Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7357 | . . 3 ⊢ Q0 = ((ω × N) / ~Q0 ) | |
2 | oveq1 5843 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~Q0 = 𝐴 → ([〈𝑥, 𝑦〉] ~Q0 ·Q0 [〈𝑧, 𝑤〉] ~Q0 ) = (𝐴 ·Q0 [〈𝑧, 𝑤〉] ~Q0 )) | |
3 | 2 | eleq1d 2233 | . . 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 5844 | . . . 4 ⊢ ([〈𝑧, 𝑤〉] ~Q0 = 𝐵 → (𝐴 ·Q0 [〈𝑧, 𝑤〉] ~Q0 ) = (𝐴 ·Q0 𝐵)) | |
5 | 4 | eleq1d 2233 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ~Q0 = 𝐵 → ((𝐴 ·Q0 [〈𝑧, 𝑤〉] ~Q0 ) ∈ ((ω × N) / ~Q0 ) ↔ (𝐴 ·Q0 𝐵) ∈ ((ω × N) / ~Q0 ))) |
6 | mulnnnq0 7382 | . . . 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 6440 | . . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝑧 ∈ ω) → (𝑥 ·o 𝑧) ∈ ω) | |
8 | mulpiord 7249 | . . . . . . . 8 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·N 𝑤) = (𝑦 ·o 𝑤)) | |
9 | mulclpi 7260 | . . . . . . . 8 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·N 𝑤) ∈ N) | |
10 | 8, 9 | eqeltrrd 2242 | . . . . . . 7 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·o 𝑤) ∈ N) |
11 | 7, 10 | anim12i 336 | . . . . . 6 ⊢ (((𝑥 ∈ ω ∧ 𝑧 ∈ ω) ∧ (𝑦 ∈ N ∧ 𝑤 ∈ N)) → ((𝑥 ·o 𝑧) ∈ ω ∧ (𝑦 ·o 𝑤) ∈ N)) |
12 | 11 | an4s 578 | . . . . 5 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) → ((𝑥 ·o 𝑧) ∈ ω ∧ (𝑦 ·o 𝑤) ∈ N)) |
13 | opelxpi 4630 | . . . . 5 ⊢ (((𝑥 ·o 𝑧) ∈ ω ∧ (𝑦 ·o 𝑤) ∈ N) → 〈(𝑥 ·o 𝑧), (𝑦 ·o 𝑤)〉 ∈ (ω × N)) | |
14 | enq0ex 7371 | . . . . . 6 ⊢ ~Q0 ∈ V | |
15 | 14 | ecelqsi 6546 | . . . . 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 2241 | . . 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 6580 | . 2 ⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) ∈ ((ω × N) / ~Q0 )) |
19 | 18, 1 | eleqtrrdi 2258 | 1 ⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) ∈ Q0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 〈cop 3573 ωcom 4561 × cxp 4596 (class class class)co 5836 ·o comu 6373 [cec 6490 / cqs 6491 Ncnpi 7204 ·N cmi 7206 ~Q0 ceq0 7218 Q0cnq0 7219 ·Q0 cmq0 7222 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-mi 7238 df-enq0 7356 df-nq0 7357 df-mq0 7360 |
This theorem is referenced by: prarloclemcalc 7434 |
Copyright terms: Public domain | W3C validator |