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Theorem dfmq0qs 7419
Description: Multiplication on nonnegative fractions. This definition is similar to df-mq0 7418 but expands Q0. (Contributed by Jim Kingdon, 22-Nov-2019.)
Assertion
Ref Expression
dfmq0qs ·Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 ))}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓

Proof of Theorem dfmq0qs
StepHypRef Expression
1 df-mq0 7418 . 2 ·Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q0𝑦Q0) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 ))}
2 df-nq0 7415 . . . . . 6 Q0 = ((ω × N) / ~Q0 )
32eleq2i 2244 . . . . 5 (𝑥Q0𝑥 ∈ ((ω × N) / ~Q0 ))
42eleq2i 2244 . . . . 5 (𝑦Q0𝑦 ∈ ((ω × N) / ~Q0 ))
53, 4anbi12i 460 . . . 4 ((𝑥Q0𝑦Q0) ↔ (𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )))
65anbi1i 458 . . 3 (((𝑥Q0𝑦Q0) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 )) ↔ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 )))
76oprabbii 5924 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q0𝑦Q0) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 ))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 ))}
81, 7eqtri 2198 1 ·Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 ))}
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wex 1492  wcel 2148  cop 3594  ωcom 4586   × cxp 4621  (class class class)co 5869  {coprab 5870   ·o comu 6409  [cec 6527   / cqs 6528  Ncnpi 7262   ~Q0 ceq0 7276  Q0cnq0 7277   ·Q0 cmq0 7280
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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  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-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-oprab 5873  df-nq0 7415  df-mq0 7418
This theorem is referenced by:  mulnnnq0  7440
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