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Theorem enqbreq2 9702
Description: Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
enqbreq2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))

Proof of Theorem enqbreq2
StepHypRef Expression
1 1st2nd2 7165 . . 3 (𝐴 ∈ (N × N) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 1st2nd2 7165 . . 3 (𝐵 ∈ (N × N) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
31, 2breqan12d 4639 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩))
4 xp1st 7158 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
5 xp2nd 7159 . . . 4 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
64, 5jca 554 . . 3 (𝐴 ∈ (N × N) → ((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N))
7 xp1st 7158 . . . 4 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
8 xp2nd 7159 . . . 4 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
97, 8jca 554 . . 3 (𝐵 ∈ (N × N) → ((1st𝐵) ∈ N ∧ (2nd𝐵) ∈ N))
10 enqbreq 9701 . . 3 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ ((1st𝐵) ∈ N ∧ (2nd𝐵) ∈ N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) = ((2nd𝐴) ·N (1st𝐵))))
116, 9, 10syl2an 494 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) = ((2nd𝐴) ·N (1st𝐵))))
12 mulcompi 9678 . . . 4 ((2nd𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (2nd𝐴))
1312eqeq2i 2633 . . 3 (((1st𝐴) ·N (2nd𝐵)) = ((2nd𝐴) ·N (1st𝐵)) ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴)))
1413a1i 11 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st𝐴) ·N (2nd𝐵)) = ((2nd𝐴) ·N (1st𝐵)) ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
153, 11, 143bitrd 294 1 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cop 4161   class class class wbr 4623   × cxp 5082  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  Ncnpi 9626   ·N cmi 9628   ~Q ceq 9633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-oadd 7524  df-omul 7525  df-ni 9654  df-mi 9656  df-enq 9693
This theorem is referenced by:  adderpqlem  9736  mulerpqlem  9737  ltsonq  9751  lterpq  9752
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