Theorem dfmq0qs 7370

Description: Multiplication on nonnegative fractions. This definition is similar to df-mq0 7369 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 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((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

Step	Hyp	Ref	Expression
1		df-mq0 7369	. 2 ⊢ ·Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 ))}
2		df-nq0 7366	. . . . . 6 ⊢ Q0 = ((ω × N) / ~Q0 )
3	2	eleq2i 2233	. . . . 5 ⊢ (𝑥 ∈ Q0 ↔ 𝑥 ∈ ((ω × N) / ~Q0 ))
4	2	eleq2i 2233	. . . . 5 ⊢ (𝑦 ∈ Q0 ↔ 𝑦 ∈ ((ω × N) / ~Q0 ))
5	3, 4	anbi12i 456	. . . 4 ⊢ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ↔ (𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )))
6	5	anbi1i 454	. . 3 ⊢ (((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 )) ↔ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 )))
7	6	oprabbii 5897	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((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 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 ))}
8	1, 7	eqtri 2186	1 ⊢ ·Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 ))}

Syntax hints:   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ⟨cop 3579  ωcom 4567   × cxp 4602  (class class class)co 5842  {coprab 5843   ·_o comu 6382  [cec 6499   / cqs 6500  Ncnpi 7213   ~_Q0 ceq0 7227  Q₀cnq0 7228   ·_Q0 cmq0 7231

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-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-oprab 5846  df-nq0 7366  df-mq0 7369

This theorem is referenced by:  mulnnnq0  7391