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Theorem enqbreq2 10961
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 8054 . . 3 (𝐴 ∈ (N × N) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 1st2nd2 8054 . . 3 (𝐵 ∈ (N × N) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
31, 2breqan12d 5158 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩))
4 xp1st 8047 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
5 xp2nd 8048 . . . 4 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
64, 5jca 511 . . 3 (𝐴 ∈ (N × N) → ((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N))
7 xp1st 8047 . . . 4 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
8 xp2nd 8048 . . . 4 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
97, 8jca 511 . . 3 (𝐵 ∈ (N × N) → ((1st𝐵) ∈ N ∧ (2nd𝐵) ∈ N))
10 enqbreq 10960 . . 3 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ ((1st𝐵) ∈ N ∧ (2nd𝐵) ∈ N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) = ((2nd𝐴) ·N (1st𝐵))))
116, 9, 10syl2an 596 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) = ((2nd𝐴) ·N (1st𝐵))))
12 mulcompi 10937 . . . 4 ((2nd𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (2nd𝐴))
1312eqeq2i 2749 . . 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 305 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 206  wa 395   = wceq 1539  wcel 2107  cop 4631   class class class wbr 5142   × cxp 5682  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014  Ncnpi 10885   ·N cmi 10887   ~Q ceq 10892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-oadd 8511  df-omul 8512  df-ni 10913  df-mi 10915  df-enq 10952
This theorem is referenced by:  adderpqlem  10995  mulerpqlem  10996  ltsonq  11010  lterpq  11011
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