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Theorem nqpi 6534
Description: Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 6533 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 6189 . . 3 (𝐴 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝐴 = [𝑎] ~Q )
2 elxpi 4389 . . . . . . 7 (𝑎 ∈ (N × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)))
32anim1i 327 . . . . . 6 ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
4 19.41vv 1799 . . . . . 6 (∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) ↔ (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
53, 4sylibr 141 . . . . 5 ((𝑎 ∈ (N × N) ∧ 𝐴 = [𝑎] ~Q ) → ∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ))
6 simplr 490 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → (𝑤N𝑣N))
7 simpr 107 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [𝑎] ~Q )
8 eceq1 6172 . . . . . . . . 9 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
98ad2antrr 465 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
107, 9eqtrd 2088 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q )
116, 10jca 294 . . . . . 6 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) ∧ 𝐴 = [𝑎] ~Q ) → ((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
12112eximi 1508 . . . . 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 2445 . . 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 6504 . 2 Q = ((N × N) / ~Q )
1715, 16eleq2s 2148 1 (𝐴Q → ∃𝑤𝑣((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  wrex 2324  cop 3406   × cxp 4371  [cec 6135   / cqs 6136  Ncnpi 6428   ~Q ceq 6435  Qcnq 6436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-ec 6139  df-qs 6143  df-nqqs 6504
This theorem is referenced by:  ltdcnq  6553  archnqq  6573  nqpnq0nq  6609  nqnq0a  6610  nqnq0m  6611
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