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Theorem enq0breq 6291
Description: Equivalence relation for non-negative fractions in terms of natural numbers. (Contributed by NM, 27-Aug-1995.)
Assertion
Ref Expression
enq0breq (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → (⟨A, B⟩ ~Q0𝐶, 𝐷⟩ ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))

Proof of Theorem enq0breq
Dummy variables x y z w v u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 5445 . . . . . 6 ((z = A u = 𝐷) → (z ·𝑜 u) = (A ·𝑜 𝐷))
2 oveq12 5445 . . . . . 6 ((w = B v = 𝐶) → (w ·𝑜 v) = (B ·𝑜 𝐶))
31, 2eqeqan12d 2037 . . . . 5 (((z = A u = 𝐷) (w = B v = 𝐶)) → ((z ·𝑜 u) = (w ·𝑜 v) ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))
43an42s 510 . . . 4 (((z = A w = B) (v = 𝐶 u = 𝐷)) → ((z ·𝑜 u) = (w ·𝑜 v) ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))
54copsex4g 3958 . . 3 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → (zwvu((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v)) ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))
65anbi2d 440 . 2 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → (((⟨A, B (𝜔 × N) 𝐶, 𝐷 (𝜔 × N)) zwvu((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v))) ↔ ((⟨A, B (𝜔 × N) 𝐶, 𝐷 (𝜔 × N)) (A ·𝑜 𝐷) = (B ·𝑜 𝐶))))
7 opexg 3938 . . 3 ((A 𝜔 B N) → ⟨A, B V)
8 opexg 3938 . . 3 ((𝐶 𝜔 𝐷 N) → ⟨𝐶, 𝐷 V)
9 eleq1 2082 . . . . . 6 (x = ⟨A, B⟩ → (x (𝜔 × N) ↔ ⟨A, B (𝜔 × N)))
109anbi1d 441 . . . . 5 (x = ⟨A, B⟩ → ((x (𝜔 × N) y (𝜔 × N)) ↔ (⟨A, B (𝜔 × N) y (𝜔 × N))))
11 eqeq1 2028 . . . . . . . 8 (x = ⟨A, B⟩ → (x = ⟨z, w⟩ ↔ ⟨A, B⟩ = ⟨z, w⟩))
1211anbi1d 441 . . . . . . 7 (x = ⟨A, B⟩ → ((x = ⟨z, w y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩)))
1312anbi1d 441 . . . . . 6 (x = ⟨A, B⟩ → (((x = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v)) ↔ ((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v))))
14134exbidv 1732 . . . . 5 (x = ⟨A, B⟩ → (zwvu((x = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v)) ↔ zwvu((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v))))
1510, 14anbi12d 445 . . . 4 (x = ⟨A, B⟩ → (((x (𝜔 × N) y (𝜔 × N)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v))) ↔ ((⟨A, B (𝜔 × N) y (𝜔 × N)) zwvu((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v)))))
16 eleq1 2082 . . . . . 6 (y = ⟨𝐶, 𝐷⟩ → (y (𝜔 × N) ↔ ⟨𝐶, 𝐷 (𝜔 × N)))
1716anbi2d 440 . . . . 5 (y = ⟨𝐶, 𝐷⟩ → ((⟨A, B (𝜔 × N) y (𝜔 × N)) ↔ (⟨A, B (𝜔 × N) 𝐶, 𝐷 (𝜔 × N))))
18 eqeq1 2028 . . . . . . . 8 (y = ⟨𝐶, 𝐷⟩ → (y = ⟨v, u⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩))
1918anbi2d 440 . . . . . . 7 (y = ⟨𝐶, 𝐷⟩ → ((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩)))
2019anbi1d 441 . . . . . 6 (y = ⟨𝐶, 𝐷⟩ → (((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v)) ↔ ((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v))))
21204exbidv 1732 . . . . 5 (y = ⟨𝐶, 𝐷⟩ → (zwvu((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v)) ↔ zwvu((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v))))
2217, 21anbi12d 445 . . . 4 (y = ⟨𝐶, 𝐷⟩ → (((⟨A, B (𝜔 × N) y (𝜔 × N)) zwvu((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v))) ↔ ((⟨A, B (𝜔 × N) 𝐶, 𝐷 (𝜔 × N)) zwvu((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v)))))
23 df-enq0 6279 . . . 4 ~Q0 = {⟨x, y⟩ ∣ ((x (𝜔 × N) y (𝜔 × N)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v)))}
2415, 22, 23brabg 3980 . . 3 ((⟨A, B V 𝐶, 𝐷 V) → (⟨A, B⟩ ~Q0𝐶, 𝐷⟩ ↔ ((⟨A, B (𝜔 × N) 𝐶, 𝐷 (𝜔 × N)) zwvu((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v)))))
257, 8, 24syl2an 273 . 2 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → (⟨A, B⟩ ~Q0𝐶, 𝐷⟩ ↔ ((⟨A, B (𝜔 × N) 𝐶, 𝐷 (𝜔 × N)) zwvu((⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v)))))
26 opelxpi 4303 . . . 4 ((A 𝜔 B N) → ⟨A, B (𝜔 × N))
27 opelxpi 4303 . . . 4 ((𝐶 𝜔 𝐷 N) → ⟨𝐶, 𝐷 (𝜔 × N))
2826, 27anim12i 321 . . 3 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → (⟨A, B (𝜔 × N) 𝐶, 𝐷 (𝜔 × N)))
2928biantrurd 289 . 2 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → ((A ·𝑜 𝐷) = (B ·𝑜 𝐶) ↔ ((⟨A, B (𝜔 × N) 𝐶, 𝐷 (𝜔 × N)) (A ·𝑜 𝐷) = (B ·𝑜 𝐶))))
306, 25, 293bitr4d 209 1 (((A 𝜔 B N) (𝐶 𝜔 𝐷 N)) → (⟨A, B⟩ ~Q0𝐶, 𝐷⟩ ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  Vcvv 2535  cop 3353   class class class wbr 3738  𝜔com 4240   × cxp 4270  (class class class)co 5436   ·𝑜 comu 5914  Ncnpi 6130   ~Q0 ceq0 6144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-xp 4278  df-iota 4794  df-fv 4837  df-ov 5439  df-enq0 6279
This theorem is referenced by:  enq0eceq  6292  nqnq0pi  6293  addcmpblnq0  6298  mulcmpblnq0  6299  mulcanenq0ec  6300  nnnq0lem1  6301
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