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Theorem enqbreq2 9727
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 7190 . . 3 (𝐴 ∈ (N × N) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 1st2nd2 7190 . . 3 (𝐵 ∈ (N × N) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
31, 2breqan12d 4660 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩))
4 xp1st 7183 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
5 xp2nd 7184 . . . 4 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
64, 5jca 554 . . 3 (𝐴 ∈ (N × N) → ((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N))
7 xp1st 7183 . . . 4 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
8 xp2nd 7184 . . . 4 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
97, 8jca 554 . . 3 (𝐵 ∈ (N × N) → ((1st𝐵) ∈ N ∧ (2nd𝐵) ∈ N))
10 enqbreq 9726 . . 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 9703 . . . 4 ((2nd𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (2nd𝐴))
1312eqeq2i 2632 . . 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 1481  wcel 1988  cop 4174   class class class wbr 4644   × cxp 5102  cfv 5876  (class class class)co 6635  1st c1st 7151  2nd c2nd 7152  Ncnpi 9651   ·N cmi 9653   ~Q ceq 9658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549  df-omul 7550  df-ni 9679  df-mi 9681  df-enq 9718
This theorem is referenced by:  adderpqlem  9761  mulerpqlem  9762  ltsonq  9776  lterpq  9777
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