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Mirrors > Home > ILE Home > Th. List > dfmq0qs | GIF version |
Description: Multiplication on nonnegative fractions. This definition is similar to df-mq0 7084 but expands Q0 (Contributed by Jim Kingdon, 22-Nov-2019.) |
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 ))} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mq0 7084 | . 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 7081 | . . . . . 6 ⊢ Q0 = ((ω × N) / ~Q0 ) | |
3 | 2 | eleq2i 2161 | . . . . 5 ⊢ (𝑥 ∈ Q0 ↔ 𝑥 ∈ ((ω × N) / ~Q0 )) |
4 | 2 | eleq2i 2161 | . . . . 5 ⊢ (𝑦 ∈ Q0 ↔ 𝑦 ∈ ((ω × N) / ~Q0 )) |
5 | 3, 4 | anbi12i 449 | . . . 4 ⊢ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ↔ (𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 ))) |
6 | 5 | anbi1i 447 | . . 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 5742 | . 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 2115 | 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 ))} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1296 ∃wex 1433 ∈ wcel 1445 〈cop 3469 ωcom 4433 × cxp 4465 (class class class)co 5690 {coprab 5691 ·o comu 6217 [cec 6330 / cqs 6331 Ncnpi 6928 ~Q0 ceq0 6942 Q0cnq0 6943 ·Q0 cmq0 6946 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-11 1449 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-oprab 5694 df-nq0 7081 df-mq0 7084 |
This theorem is referenced by: mulnnnq0 7106 |
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