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Theorem dmaddpqlem 6997
Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6999. (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 6358 . . 3 (𝑥 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝑥 = [𝑎] ~Q )
2 elxpi 4468 . . . . . . . 8 (𝑎 ∈ (N × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)))
3 simpl 108 . . . . . . . . 9 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) → 𝑎 = ⟨𝑤, 𝑣⟩)
432eximi 1538 . . . . . . . 8 (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) → ∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩)
52, 4syl 14 . . . . . . 7 (𝑎 ∈ (N × N) → ∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩)
65anim1i 334 . . . . . 6 ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → (∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
7 19.41vv 1832 . . . . . 6 (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) ↔ (∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
86, 7sylibr 133 . . . . 5 ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
9 simpr 109 . . . . . . 7 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [𝑎] ~Q )
10 eceq1 6341 . . . . . . . 8 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
1110adantr 271 . . . . . . 7 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
129, 11eqtrd 2121 . . . . . 6 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
13122eximi 1538 . . . . 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 2485 . . 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 6968 . 2 Q = ((N × N) / ~Q )
1816, 17eleq2s 2183 1 (𝑥Q → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1290  wex 1427  wcel 1439  wrex 2361  cop 3453   × cxp 4450  [cec 6304   / cqs 6305  Ncnpi 6892   ~Q ceq 6899  Qcnq 6900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-ec 6308  df-qs 6312  df-nqqs 6968
This theorem is referenced by:  dmaddpq  6999  dmmulpq  7000
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