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Theorem nq0nn 6764
Description: Decomposition of a non-negative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
Assertion
Ref Expression
nq0nn (𝐴Q0 → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
Distinct variable group:   𝑣,𝐴,𝑤

Proof of Theorem nq0nn
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 elqsi 6246 . . 3 (𝐴 ∈ ((ω × N) / ~Q0 ) → ∃𝑎 ∈ (ω × N)𝐴 = [𝑎] ~Q0 )
2 elxpi 4407 . . . . . . 7 (𝑎 ∈ (ω × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)))
32anim1i 333 . . . . . 6 ((𝑎 ∈ (ω × N) ∧ 𝐴 = [𝑎] ~Q0 ) → (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ))
4 19.41vv 1826 . . . . . 6 (∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) ↔ (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ))
53, 4sylibr 132 . . . . 5 ((𝑎 ∈ (ω × N) ∧ 𝐴 = [𝑎] ~Q0 ) → ∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ))
6 simplr 497 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → (𝑤 ∈ ω ∧ 𝑣N))
7 simpr 108 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → 𝐴 = [𝑎] ~Q0 )
8 eceq1 6229 . . . . . . . . 9 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 )
98ad2antrr 472 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → [𝑎] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 )
107, 9eqtrd 2115 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 )
116, 10jca 300 . . . . . 6 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → ((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
12112eximi 1533 . . . . 5 (∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
135, 12syl 14 . . . 4 ((𝑎 ∈ (ω × N) ∧ 𝐴 = [𝑎] ~Q0 ) → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
1413rexlimiva 2477 . . 3 (∃𝑎 ∈ (ω × N)𝐴 = [𝑎] ~Q0 → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
151, 14syl 14 . 2 (𝐴 ∈ ((ω × N) / ~Q0 ) → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
16 df-nq0 6747 . 2 Q0 = ((ω × N) / ~Q0 )
1715, 16eleq2s 2177 1 (𝐴Q0 → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  wrex 2354  cop 3419  ωcom 4359   × cxp 4389  [cec 6192   / cqs 6193  Ncnpi 6594   ~Q0 ceq0 6608  Q0cnq0 6609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-ec 6196  df-qs 6200  df-nq0 6747
This theorem is referenced by:  nqpnq0nq  6775  nq0m0r  6778  nq0a0  6779  nq02m  6787
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