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Theorem dmaddpqlem 6418
Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6420. (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 6117 . . 3 (𝑥 ∈ ((N × N) / ~Q ) → ∃𝑎 ∈ (N × N)𝑥 = [𝑎] ~Q )
2 elxpi 4322 . . . . . . . 8 (𝑎 ∈ (N × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)))
3 simpl 102 . . . . . . . . 9 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) → 𝑎 = ⟨𝑤, 𝑣⟩)
432eximi 1492 . . . . . . . 8 (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤N𝑣N)) → ∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩)
52, 4syl 14 . . . . . . 7 (𝑎 ∈ (N × N) → ∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩)
65anim1i 323 . . . . . 6 ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → (∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
7 19.41vv 1783 . . . . . 6 (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) ↔ (∃𝑤𝑣 𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
86, 7sylibr 137 . . . . 5 ((𝑎 ∈ (N × N) ∧ 𝑥 = [𝑎] ~Q ) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ))
9 simpr 103 . . . . . . 7 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [𝑎] ~Q )
10 eceq1 6100 . . . . . . . 8 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
1110adantr 261 . . . . . . 7 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → [𝑎] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
129, 11eqtrd 2072 . . . . . 6 ((𝑎 = ⟨𝑤, 𝑣⟩ ∧ 𝑥 = [𝑎] ~Q ) → 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
13122eximi 1492 . . . . 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 2425 . . 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 6389 . 2 Q = ((N × N) / ~Q )
1816, 17eleq2s 2132 1 (𝑥Q → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wex 1381  wcel 1393  wrex 2304  cop 3375   × cxp 4304  [cec 6063   / cqs 6064  Ncnpi 6313   ~Q ceq 6320  Qcnq 6321
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 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-sn 3378  df-pr 3379  df-op 3381  df-br 3761  df-opab 3815  df-xp 4312  df-cnv 4314  df-dm 4316  df-rn 4317  df-res 4318  df-ima 4319  df-ec 6067  df-qs 6071  df-nqqs 6389
This theorem is referenced by:  dmaddpq  6420  dmmulpq  6421
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