Theorem enqdc1 7324

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 6144	. . . 4 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
2		xp2nd 6145	. . . 4 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
3	1, 2	jca 304	. . 3 ⊢ (𝐶 ∈ (N × N) → ((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N))
4		enqdc 7323	. . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ ((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N)) → DECID ⟨𝐴, 𝐵⟩ ~Q ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
5	3, 4	sylan2 284	. 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ (N × N)) → DECID ⟨𝐴, 𝐵⟩ ~Q ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
6		1st2nd2 6154	. . . . 5 ⊢ (𝐶 ∈ (N × N) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
7	6	breq2d 4001	. . . 4 ⊢ (𝐶 ∈ (N × N) → (⟨𝐴, 𝐵⟩ ~Q 𝐶 ↔ ⟨𝐴, 𝐵⟩ ~Q ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩))
8	7	dcbid 833	. . 3 ⊢ (𝐶 ∈ (N × N) → (DECID ⟨𝐴, 𝐵⟩ ~Q 𝐶 ↔ DECID ⟨𝐴, 𝐵⟩ ~Q ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩))
9	8	adantl 275	. 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 166	1 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ (N × N)) → DECID ⟨𝐴, 𝐵⟩ ~Q 𝐶)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 829   ∈ wcel 2141  ⟨cop 3586   class class class wbr 3989   × cxp 4609  ‘cfv 5198  1^st c1st 6117  2^nd c2nd 6118  Ncnpi 7234   ~_Q ceq 7241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-oadd 6399  df-omul 6400  df-ni 7266  df-mi 7268  df-enq 7309

This theorem is referenced by: (None)