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Mirrors > Home > ILE Home > Th. List > mulresr | GIF version |
Description: Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) |
Ref | Expression |
---|---|
mulresr | ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 · 〈𝐵, 0R〉) = 〈(𝐴 ·R 𝐵), 0R〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0r 7526 | . . 3 ⊢ 0R ∈ R | |
2 | mulcnsr 7611 | . . . 4 ⊢ (((𝐴 ∈ R ∧ 0R ∈ R) ∧ (𝐵 ∈ R ∧ 0R ∈ R)) → (〈𝐴, 0R〉 · 〈𝐵, 0R〉) = 〈((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))), ((0R ·R 𝐵) +R (𝐴 ·R 0R))〉) | |
3 | 2 | an4s 562 | . . 3 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (0R ∈ R ∧ 0R ∈ R)) → (〈𝐴, 0R〉 · 〈𝐵, 0R〉) = 〈((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))), ((0R ·R 𝐵) +R (𝐴 ·R 0R))〉) |
4 | 1, 1, 3 | mpanr12 435 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 · 〈𝐵, 0R〉) = 〈((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))), ((0R ·R 𝐵) +R (𝐴 ·R 0R))〉) |
5 | 00sr 7545 | . . . . . . . 8 ⊢ (0R ∈ R → (0R ·R 0R) = 0R) | |
6 | 1, 5 | ax-mp 5 | . . . . . . 7 ⊢ (0R ·R 0R) = 0R |
7 | 6 | oveq2i 5753 | . . . . . 6 ⊢ (-1R ·R (0R ·R 0R)) = (-1R ·R 0R) |
8 | m1r 7528 | . . . . . . 7 ⊢ -1R ∈ R | |
9 | 00sr 7545 | . . . . . . 7 ⊢ (-1R ∈ R → (-1R ·R 0R) = 0R) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (-1R ·R 0R) = 0R |
11 | 7, 10 | eqtri 2138 | . . . . 5 ⊢ (-1R ·R (0R ·R 0R)) = 0R |
12 | 11 | oveq2i 5753 | . . . 4 ⊢ ((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))) = ((𝐴 ·R 𝐵) +R 0R) |
13 | mulclsr 7530 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) | |
14 | 0idsr 7543 | . . . . 5 ⊢ ((𝐴 ·R 𝐵) ∈ R → ((𝐴 ·R 𝐵) +R 0R) = (𝐴 ·R 𝐵)) | |
15 | 13, 14 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((𝐴 ·R 𝐵) +R 0R) = (𝐴 ·R 𝐵)) |
16 | 12, 15 | syl5eq 2162 | . . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))) = (𝐴 ·R 𝐵)) |
17 | mulcomsrg 7533 | . . . . . . 7 ⊢ ((0R ∈ R ∧ 𝐵 ∈ R) → (0R ·R 𝐵) = (𝐵 ·R 0R)) | |
18 | 1, 17 | mpan 420 | . . . . . 6 ⊢ (𝐵 ∈ R → (0R ·R 𝐵) = (𝐵 ·R 0R)) |
19 | 00sr 7545 | . . . . . 6 ⊢ (𝐵 ∈ R → (𝐵 ·R 0R) = 0R) | |
20 | 18, 19 | eqtrd 2150 | . . . . 5 ⊢ (𝐵 ∈ R → (0R ·R 𝐵) = 0R) |
21 | 00sr 7545 | . . . . 5 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | |
22 | 20, 21 | oveqan12rd 5762 | . . . 4 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R ·R 𝐵) +R (𝐴 ·R 0R)) = (0R +R 0R)) |
23 | 0idsr 7543 | . . . . 5 ⊢ (0R ∈ R → (0R +R 0R) = 0R) | |
24 | 1, 23 | ax-mp 5 | . . . 4 ⊢ (0R +R 0R) = 0R |
25 | 22, 24 | syl6eq 2166 | . . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R ·R 𝐵) +R (𝐴 ·R 0R)) = 0R) |
26 | 16, 25 | opeq12d 3683 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → 〈((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))), ((0R ·R 𝐵) +R (𝐴 ·R 0R))〉 = 〈(𝐴 ·R 𝐵), 0R〉) |
27 | 4, 26 | eqtrd 2150 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 · 〈𝐵, 0R〉) = 〈(𝐴 ·R 𝐵), 0R〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 〈cop 3500 (class class class)co 5742 Rcnr 7073 0Rc0r 7074 -1Rcm1r 7076 +R cplr 7077 ·R cmr 7078 · cmul 7593 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-2o 6282 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 df-enq0 7200 df-nq0 7201 df-0nq0 7202 df-plq0 7203 df-mq0 7204 df-inp 7242 df-i1p 7243 df-iplp 7244 df-imp 7245 df-enr 7502 df-nr 7503 df-plr 7504 df-mr 7505 df-0r 7507 df-m1r 7509 df-c 7594 df-mul 7600 |
This theorem is referenced by: recidpirq 7634 axmulrcl 7643 ax1rid 7653 axprecex 7656 axpre-mulgt0 7663 axpre-mulext 7664 |
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