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Mirrors > Home > ILE Home > Th. List > mulclsr | GIF version |
Description: Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) |
Ref | Expression |
---|---|
mulclsr | ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 6955 | . . 3 ⊢ R = ((P × P) / ~R ) | |
2 | oveq1 5544 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → ([〈𝑥, 𝑦〉] ~R ·R [〈𝑧, 𝑤〉] ~R ) = (𝐴 ·R [〈𝑧, 𝑤〉] ~R )) | |
3 | 2 | eleq1d 2148 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → (([〈𝑥, 𝑦〉] ~R ·R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 ·R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R ))) |
4 | oveq2 5545 | . . . 4 ⊢ ([〈𝑧, 𝑤〉] ~R = 𝐵 → (𝐴 ·R [〈𝑧, 𝑤〉] ~R ) = (𝐴 ·R 𝐵)) | |
5 | 4 | eleq1d 2148 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ~R = 𝐵 → ((𝐴 ·R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 ·R 𝐵) ∈ ((P × P) / ~R ))) |
6 | mulsrpr 6974 | . . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([〈𝑥, 𝑦〉] ~R ·R [〈𝑧, 𝑤〉] ~R ) = [〈((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))〉] ~R ) | |
7 | mulclpr 6813 | . . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 ·P 𝑧) ∈ P) | |
8 | mulclpr 6813 | . . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 ·P 𝑤) ∈ P) | |
9 | addclpr 6778 | . . . . . . . 8 ⊢ (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) | |
10 | 7, 8, 9 | syl2an 283 | . . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) |
11 | 10 | an4s 553 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) |
12 | mulclpr 6813 | . . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 ·P 𝑤) ∈ P) | |
13 | mulclpr 6813 | . . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 ·P 𝑧) ∈ P) | |
14 | addclpr 6778 | . . . . . . . 8 ⊢ (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) | |
15 | 12, 13, 14 | syl2an 283 | . . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) |
16 | 15 | an42s 554 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) |
17 | 11, 16 | jca 300 | . . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)) |
18 | opelxpi 4396 | . . . . 5 ⊢ ((((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) → 〈((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))〉 ∈ (P × P)) | |
19 | enrex 6965 | . . . . . 6 ⊢ ~R ∈ V | |
20 | 19 | ecelqsi 6219 | . . . . 5 ⊢ (〈((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))〉 ∈ (P × P) → [〈((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))〉] ~R ∈ ((P × P) / ~R )) |
21 | 17, 18, 20 | 3syl 17 | . . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → [〈((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))〉] ~R ∈ ((P × P) / ~R )) |
22 | 6, 21 | eqeltrd 2156 | . . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([〈𝑥, 𝑦〉] ~R ·R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R )) |
23 | 1, 3, 5, 22 | 2ecoptocl 6253 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ ((P × P) / ~R )) |
24 | 23, 1 | syl6eleqr 2173 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 〈cop 3403 × cxp 4363 (class class class)co 5537 [cec 6163 / cqs 6164 Pcnp 6532 +P cpp 6534 ·P cmp 6535 ~R cer 6537 Rcnr 6538 ·R cmr 6543 |
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 3895 ax-sep 3898 ax-nul 3906 ax-pow 3950 ax-pr 3966 ax-un 4190 ax-setind 4282 ax-iinf 4331 |
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 3253 df-pw 3386 df-sn 3406 df-pr 3407 df-op 3409 df-uni 3604 df-int 3639 df-iun 3682 df-br 3788 df-opab 3842 df-mpt 3843 df-tr 3878 df-eprel 4046 df-id 4050 df-po 4053 df-iso 4054 df-iord 4123 df-on 4125 df-suc 4128 df-iom 4334 df-xp 4371 df-rel 4372 df-cnv 4373 df-co 4374 df-dm 4375 df-rn 4376 df-res 4377 df-ima 4378 df-iota 4891 df-fun 4928 df-fn 4929 df-f 4930 df-f1 4931 df-fo 4932 df-f1o 4933 df-fv 4934 df-ov 5540 df-oprab 5541 df-mpt2 5542 df-1st 5792 df-2nd 5793 df-recs 5948 df-irdg 6013 df-1o 6059 df-2o 6060 df-oadd 6063 df-omul 6064 df-er 6165 df-ec 6167 df-qs 6171 df-ni 6545 df-pli 6546 df-mi 6547 df-lti 6548 df-plpq 6585 df-mpq 6586 df-enq 6588 df-nqqs 6589 df-plqqs 6590 df-mqqs 6591 df-1nqqs 6592 df-rq 6593 df-ltnqqs 6594 df-enq0 6665 df-nq0 6666 df-0nq0 6667 df-plq0 6668 df-mq0 6669 df-inp 6707 df-iplp 6709 df-imp 6710 df-enr 6954 df-nr 6955 df-mr 6957 |
This theorem is referenced by: pn0sr 6999 negexsr 7000 caucvgsrlemoffval 7023 caucvgsrlemofff 7024 mulcnsr 7054 mulresr 7057 mulcnsrec 7062 axmulcl 7085 axmulrcl 7086 axmulcom 7088 axmulass 7090 axdistr 7091 axrnegex 7096 |
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