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Theorem enqbreq2 9944
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 7354 . . 3 (𝐴 ∈ (N × N) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 1st2nd2 7354 . . 3 (𝐵 ∈ (N × N) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
31, 2breqan12d 4802 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩))
4 xp1st 7347 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
5 xp2nd 7348 . . . 4 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
64, 5jca 501 . . 3 (𝐴 ∈ (N × N) → ((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N))
7 xp1st 7347 . . . 4 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
8 xp2nd 7348 . . . 4 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
97, 8jca 501 . . 3 (𝐵 ∈ (N × N) → ((1st𝐵) ∈ N ∧ (2nd𝐵) ∈ N))
10 enqbreq 9943 . . 3 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ ((1st𝐵) ∈ N ∧ (2nd𝐵) ∈ N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) = ((2nd𝐴) ·N (1st𝐵))))
116, 9, 10syl2an 583 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) = ((2nd𝐴) ·N (1st𝐵))))
12 mulcompi 9920 . . . 4 ((2nd𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (2nd𝐴))
1312eqeq2i 2783 . . 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 382   = wceq 1631  wcel 2145  cop 4322   class class class wbr 4786   × cxp 5247  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  Ncnpi 9868   ·N cmi 9870   ~Q ceq 9875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-omul 7718  df-ni 9896  df-mi 9898  df-enq 9935
This theorem is referenced by:  adderpqlem  9978  mulerpqlem  9979  ltsonq  9993  lterpq  9994
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