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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 7744 | . . 3 ⊢ R = ((P × P) / ~R ) | |
2 | oveq1 5898 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → ([〈𝑥, 𝑦〉] ~R ·R [〈𝑧, 𝑤〉] ~R ) = (𝐴 ·R [〈𝑧, 𝑤〉] ~R )) | |
3 | 2 | eleq1d 2258 | . . 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 5899 | . . . 4 ⊢ ([〈𝑧, 𝑤〉] ~R = 𝐵 → (𝐴 ·R [〈𝑧, 𝑤〉] ~R ) = (𝐴 ·R 𝐵)) | |
5 | 4 | eleq1d 2258 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ~R = 𝐵 → ((𝐴 ·R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 ·R 𝐵) ∈ ((P × P) / ~R ))) |
6 | mulsrpr 7763 | . . . 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 7589 | . . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 ·P 𝑧) ∈ P) | |
8 | mulclpr 7589 | . . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 ·P 𝑤) ∈ P) | |
9 | addclpr 7554 | . . . . . . . 8 ⊢ (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) | |
10 | 7, 8, 9 | syl2an 289 | . . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) |
11 | 10 | an4s 588 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) |
12 | mulclpr 7589 | . . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 ·P 𝑤) ∈ P) | |
13 | mulclpr 7589 | . . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 ·P 𝑧) ∈ P) | |
14 | addclpr 7554 | . . . . . . . 8 ⊢ (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) | |
15 | 12, 13, 14 | syl2an 289 | . . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) |
16 | 15 | an42s 589 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) |
17 | 11, 16 | jca 306 | . . . . 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 4673 | . . . . 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 7754 | . . . . . 6 ⊢ ~R ∈ V | |
20 | 19 | ecelqsi 6607 | . . . . 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 2266 | . . 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 6641 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ ((P × P) / ~R )) |
24 | 23, 1 | eleqtrrdi 2283 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 〈cop 3610 × cxp 4639 (class class class)co 5891 [cec 6551 / cqs 6552 Pcnp 7308 +P cpp 7310 ·P cmp 7311 ~R cer 7313 Rcnr 7314 ·R cmr 7319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4304 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-1o 6435 df-2o 6436 df-oadd 6439 df-omul 6440 df-er 6553 df-ec 6555 df-qs 6559 df-ni 7321 df-pli 7322 df-mi 7323 df-lti 7324 df-plpq 7361 df-mpq 7362 df-enq 7364 df-nqqs 7365 df-plqqs 7366 df-mqqs 7367 df-1nqqs 7368 df-rq 7369 df-ltnqqs 7370 df-enq0 7441 df-nq0 7442 df-0nq0 7443 df-plq0 7444 df-mq0 7445 df-inp 7483 df-iplp 7485 df-imp 7486 df-enr 7743 df-nr 7744 df-mr 7746 |
This theorem is referenced by: pn0sr 7788 negexsr 7789 caucvgsrlemoffval 7813 caucvgsrlemofff 7814 map2psrprg 7822 mulcnsr 7852 mulresr 7855 mulcnsrec 7860 axmulcl 7883 axmulrcl 7884 axmulcom 7888 axmulass 7890 axdistr 7891 axrnegex 7896 |
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