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Theorem dfmq0qs 6411
Description: Multiplication on non-negative fractions. This definition is similar to df-mq0 6410 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 6410 . 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 6407 . . . . . 6 Q0 = ((𝜔 × N) / ~Q0 )
32eleq2i 2101 . . . . 5 (x Q0x ((𝜔 × N) / ~Q0 ))
42eleq2i 2101 . . . . 5 (y Q0y ((𝜔 × N) / ~Q0 ))
53, 4anbi12i 433 . . . 4 ((x Q0 y Q0) ↔ (x ((𝜔 × N) / ~Q0 ) y ((𝜔 × N) / ~Q0 )))
65anbi1i 431 . . 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 5502 . 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 2057 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 1242  wex 1378   wcel 1390  cop 3370  𝜔com 4256   × cxp 4286  (class class class)co 5455  {coprab 5456   ·𝑜 comu 5938  [cec 6040   / cqs 6041  Ncnpi 6256   ~Q0 ceq0 6270  Q0cnq0 6271   ·Q0 cmq0 6274
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-oprab 5459  df-nq0 6407  df-mq0 6410
This theorem is referenced by:  mulnnnq0  6432
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