Theorem enqdc1 7673

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 6358	. . . 4 |- ( C e. ( N. X. N. ) -> ( 1st ` C ) e. N. )
2		xp2nd 6359	. . . 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 7672	. . 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 6368	. . . . 5 |- ( C e. ( N. X. N. ) -> C = <. ( 1st ` C ) , ( 2nd ` C ) >. )
7	6	breq2d 4120	. . . 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 2203   <.cop 3691   class class class wbr 4108   X. cxp 4746   ` cfv 5351   1stc1st 6331   2ndc2nd 6332   N.cnpi 7583   ~Q ceq 7590

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-oadd 6650  df-omul 6651  df-ni 7615  df-mi 7617  df-enq 7658

This theorem is referenced by: (None)