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Theorem nq0nn 7364
Description: Decomposition of a nonnegative 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 6534 . . 3 (𝐴 ∈ ((ω × N) / ~Q0 ) → ∃𝑎 ∈ (ω × N)𝐴 = [𝑎] ~Q0 )
2 elxpi 4604 . . . . . . 7 (𝑎 ∈ (ω × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)))
32anim1i 338 . . . . . 6 ((𝑎 ∈ (ω × N) ∧ 𝐴 = [𝑎] ~Q0 ) → (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ))
4 19.41vv 1883 . . . . . 6 (∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) ↔ (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ))
53, 4sylibr 133 . . . . 5 ((𝑎 ∈ (ω × N) ∧ 𝐴 = [𝑎] ~Q0 ) → ∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ))
6 simplr 520 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → (𝑤 ∈ ω ∧ 𝑣N))
7 simpr 109 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → 𝐴 = [𝑎] ~Q0 )
8 eceq1 6517 . . . . . . . . 9 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 )
98ad2antrr 480 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → [𝑎] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 )
107, 9eqtrd 2190 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 )
116, 10jca 304 . . . . . 6 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → ((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
12112eximi 1581 . . . . 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 2569 . . 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 7347 . 2 Q0 = ((ω × N) / ~Q0 )
1715, 16eleq2s 2252 1 (𝐴Q0 → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wex 1472  wcel 2128  wrex 2436  cop 3564  ωcom 4551   × cxp 4586  [cec 6480   / cqs 6481  Ncnpi 7194   ~Q0 ceq0 7208  Q0cnq0 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-ec 6484  df-qs 6488  df-nq0 7347
This theorem is referenced by:  nqpnq0nq  7375  nq0m0r  7378  nq0a0  7379  nq02m  7387
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