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Theorem dmaddpqlem 7364
Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7366. (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 6581 . . 3 (𝑥 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝑥 = [𝑎] ~Q )
2 elxpi 4639 . . . . . . . 8 (𝑎 ∈ (N × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)))
3 simpl 109 . . . . . . . . 9 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) → 𝑎 = ⟨𝑤, 𝑣⟩)
432eximi 1601 . . . . . . . 8 (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) → ∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩)
52, 4syl 14 . . . . . . 7 (𝑎 ∈ (N × N) → ∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩)
65anim1i 340 . . . . . 6 ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → (∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
7 19.41vv 1903 . . . . . 6 (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) ↔ (∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
86, 7sylibr 134 . . . . 5 ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
9 simpr 110 . . . . . . 7 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [𝑎] ~Q )
10 eceq1 6564 . . . . . . . 8 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
1110adantr 276 . . . . . . 7 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
129, 11eqtrd 2210 . . . . . 6 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
13122eximi 1601 . . . . 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 2589 . . 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 7335 . 2 Q = ((N × N) / ~Q )
1816, 17eleq2s 2272 1 (𝑥Q → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wex 1492  wcel 2148  wrex 2456  cop 3594   × cxp 4621  [cec 6527   / cqs 6528  Ncnpi 7259   ~Q ceq 7266  Qcnq 7267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-cnv 4631  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-ec 6531  df-qs 6535  df-nqqs 7335
This theorem is referenced by:  dmaddpq  7366  dmmulpq  7367
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