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Mirrors > Home > ILE Home > Th. List > dfmq0qs | GIF version |
Description: Multiplication on non-negative fractions. This definition is similar to df-mq0 6526 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 6526 | . 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 6523 | . . . . . 6 ⊢ Q0 = ((ω × N) / ~Q0 ) | |
3 | 2 | eleq2i 2104 | . . . . 5 ⊢ (𝑥 ∈ Q0 ↔ 𝑥 ∈ ((ω × N) / ~Q0 )) |
4 | 2 | eleq2i 2104 | . . . . 5 ⊢ (𝑦 ∈ Q0 ↔ 𝑦 ∈ ((ω × N) / ~Q0 )) |
5 | 3, 4 | anbi12i 433 | . . . 4 ⊢ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ↔ (𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 ))) |
6 | 5 | anbi1i 431 | . . 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 5560 | . 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 2060 | 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 97 = wceq 1243 ∃wex 1381 ∈ wcel 1393 〈cop 3378 ωcom 4313 × cxp 4343 (class class class)co 5512 {coprab 5513 ·𝑜 comu 5999 [cec 6104 / cqs 6105 Ncnpi 6370 ~Q0 ceq0 6384 Q0cnq0 6385 ·Q0 cmq0 6388 |
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 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-oprab 5516 df-nq0 6523 df-mq0 6526 |
This theorem is referenced by: mulnnnq0 6548 |
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