Theorem enqdc1 7693

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

Step	Hyp	Ref	Expression
1		xp1st 6372	. . . 4 |- ( C e. ( N. X. N. ) -> ( 1st ` C ) e. N. )
2		xp2nd 6373	. . . 4 |- ( C e. ( N. X. N. ) -> ( 2nd ` C ) e. N. )
3	1, 2	jca 306	. . 3 |- ( C e. ( N. X. N. ) -> ( ( 1st ` C ) e. N. /\ ( 2nd ` C ) e. N. ) )
4		enqdc 7692	. . 3 |- ( ( ( A e. N. /\ B e. N. ) /\ ( ( 1st ` C ) e. N. /\ ( 2nd ` C ) e. N. ) ) -> DECID <. A , B >. ~Q <. ( 1st ` C ) , ( 2nd ` C ) >. )
5	3, 4	sylan2 286	. 2 |- ( ( ( A e. N. /\ B e. N. ) /\ C e. ( N. X. N. ) ) -> DECID <. A , B >. ~Q <. ( 1st ` C ) , ( 2nd ` C ) >. )
6		1st2nd2 6382	. . . . 5 |- ( C e. ( N. X. N. ) -> C = <. ( 1st ` C ) , ( 2nd ` C ) >. )
7	6	breq2d 4126	. . . 4 |- ( C e. ( N. X. N. ) -> ( <. A , B >. ~Q C <-> <. A , B >. ~Q <. ( 1st ` C ) , ( 2nd ` C ) >. ) )
8	7	dcbid 846	. . 3 |- ( C e. ( N. X. N. ) -> (DECID <. A , B >. ~Q C <-> DECID <. A , B >. ~Q <. ( 1st ` C ) , ( 2nd ` C ) >. ) )
9	8	adantl 277	. 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 ) >. ) )
10	5, 9	mpbird 167	1 |- ( ( ( A e. N. /\ B e. N. ) /\ C e. ( N. X. N. ) ) -> DECID <. A , B >. ~Q C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   e. wcel 2205   <.cop 3697   class class class wbr 4114   X. cxp 4752   ` cfv 5357   1stc1st 6345   2ndc2nd 6346   N.cnpi 7603   ~Q ceq 7610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-oadd 6664  df-omul 6665  df-ni 7635  df-mi 7637  df-enq 7678

This theorem is referenced by: (None)