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Theorem nqpi 7391
Description: Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7390 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 6601 . . 3 (𝐴 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝐴 = [𝑎] ~Q )
2 elxpi 4654 . . . . . . 7 (𝑎 ∈ (N × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)))
32anim1i 340 . . . . . 6 ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
4 19.41vv 1913 . . . . . 6 (∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) ↔ (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
53, 4sylibr 134 . . . . 5 ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → ∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
6 simplr 528 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → (𝑤N𝑣N))
7 simpr 110 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [𝑎] ~Q )
8 eceq1 6584 . . . . . . . . 9 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
98ad2antrr 488 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
107, 9eqtrd 2220 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q )
116, 10jca 306 . . . . . 6 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → ((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
12112eximi 1611 . . . . 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 2599 . . 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 7361 . 2 Q = ((N × N) / ~Q )
1715, 16eleq2s 2282 1 (𝐴Q → ∃𝑤𝑣((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wex 1502  wcel 2158  wrex 2466  cop 3607   × cxp 4636  [cec 6547   / cqs 6548  Ncnpi 7285   ~Q ceq 7292  Qcnq 7293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-ec 6551  df-qs 6555  df-nqqs 7361
This theorem is referenced by:  ltdcnq  7410  archnqq  7430  nqpnq0nq  7466  nqnq0a  7467  nqnq0m  7468
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