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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 7229 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 7229 | . 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 7226 | . . . . . 6 ⊢ Q0 = ((ω × N) / ~Q0 ) | |
3 | 2 | eleq2i 2204 | . . . . 5 ⊢ (𝑥 ∈ Q0 ↔ 𝑥 ∈ ((ω × N) / ~Q0 )) |
4 | 2 | eleq2i 2204 | . . . . 5 ⊢ (𝑦 ∈ Q0 ↔ 𝑦 ∈ ((ω × N) / ~Q0 )) |
5 | 3, 4 | anbi12i 455 | . . . 4 ⊢ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ↔ (𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 ))) |
6 | 5 | anbi1i 453 | . . 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 5819 | . 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 2158 | 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 1331 ∃wex 1468 ∈ wcel 1480 〈cop 3525 ωcom 4499 × cxp 4532 (class class class)co 5767 {coprab 5768 ·o comu 6304 [cec 6420 / cqs 6421 Ncnpi 7073 ~Q0 ceq0 7087 Q0cnq0 7088 ·Q0 cmq0 7091 |
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 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-oprab 5771 df-nq0 7226 df-mq0 7229 |
This theorem is referenced by: mulnnnq0 7251 |
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