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Theorem nqpi 7154
Description: Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7153 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 6449 . . 3 (𝐴 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝐴 = [𝑎] ~Q )
2 elxpi 4525 . . . . . . 7 (𝑎 ∈ (N × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)))
32anim1i 338 . . . . . 6 ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
4 19.41vv 1859 . . . . . 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 504 . . . . . . 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 6432 . . . . . . . . 9 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
98ad2antrr 479 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
107, 9eqtrd 2150 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q )
116, 10jca 304 . . . . . 6 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → ((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
12112eximi 1565 . . . . 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 2521 . . 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 7124 . 2 Q = ((N × N) / ~Q )
1715, 16eleq2s 2212 1 (𝐴Q → ∃𝑤𝑣((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wex 1453  wcel 1465  wrex 2394  cop 3500   × cxp 4507  [cec 6395   / cqs 6396  Ncnpi 7048   ~Q ceq 7055  Qcnq 7056
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-ec 6399  df-qs 6403  df-nqqs 7124
This theorem is referenced by:  ltdcnq  7173  archnqq  7193  nqpnq0nq  7229  nqnq0a  7230  nqnq0m  7231
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