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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 6600 | . . 3 ⊢ Q = ((N × N) / ~Q ) | |
2 | oveq1 5550 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~Q = 𝐴 → ([〈𝑥, 𝑦〉] ~Q ·Q [〈𝑧, 𝑤〉] ~Q ) = (𝐴 ·Q [〈𝑧, 𝑤〉] ~Q )) | |
3 | 2 | eleq1d 2148 | . . 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 5551 | . . . 4 ⊢ ([〈𝑧, 𝑤〉] ~Q = 𝐵 → (𝐴 ·Q [〈𝑧, 𝑤〉] ~Q ) = (𝐴 ·Q 𝐵)) | |
5 | 4 | eleq1d 2148 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ~Q = 𝐵 → ((𝐴 ·Q [〈𝑧, 𝑤〉] ~Q ) ∈ ((N × N) / ~Q ) ↔ (𝐴 ·Q 𝐵) ∈ ((N × N) / ~Q ))) |
6 | mulpipqqs 6625 | . . . 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 6580 | . . . . . . 7 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) ∈ N) | |
8 | mulclpi 6580 | . . . . . . 7 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·N 𝑤) ∈ N) | |
9 | 7, 8 | anim12i 331 | . . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑦 ∈ N ∧ 𝑤 ∈ N)) → ((𝑥 ·N 𝑧) ∈ N ∧ (𝑦 ·N 𝑤) ∈ N)) |
10 | 9 | an4s 553 | . . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → ((𝑥 ·N 𝑧) ∈ N ∧ (𝑦 ·N 𝑤) ∈ N)) |
11 | opelxpi 4402 | . . . . 5 ⊢ (((𝑥 ·N 𝑧) ∈ N ∧ (𝑦 ·N 𝑤) ∈ N) → 〈(𝑥 ·N 𝑧), (𝑦 ·N 𝑤)〉 ∈ (N × N)) | |
12 | enqex 6612 | . . . . . 6 ⊢ ~Q ∈ V | |
13 | 12 | ecelqsi 6226 | . . . . 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 2156 | . . 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 6260 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ ((N × N) / ~Q )) |
17 | 16, 1 | syl6eleqr 2173 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 〈cop 3409 × cxp 4369 (class class class)co 5543 [cec 6170 / cqs 6171 Ncnpi 6524 ·N cmi 6526 ~Q ceq 6531 Qcnq 6532 ·Q cmq 6535 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-coll 3901 ax-sep 3904 ax-nul 3912 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-iinf 4337 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-csb 2910 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3259 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-iun 3688 df-br 3794 df-opab 3848 df-mpt 3849 df-tr 3884 df-id 4056 df-iord 4129 df-on 4131 df-suc 4134 df-iom 4340 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-f1 4937 df-fo 4938 df-f1o 4939 df-fv 4940 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-1st 5798 df-2nd 5799 df-recs 5954 df-irdg 6019 df-oadd 6069 df-omul 6070 df-er 6172 df-ec 6174 df-qs 6178 df-ni 6556 df-mi 6558 df-mpq 6597 df-enq 6599 df-nqqs 6600 df-mqqs 6602 |
This theorem is referenced by: halfnqq 6662 prarloclemarch 6670 prarloclemarch2 6671 ltrnqg 6672 prarloclemlt 6745 prarloclemlo 6746 prarloclemcalc 6754 addnqprllem 6779 addnqprulem 6780 addnqprl 6781 addnqpru 6782 mpvlu 6791 dmmp 6793 appdivnq 6815 prmuloclemcalc 6817 prmuloc 6818 mulnqprl 6820 mulnqpru 6821 mullocprlem 6822 mullocpr 6823 mulclpr 6824 mulnqprlemrl 6825 mulnqprlemru 6826 mulnqprlemfl 6827 mulnqprlemfu 6828 mulnqpr 6829 mulassprg 6833 distrlem1prl 6834 distrlem1pru 6835 distrlem4prl 6836 distrlem4pru 6837 distrlem5prl 6838 distrlem5pru 6839 1idprl 6842 1idpru 6843 recexprlem1ssl 6885 recexprlem1ssu 6886 recexprlemss1l 6887 recexprlemss1u 6888 |
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