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Theorem enq0breq 6534
 Description: Equivalence relation for non-negative fractions in terms of natural numbers. (Contributed by NM, 27-Aug-1995.)
Assertion
Ref Expression
enq0breq (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → (⟨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 5521 . . . . . 6 ((𝑧 = 𝐴𝑢 = 𝐷) → (𝑧 ·𝑜 𝑢) = (𝐴 ·𝑜 𝐷))
2 oveq12 5521 . . . . . 6 ((𝑤 = 𝐵𝑣 = 𝐶) → (𝑤 ·𝑜 𝑣) = (𝐵 ·𝑜 𝐶))
31, 2eqeqan12d 2055 . . . . 5 (((𝑧 = 𝐴𝑢 = 𝐷) ∧ (𝑤 = 𝐵𝑣 = 𝐶)) → ((𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣) ↔ (𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶)))
43an42s 523 . . . 4 (((𝑧 = 𝐴𝑤 = 𝐵) ∧ (𝑣 = 𝐶𝑢 = 𝐷)) → ((𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣) ↔ (𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶)))
54copsex4g 3984 . . 3 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → (∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ (𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶)))
65anbi2d 437 . 2 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → (((⟨𝐴, 𝐵⟩ ∈ (ω × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 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 3964 . . 3 ((𝐴 ∈ ω ∧ 𝐵N) → ⟨𝐴, 𝐵⟩ ∈ V)
8 opexg 3964 . . 3 ((𝐶 ∈ ω ∧ 𝐷N) → ⟨𝐶, 𝐷⟩ ∈ V)
9 eleq1 2100 . . . . . 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 2046 . . . . . . . 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 1750 . . . . 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 2100 . . . . . 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 2046 . . . . . . . 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, 𝑤⟩ ∧ ⟨𝐶, 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, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
21204exbidv 1750 . . . . 5 (𝑦 = ⟨𝐶, 𝐷⟩ → (∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ ∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 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, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))))
23 df-enq0 6522 . . . 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 4006 . . 3 ((⟨𝐴, 𝐵⟩ ∈ V ∧ ⟨𝐶, 𝐷⟩ ∈ V) → (⟨𝐴, 𝐵⟩ ~Q0𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (ω × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))))
257, 8, 24syl2an 273 . 2 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → (⟨𝐴, 𝐵⟩ ~Q0𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (ω × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))))
26 opelxpi 4376 . . . 4 ((𝐴 ∈ ω ∧ 𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (ω × N))
27 opelxpi 4376 . . . 4 ((𝐶 ∈ ω ∧ 𝐷N) → ⟨𝐶, 𝐷⟩ ∈ (ω × N))
2826, 27anim12i 321 . . 3 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → (⟨𝐴, 𝐵⟩ ∈ (ω × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (ω × N)))
2928biantrurd 289 . 2 (((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ((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 ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → (⟨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 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  ⟨cop 3378   class class class wbr 3764  ωcom 4313   × cxp 4343  (class class class)co 5512   ·𝑜 comu 5999  Ncnpi 6370   ~Q0 ceq0 6384 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-iota 4867  df-fv 4910  df-ov 5515  df-enq0 6522 This theorem is referenced by:  enq0eceq  6535  nqnq0pi  6536  addcmpblnq0  6541  mulcmpblnq0  6542  mulcanenq0ec  6543  nnnq0lem1  6544
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