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Theorem nqpi 6476
 Description: Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 6475 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 6158 . . 3 (𝐴 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝐴 = [𝑎] ~Q )
2 elxpi 4361 . . . . . . 7 (𝑎 ∈ (N × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)))
32anim1i 323 . . . . . 6 ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
4 19.41vv 1783 . . . . . 6 (∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) ↔ (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
53, 4sylibr 137 . . . . 5 ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → ∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
6 simplr 482 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → (𝑤N𝑣N))
7 simpr 103 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [𝑎] ~Q )
8 eceq1 6141 . . . . . . . . 9 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
98ad2antrr 457 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
107, 9eqtrd 2072 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q )
116, 10jca 290 . . . . . 6 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → ((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
12112eximi 1492 . . . . 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 2428 . . 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 6446 . 2 Q = ((N × N) / ~Q )
1715, 16eleq2s 2132 1 (𝐴Q → ∃𝑤𝑣((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  ⟨cop 3378   × cxp 4343  [cec 6104   / cqs 6105  Ncnpi 6370   ~Q ceq 6377  Qcnq 6378 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-ec 6108  df-qs 6112  df-nqqs 6446 This theorem is referenced by:  ltdcnq  6495  archnqq  6515  nqpnq0nq  6551  nqnq0a  6552  nqnq0m  6553
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