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Theorem nqpi 7034
Description: Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7033 but also shows that the numerator and denominator are positive integers. (Contributed by Jim Kingdon, 20-Sep-2019.)
Assertion
Ref Expression
nqpi (𝐴Q → ∃𝑤𝑣((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
Distinct variable group:   𝑣,𝐴,𝑤

Proof of Theorem nqpi
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 elqsi 6384 . . 3 (𝐴 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝐴 = [𝑎] ~Q )
2 elxpi 4483 . . . . . . 7 (𝑎 ∈ (N × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)))
32anim1i 334 . . . . . 6 ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
4 19.41vv 1838 . . . . . 6 (∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) ↔ (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
53, 4sylibr 133 . . . . 5 ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → ∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
6 simplr 498 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → (𝑤N𝑣N))
7 simpr 109 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [𝑎] ~Q )
8 eceq1 6367 . . . . . . . . 9 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
98ad2antrr 473 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
107, 9eqtrd 2127 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q )
116, 10jca 301 . . . . . 6 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → ((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
12112eximi 1544 . . . . 5 (∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → ∃𝑤𝑣((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
135, 12syl 14 . . . 4 ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → ∃𝑤𝑣((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
1413rexlimiva 2497 . . 3 (∃𝑎 ∈ (N × N)𝐴 = [𝑎] ~Q → ∃𝑤𝑣((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
151, 14syl 14 . 2 (𝐴 ∈ ((N × N) / ~Q ) → ∃𝑤𝑣((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
16 df-nqqs 7004 . 2 Q = ((N × N) / ~Q )
1715, 16eleq2s 2189 1 (𝐴Q → ∃𝑤𝑣((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wex 1433  wcel 1445  wrex 2371  cop 3469   × cxp 4465  [cec 6330   / cqs 6331  Ncnpi 6928   ~Q ceq 6935  Qcnq 6936
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 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-xp 4473  df-cnv 4475  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-ec 6334  df-qs 6338  df-nqqs 7004
This theorem is referenced by:  ltdcnq  7053  archnqq  7073  nqpnq0nq  7109  nqnq0a  7110  nqnq0m  7111
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