Theorem enqdc1 7363

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 6168	. . . 4 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
2		xp2nd 6169	. . . 4 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
3	1, 2	jca 306	. . 3 ⊢ (𝐶 ∈ (N × N) → ((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N))
4		enqdc 7362	. . 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 6178	. . . . 5 ⊢ (𝐶 ∈ (N × N) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
7	6	breq2d 4017	. . . 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 2148  ⟨cop 3597   class class class wbr 4005   × cxp 4626  ‘cfv 5218  1^st c1st 6141  2^nd c2nd 6142  Ncnpi 7273   ~_Q ceq 7280

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-oadd 6423  df-omul 6424  df-ni 7305  df-mi 7307  df-enq 7348

This theorem is referenced by: (None)