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Theorem enqbreq2 10319
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 7703 . . 3 (𝐴 ∈ (N × N) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 1st2nd2 7703 . . 3 (𝐵 ∈ (N × N) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
31, 2breqan12d 5055 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩))
4 xp1st 7696 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
5 xp2nd 7697 . . . 4 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
64, 5jca 515 . . 3 (𝐴 ∈ (N × N) → ((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N))
7 xp1st 7696 . . . 4 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
8 xp2nd 7697 . . . 4 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
97, 8jca 515 . . 3 (𝐵 ∈ (N × N) → ((1st𝐵) ∈ N ∧ (2nd𝐵) ∈ N))
10 enqbreq 10318 . . 3 ((((1st𝐴) ∈ N ∧ (2nd𝐴) ∈ N) ∧ ((1st𝐵) ∈ N ∧ (2nd𝐵) ∈ N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) = ((2nd𝐴) ·N (1st𝐵))))
116, 9, 10syl2an 598 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (⟨(1st𝐴), (2nd𝐴)⟩ ~Q ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) = ((2nd𝐴) ·N (1st𝐵))))
12 mulcompi 10295 . . . 4 ((2nd𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (2nd𝐴))
1312eqeq2i 2834 . . 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 308 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 209  wa 399   = wceq 1538  wcel 2115  cop 4546   class class class wbr 5039   × cxp 5526  cfv 6328  (class class class)co 7130  1st c1st 7662  2nd c2nd 7663  Ncnpi 10243   ·N cmi 10245   ~Q ceq 10250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-oadd 8081  df-omul 8082  df-ni 10271  df-mi 10273  df-enq 10310
This theorem is referenced by:  adderpqlem  10353  mulerpqlem  10354  ltsonq  10368  lterpq  10369
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