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Theorem dmaddpqlem 6629
Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6631. (Contributed by Jim Kingdon, 15-Sep-2019.)
Assertion
Ref Expression
dmaddpqlem (𝑥Q → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
Distinct variable group:   𝑤,𝑣,𝑥

Proof of Theorem dmaddpqlem
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 elqsi 6224 . . 3 (𝑥 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝑥 = [𝑎] ~Q )
2 elxpi 4387 . . . . . . . 8 (𝑎 ∈ (N × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)))
3 simpl 107 . . . . . . . . 9 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) → 𝑎 = ⟨𝑤, 𝑣⟩)
432eximi 1533 . . . . . . . 8 (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) → ∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩)
52, 4syl 14 . . . . . . 7 (𝑎 ∈ (N × N) → ∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩)
65anim1i 333 . . . . . 6 ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → (∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
7 19.41vv 1825 . . . . . 6 (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) ↔ (∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
86, 7sylibr 132 . . . . 5 ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
9 simpr 108 . . . . . . 7 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [𝑎] ~Q )
10 eceq1 6207 . . . . . . . 8 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
1110adantr 270 . . . . . . 7 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
129, 11eqtrd 2114 . . . . . 6 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
13122eximi 1533 . . . . 5 (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
148, 13syl 14 . . . 4 ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
1514rexlimiva 2473 . . 3 (∃𝑎 ∈ (N × N)𝑥 = [𝑎] ~Q → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
161, 15syl 14 . 2 (𝑥 ∈ ((N × N) / ~Q ) → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
17 df-nqqs 6600 . 2 Q = ((N × N) / ~Q )
1816, 17eleq2s 2174 1 (𝑥Q → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  wrex 2350  cop 3409   × cxp 4369  [cec 6170   / cqs 6171  Ncnpi 6524   ~Q ceq 6531  Qcnq 6532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-ec 6174  df-qs 6178  df-nqqs 6600
This theorem is referenced by:  dmaddpq  6631  dmmulpq  6632
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