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Theorem dfmq0qs 6284
 Description: Multiplication on non-negative fractions. This definition is similar to df-mq0 6283 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⟩] ~Q0 y = [⟨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 6283 . 2 ·Q0 = {⟨⟨x, y⟩, z⟩ ∣ ((x Q0 y Q0) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨(w ·𝑜 u), (v ·𝑜 f)⟩] ~Q0 ))}
2 df-nq0 6280 . . . . . 6 Q0 = ((𝜔 × N) / ~Q0 )
32eleq2i 2086 . . . . 5 (x Q0x ((𝜔 × N) / ~Q0 ))
42eleq2i 2086 . . . . 5 (y Q0y ((𝜔 × N) / ~Q0 ))
53, 4anbi12i 436 . . . 4 ((x Q0 y Q0) ↔ (x ((𝜔 × N) / ~Q0 ) y ((𝜔 × N) / ~Q0 )))
65anbi1i 434 . . 3 (((x Q0 y Q0) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨(w ·𝑜 u), (v ·𝑜 f)⟩] ~Q0 )) ↔ ((x ((𝜔 × N) / ~Q0 ) y ((𝜔 × N) / ~Q0 )) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨(w ·𝑜 u), (v ·𝑜 f)⟩] ~Q0 )))
76oprabbii 5483 . 2 {⟨⟨x, y⟩, z⟩ ∣ ((x Q0 y Q0) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨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⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨(w ·𝑜 u), (v ·𝑜 f)⟩] ~Q0 ))}
81, 7eqtri 2042 1 ·Q0 = {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝜔 × N) / ~Q0 ) y ((𝜔 × N) / ~Q0 )) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨(w ·𝑜 u), (v ·𝑜 f)⟩] ~Q0 ))}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ⟨cop 3353  𝜔com 4240   × cxp 4270  (class class class)co 5436  {coprab 5437   ·𝑜 comu 5914  [cec 6015   / cqs 6016  Ncnpi 6130   ~Q0 ceq0 6144  Q0cnq0 6145   ·Q0 cmq0 6148 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-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-oprab 5440  df-nq0 6280  df-mq0 6283 This theorem is referenced by:  mulnnnq0  6305
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