 3  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. ) )  -> DECID  <. A ,  B >.  ~Q 
<. ( 1st `  C
) ,  ( 2nd `  C ) >. )
53, 4sylan2 284 . 2  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  ( N.  X.  N. ) )  -> DECID  <. A ,  B >.  ~Q 
<. ( 1st `  C
) ,  ( 2nd `  C ) >. )
6 1st2nd2 6073 . . . . 5  |-  ( C  e.  ( N.  X.  N. )  ->  C  = 
<. ( 1st `  C
) ,  ( 2nd `  C ) >. )
76breq2d 3941 . . . 4  |-  ( C  e.  ( N.  X.  N. )  ->  ( <. A ,  B >.  ~Q  C  <->  <. A ,  B >.  ~Q  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
)
87dcbid 823 . . 3  |-  ( C  e.  ( N.  X.  N. )  ->  (DECID  <. A ,  B >.  ~Q  C  <-> DECID  <. A ,  B >.  ~Q  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
)
98adantl 275 . 2  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  (DECID 
<. A ,  B >.  ~Q  C  <-> DECID  <. A ,  B >.  ~Q 
<. ( 1st `  C
) ,  ( 2nd `  C ) >. )
)
105, 9mpbird 166 1  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  ( N.  X.  N. ) )  -> DECID  <. A ,  B >.  ~Q  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 819    e. wcel 1480   <.cop 3530   class class class wbr 3929    X. cxp 4537   ` cfv 5123   1stc1st 6036   2ndc2nd 6037   N.cnpi 7087    ~Q ceq 7094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318  df-ni 7119  df-mi 7121  df-enq 7162
This theorem is referenced by: (None)
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