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Mirrors > Home > ILE Home > Th. List > mulclnq0 | GIF version |
Description: Closure of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.) |
Ref | Expression |
---|---|
mulclnq0 | ⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) ∈ Q0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nq0 7134 | . . 3 ⊢ Q0 = ((ω × N) / ~Q0 ) | |
2 | oveq1 5713 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~Q0 = 𝐴 → ([〈𝑥, 𝑦〉] ~Q0 ·Q0 [〈𝑧, 𝑤〉] ~Q0 ) = (𝐴 ·Q0 [〈𝑧, 𝑤〉] ~Q0 )) | |
3 | 2 | eleq1d 2168 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~Q0 = 𝐴 → (([〈𝑥, 𝑦〉] ~Q0 ·Q0 [〈𝑧, 𝑤〉] ~Q0 ) ∈ ((ω × N) / ~Q0 ) ↔ (𝐴 ·Q0 [〈𝑧, 𝑤〉] ~Q0 ) ∈ ((ω × N) / ~Q0 ))) |
4 | oveq2 5714 | . . . 4 ⊢ ([〈𝑧, 𝑤〉] ~Q0 = 𝐵 → (𝐴 ·Q0 [〈𝑧, 𝑤〉] ~Q0 ) = (𝐴 ·Q0 𝐵)) | |
5 | 4 | eleq1d 2168 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ~Q0 = 𝐵 → ((𝐴 ·Q0 [〈𝑧, 𝑤〉] ~Q0 ) ∈ ((ω × N) / ~Q0 ) ↔ (𝐴 ·Q0 𝐵) ∈ ((ω × N) / ~Q0 ))) |
6 | mulnnnq0 7159 | . . . 4 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) → ([〈𝑥, 𝑦〉] ~Q0 ·Q0 [〈𝑧, 𝑤〉] ~Q0 ) = [〈(𝑥 ·o 𝑧), (𝑦 ·o 𝑤)〉] ~Q0 ) | |
7 | nnmcl 6307 | . . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝑧 ∈ ω) → (𝑥 ·o 𝑧) ∈ ω) | |
8 | mulpiord 7026 | . . . . . . . 8 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·N 𝑤) = (𝑦 ·o 𝑤)) | |
9 | mulclpi 7037 | . . . . . . . 8 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·N 𝑤) ∈ N) | |
10 | 8, 9 | eqeltrrd 2177 | . . . . . . 7 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·o 𝑤) ∈ N) |
11 | 7, 10 | anim12i 334 | . . . . . 6 ⊢ (((𝑥 ∈ ω ∧ 𝑧 ∈ ω) ∧ (𝑦 ∈ N ∧ 𝑤 ∈ N)) → ((𝑥 ·o 𝑧) ∈ ω ∧ (𝑦 ·o 𝑤) ∈ N)) |
12 | 11 | an4s 558 | . . . . 5 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) → ((𝑥 ·o 𝑧) ∈ ω ∧ (𝑦 ·o 𝑤) ∈ N)) |
13 | opelxpi 4509 | . . . . 5 ⊢ (((𝑥 ·o 𝑧) ∈ ω ∧ (𝑦 ·o 𝑤) ∈ N) → 〈(𝑥 ·o 𝑧), (𝑦 ·o 𝑤)〉 ∈ (ω × N)) | |
14 | enq0ex 7148 | . . . . . 6 ⊢ ~Q0 ∈ V | |
15 | 14 | ecelqsi 6413 | . . . . 5 ⊢ (〈(𝑥 ·o 𝑧), (𝑦 ·o 𝑤)〉 ∈ (ω × N) → [〈(𝑥 ·o 𝑧), (𝑦 ·o 𝑤)〉] ~Q0 ∈ ((ω × N) / ~Q0 )) |
16 | 12, 13, 15 | 3syl 17 | . . . 4 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) → [〈(𝑥 ·o 𝑧), (𝑦 ·o 𝑤)〉] ~Q0 ∈ ((ω × N) / ~Q0 )) |
17 | 6, 16 | eqeltrd 2176 | . . 3 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) → ([〈𝑥, 𝑦〉] ~Q0 ·Q0 [〈𝑧, 𝑤〉] ~Q0 ) ∈ ((ω × N) / ~Q0 )) |
18 | 1, 3, 5, 17 | 2ecoptocl 6447 | . 2 ⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) ∈ ((ω × N) / ~Q0 )) |
19 | 18, 1 | syl6eleqr 2193 | 1 ⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) ∈ Q0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1299 ∈ wcel 1448 〈cop 3477 ωcom 4442 × cxp 4475 (class class class)co 5706 ·o comu 6241 [cec 6357 / cqs 6358 Ncnpi 6981 ·N cmi 6983 ~Q0 ceq0 6995 Q0cnq0 6996 ·Q0 cmq0 6999 |
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-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-iord 4226 df-on 4228 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-irdg 6197 df-oadd 6247 df-omul 6248 df-er 6359 df-ec 6361 df-qs 6365 df-ni 7013 df-mi 7015 df-enq0 7133 df-nq0 7134 df-mq0 7137 |
This theorem is referenced by: prarloclemcalc 7211 |
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