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Theorem enq0breq 6418
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 5464 . . . . . 6 ((z = A u = 𝐷) → (z ·𝑜 u) = (A ·𝑜 𝐷))
2 oveq12 5464 . . . . . 6 ((w = B v = 𝐶) → (w ·𝑜 v) = (B ·𝑜 𝐶))
31, 2eqeqan12d 2052 . . . . 5 (((z = A u = 𝐷) (w = B v = 𝐶)) → ((z ·𝑜 u) = (w ·𝑜 v) ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))
43an42s 523 . . . 4 (((z = A w = B) (v = 𝐶 u = 𝐷)) → ((z ·𝑜 u) = (w ·𝑜 v) ↔ (A ·𝑜 𝐷) = (B ·𝑜 𝐶)))
54copsex4g 3975 . . 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 437 . 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 3955 . . 3 ((A 𝜔 B N) → ⟨A, B V)
8 opexg 3955 . . 3 ((𝐶 𝜔 𝐷 N) → ⟨𝐶, 𝐷 V)
9 eleq1 2097 . . . . . 6 (x = ⟨A, B⟩ → (x (𝜔 × N) ↔ ⟨A, B (𝜔 × N)))
109anbi1d 438 . . . . 5 (x = ⟨A, B⟩ → ((x (𝜔 × N) y (𝜔 × N)) ↔ (⟨A, B (𝜔 × N) y (𝜔 × N))))
11 eqeq1 2043 . . . . . . . 8 (x = ⟨A, B⟩ → (x = ⟨z, w⟩ ↔ ⟨A, B⟩ = ⟨z, w⟩))
1211anbi1d 438 . . . . . . 7 (x = ⟨A, B⟩ → ((x = ⟨z, w y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩)))
1312anbi1d 438 . . . . . 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 1747 . . . . 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 442 . . . 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 2097 . . . . . 6 (y = ⟨𝐶, 𝐷⟩ → (y (𝜔 × N) ↔ ⟨𝐶, 𝐷 (𝜔 × N)))
1716anbi2d 437 . . . . 5 (y = ⟨𝐶, 𝐷⟩ → ((⟨A, B (𝜔 × N) y (𝜔 × N)) ↔ (⟨A, B (𝜔 × N) 𝐶, 𝐷 (𝜔 × N))))
18 eqeq1 2043 . . . . . . . 8 (y = ⟨𝐶, 𝐷⟩ → (y = ⟨v, u⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨v, u⟩))
1918anbi2d 437 . . . . . . 7 (y = ⟨𝐶, 𝐷⟩ → ((⟨A, B⟩ = ⟨z, w y = ⟨v, u⟩) ↔ (⟨A, B⟩ = ⟨z, w𝐶, 𝐷⟩ = ⟨v, u⟩)))
2019anbi1d 438 . . . . . 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 1747 . . . . 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 442 . . . 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 6406 . . . 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 3997 . . 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 4319 . . . 4 ((A 𝜔 B N) → ⟨A, B (𝜔 × N))
27 opelxpi 4319 . . . 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 1242  wex 1378   wcel 1390  Vcvv 2551  cop 3370   class class class wbr 3755  𝜔com 4256   × cxp 4286  (class class class)co 5455   ·𝑜 comu 5938  Ncnpi 6256   ~Q0 ceq0 6270
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-iota 4810  df-fv 4853  df-ov 5458  df-enq0 6406
This theorem is referenced by:  enq0eceq  6419  nqnq0pi  6420  addcmpblnq0  6425  mulcmpblnq0  6426  mulcanenq0ec  6427  nnnq0lem1  6428
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