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Mirrors > Home > ILE Home > Th. List > mulclnq | GIF version |
Description: Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) |
Ref | Expression |
---|---|
mulclnq | ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nqqs 7156 | . . 3 ⊢ Q = ((N × N) / ~Q ) | |
2 | oveq1 5781 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~Q = 𝐴 → ([〈𝑥, 𝑦〉] ~Q ·Q [〈𝑧, 𝑤〉] ~Q ) = (𝐴 ·Q [〈𝑧, 𝑤〉] ~Q )) | |
3 | 2 | eleq1d 2208 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~Q = 𝐴 → (([〈𝑥, 𝑦〉] ~Q ·Q [〈𝑧, 𝑤〉] ~Q ) ∈ ((N × N) / ~Q ) ↔ (𝐴 ·Q [〈𝑧, 𝑤〉] ~Q ) ∈ ((N × N) / ~Q ))) |
4 | oveq2 5782 | . . . 4 ⊢ ([〈𝑧, 𝑤〉] ~Q = 𝐵 → (𝐴 ·Q [〈𝑧, 𝑤〉] ~Q ) = (𝐴 ·Q 𝐵)) | |
5 | 4 | eleq1d 2208 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ~Q = 𝐵 → ((𝐴 ·Q [〈𝑧, 𝑤〉] ~Q ) ∈ ((N × N) / ~Q ) ↔ (𝐴 ·Q 𝐵) ∈ ((N × N) / ~Q ))) |
6 | mulpipqqs 7181 | . . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → ([〈𝑥, 𝑦〉] ~Q ·Q [〈𝑧, 𝑤〉] ~Q ) = [〈(𝑥 ·N 𝑧), (𝑦 ·N 𝑤)〉] ~Q ) | |
7 | mulclpi 7136 | . . . . . . 7 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) ∈ N) | |
8 | mulclpi 7136 | . . . . . . 7 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·N 𝑤) ∈ N) | |
9 | 7, 8 | anim12i 336 | . . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑦 ∈ N ∧ 𝑤 ∈ N)) → ((𝑥 ·N 𝑧) ∈ N ∧ (𝑦 ·N 𝑤) ∈ N)) |
10 | 9 | an4s 577 | . . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → ((𝑥 ·N 𝑧) ∈ N ∧ (𝑦 ·N 𝑤) ∈ N)) |
11 | opelxpi 4571 | . . . . 5 ⊢ (((𝑥 ·N 𝑧) ∈ N ∧ (𝑦 ·N 𝑤) ∈ N) → 〈(𝑥 ·N 𝑧), (𝑦 ·N 𝑤)〉 ∈ (N × N)) | |
12 | enqex 7168 | . . . . . 6 ⊢ ~Q ∈ V | |
13 | 12 | ecelqsi 6483 | . . . . 5 ⊢ (〈(𝑥 ·N 𝑧), (𝑦 ·N 𝑤)〉 ∈ (N × N) → [〈(𝑥 ·N 𝑧), (𝑦 ·N 𝑤)〉] ~Q ∈ ((N × N) / ~Q )) |
14 | 10, 11, 13 | 3syl 17 | . . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → [〈(𝑥 ·N 𝑧), (𝑦 ·N 𝑤)〉] ~Q ∈ ((N × N) / ~Q )) |
15 | 6, 14 | eqeltrd 2216 | . . 3 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → ([〈𝑥, 𝑦〉] ~Q ·Q [〈𝑧, 𝑤〉] ~Q ) ∈ ((N × N) / ~Q )) |
16 | 1, 3, 5, 15 | 2ecoptocl 6517 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ ((N × N) / ~Q )) |
17 | 16, 1 | eleqtrrdi 2233 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 〈cop 3530 × cxp 4537 (class class class)co 5774 [cec 6427 / cqs 6428 Ncnpi 7080 ·N cmi 7082 ~Q ceq 7087 Qcnq 7088 ·Q cmq 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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-mi 7114 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-mqqs 7158 |
This theorem is referenced by: halfnqq 7218 prarloclemarch 7226 prarloclemarch2 7227 ltrnqg 7228 prarloclemlt 7301 prarloclemlo 7302 prarloclemcalc 7310 addnqprllem 7335 addnqprulem 7336 addnqprl 7337 addnqpru 7338 mpvlu 7347 dmmp 7349 appdivnq 7371 prmuloclemcalc 7373 prmuloc 7374 mulnqprl 7376 mulnqpru 7377 mullocprlem 7378 mullocpr 7379 mulclpr 7380 mulnqprlemrl 7381 mulnqprlemru 7382 mulnqprlemfl 7383 mulnqprlemfu 7384 mulnqpr 7385 mulassprg 7389 distrlem1prl 7390 distrlem1pru 7391 distrlem4prl 7392 distrlem4pru 7393 distrlem5prl 7394 distrlem5pru 7395 1idprl 7398 1idpru 7399 recexprlem1ssl 7441 recexprlem1ssu 7442 recexprlemss1l 7443 recexprlemss1u 7444 |
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