Theorem enqdc1 7719

Description: The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)

Assertion

Ref	Expression
enqdc1	⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ (N × N)) → DECID ⟨𝐴, 𝐵⟩ ~Q 𝐶)

Proof of Theorem enqdc1

Step	Hyp	Ref	Expression
1		xp1st 6389	. . . 4 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
2		xp2nd 6390	. . . 4 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
3	1, 2	jca 306	. . 3 ⊢ (𝐶 ∈ (N × N) → ((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N))
4		enqdc 7718	. . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ ((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N)) → DECID ⟨𝐴, 𝐵⟩ ~Q ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
5	3, 4	sylan2 286	. 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ (N × N)) → DECID ⟨𝐴, 𝐵⟩ ~Q ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
6		1st2nd2 6399	. . . . 5 ⊢ (𝐶 ∈ (N × N) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
7	6	breq2d 4137	. . . 4 ⊢ (𝐶 ∈ (N × N) → (⟨𝐴, 𝐵⟩ ~Q 𝐶 ↔ ⟨𝐴, 𝐵⟩ ~Q ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩))
8	7	dcbid 850	. . 3 ⊢ (𝐶 ∈ (N × N) → (DECID ⟨𝐴, 𝐵⟩ ~Q 𝐶 ↔ DECID ⟨𝐴, 𝐵⟩ ~Q ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩))
9	8	adantl 277	. 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ (N × N)) → (DECID ⟨𝐴, 𝐵⟩ ~Q 𝐶 ↔ DECID ⟨𝐴, 𝐵⟩ ~Q ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩))
10	5, 9	mpbird 167	1 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ (N × N)) → DECID ⟨𝐴, 𝐵⟩ ~Q 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   ∈ wcel 2209  ⟨cop 3708   class class class wbr 4125   × cxp 4767  ‘cfv 5372  1^st c1st 6362  2^nd c2nd 6363  Ncnpi 7629   ~_Q ceq 7636

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-oadd 6681  df-omul 6682  df-ni 7661  df-mi 7663  df-enq 7704

This theorem is referenced by: (None)