Theorem enqdc1 7336

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 6156	. . . 4 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
2		xp2nd 6157	. . . 4 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
3	1, 2	jca 306	. . 3 ⊢ (𝐶 ∈ (N × N) → ((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N))
4		enqdc 7335	. . 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 6166	. . . . 5 ⊢ (𝐶 ∈ (N × N) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
7	6	breq2d 4010	. . . 4 ⊢ (𝐶 ∈ (N × N) → (⟨𝐴, 𝐵⟩ ~Q 𝐶 ↔ ⟨𝐴, 𝐵⟩ ~Q ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩))
8	7	dcbid 838	. . 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 834   ∈ wcel 2146  ⟨cop 3592   class class class wbr 3998   × cxp 4618  ‘cfv 5208  1^st c1st 6129  2^nd c2nd 6130  Ncnpi 7246   ~_Q ceq 7253

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-oadd 6411  df-omul 6412  df-ni 7278  df-mi 7280  df-enq 7321

This theorem is referenced by: (None)