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Theorem nq0nn 7404
Description: Decomposition of a nonnegative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
Assertion
Ref Expression
nq0nn (𝐴Q0 → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
Distinct variable group:   𝑣,𝐴,𝑤

Proof of Theorem nq0nn
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 elqsi 6565 . . 3 (𝐴 ∈ ((ω × N) / ~Q0 ) → ∃𝑎 ∈ (ω × N)𝐴 = [𝑎] ~Q0 )
2 elxpi 4627 . . . . . . 7 (𝑎 ∈ (ω × N) → ∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)))
32anim1i 338 . . . . . 6 ((𝑎 ∈ (ω × N) ∧ 𝐴 = [𝑎] ~Q0 ) → (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ))
4 19.41vv 1896 . . . . . 6 (∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) ↔ (∃𝑤𝑣(𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ))
53, 4sylibr 133 . . . . 5 ((𝑎 ∈ (ω × N) ∧ 𝐴 = [𝑎] ~Q0 ) → ∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ))
6 simplr 525 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → (𝑤 ∈ ω ∧ 𝑣N))
7 simpr 109 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → 𝐴 = [𝑎] ~Q0 )
8 eceq1 6548 . . . . . . . . 9 (𝑎 = ⟨𝑤, 𝑣⟩ → [𝑎] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 )
98ad2antrr 485 . . . . . . . 8 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → [𝑎] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 )
107, 9eqtrd 2203 . . . . . . 7 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 )
116, 10jca 304 . . . . . 6 (((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → ((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
12112eximi 1594 . . . . 5 (∃𝑤𝑣((𝑎 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ ω ∧ 𝑣N)) ∧ 𝐴 = [𝑎] ~Q0 ) → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
135, 12syl 14 . . . 4 ((𝑎 ∈ (ω × N) ∧ 𝐴 = [𝑎] ~Q0 ) → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
1413rexlimiva 2582 . . 3 (∃𝑎 ∈ (ω × N)𝐴 = [𝑎] ~Q0 → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
151, 14syl 14 . 2 (𝐴 ∈ ((ω × N) / ~Q0 ) → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
16 df-nq0 7387 . 2 Q0 = ((ω × N) / ~Q0 )
1715, 16eleq2s 2265 1 (𝐴Q0 → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wex 1485  wcel 2141  wrex 2449  cop 3586  ωcom 4574   × cxp 4609  [cec 6511   / cqs 6512  Ncnpi 7234   ~Q0 ceq0 7248  Q0cnq0 7249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-ec 6515  df-qs 6519  df-nq0 7387
This theorem is referenced by:  nqpnq0nq  7415  nq0m0r  7418  nq0a0  7419  nq02m  7427
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