Theorem enqdc1 7581

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 6327	. . . 4 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
2		xp2nd 6328	. . . 4 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
3	1, 2	jca 306	. . 3 ⊢ (𝐶 ∈ (N × N) → ((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N))
4		enqdc 7580	. . 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 6337	. . . . 5 ⊢ (𝐶 ∈ (N × N) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
7	6	breq2d 4100	. . . 4 ⊢ (𝐶 ∈ (N × N) → (⟨𝐴, 𝐵⟩ ~Q 𝐶 ↔ ⟨𝐴, 𝐵⟩ ~Q ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩))
8	7	dcbid 845	. . 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 841   ∈ wcel 2202  ⟨cop 3672   class class class wbr 4088   × cxp 4723  ‘cfv 5326  1^st c1st 6300  2^nd c2nd 6301  Ncnpi 7491   ~_Q ceq 7498

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-oadd 6585  df-omul 6586  df-ni 7523  df-mi 7525  df-enq 7566

This theorem is referenced by: (None)