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Theorem nq0nn 7062
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 6358 . . 3 (𝐴 ∈ ((ω × N) / ~Q0 ) → ∃𝑎 ∈ (ω × N)𝐴 = [𝑎] ~Q0 )
2 elxpi 4468 . . . . . . 7 (𝑎 ∈ (ω × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)))
32anim1i 334 . . . . . 6 ((𝑎 ∈ (ω × N) ∧ 𝐴 = [𝑎] ~Q0 ) → (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ))
4 19.41vv 1832 . . . . . 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 498 . . . . . . 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 6341 . . . . . . . . 9 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 )
98ad2antrr 473 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → [𝑎] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 )
107, 9eqtrd 2121 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 )
116, 10jca 301 . . . . . 6 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → ((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
12112eximi 1538 . . . . 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 2485 . . 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 7045 . 2 Q0 = ((ω × N) / ~Q0 )
1715, 16eleq2s 2183 1 (𝐴Q0 → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1290  wex 1427  wcel 1439  wrex 2361  cop 3453  ωcom 4418   × cxp 4450  [cec 6304   / cqs 6305  Ncnpi 6892   ~Q0 ceq0 6906  Q0cnq0 6907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-ec 6308  df-qs 6312  df-nq0 7045
This theorem is referenced by:  nqpnq0nq  7073  nq0m0r  7076  nq0a0  7077  nq02m  7085
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