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Theorem enqdc1 6518
 Description: The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
Assertion
Ref Expression
enqdc1 (((𝐴N𝐵N) ∧ 𝐶 ∈ (N × N)) → DECID𝐴, 𝐵⟩ ~Q 𝐶)

Proof of Theorem enqdc1
StepHypRef Expression
1 xp1st 5820 . . . 4 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
2 xp2nd 5821 . . . 4 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
31, 2jca 294 . . 3 (𝐶 ∈ (N × N) → ((1st𝐶) ∈ N ∧ (2nd𝐶) ∈ N))
4 enqdc 6517 . . 3 (((𝐴N𝐵N) ∧ ((1st𝐶) ∈ N ∧ (2nd𝐶) ∈ N)) → DECID𝐴, 𝐵⟩ ~Q ⟨(1st𝐶), (2nd𝐶)⟩)
53, 4sylan2 274 . 2 (((𝐴N𝐵N) ∧ 𝐶 ∈ (N × N)) → DECID𝐴, 𝐵⟩ ~Q ⟨(1st𝐶), (2nd𝐶)⟩)
6 1st2nd2 5829 . . . . 5 (𝐶 ∈ (N × N) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
76breq2d 3804 . . . 4 (𝐶 ∈ (N × N) → (⟨𝐴, 𝐵⟩ ~Q 𝐶 ↔ ⟨𝐴, 𝐵⟩ ~Q ⟨(1st𝐶), (2nd𝐶)⟩))
87dcbid 759 . . 3 (𝐶 ∈ (N × N) → (DECID𝐴, 𝐵⟩ ~Q 𝐶DECID𝐴, 𝐵⟩ ~Q ⟨(1st𝐶), (2nd𝐶)⟩))
98adantl 266 . 2 (((𝐴N𝐵N) ∧ 𝐶 ∈ (N × N)) → (DECID𝐴, 𝐵⟩ ~Q 𝐶DECID𝐴, 𝐵⟩ ~Q ⟨(1st𝐶), (2nd𝐶)⟩))
105, 9mpbird 160 1 (((𝐴N𝐵N) ∧ 𝐶 ∈ (N × N)) → DECID𝐴, 𝐵⟩ ~Q 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102  DECID wdc 753   ∈ wcel 1409  ⟨cop 3406   class class class wbr 3792   × cxp 4371  ‘cfv 4930  1st c1st 5793  2nd c2nd 5794  Ncnpi 6428   ~Q ceq 6435 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339 This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037  df-ni 6460  df-mi 6462  df-enq 6503 This theorem is referenced by: (None)
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