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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 7740 | . . 3 ⊢ 0R ∈ R | |
2 | mulcnsr 7825 | . . . 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 588 | . . 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 439 | . 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 7759 | . . . . . . . 8 ⊢ (0R ∈ R → (0R ·R 0R) = 0R) | |
6 | 1, 5 | ax-mp 5 | . . . . . . 7 ⊢ (0R ·R 0R) = 0R |
7 | 6 | oveq2i 5880 | . . . . . 6 ⊢ (-1R ·R (0R ·R 0R)) = (-1R ·R 0R) |
8 | m1r 7742 | . . . . . . 7 ⊢ -1R ∈ R | |
9 | 00sr 7759 | . . . . . . 7 ⊢ (-1R ∈ R → (-1R ·R 0R) = 0R) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (-1R ·R 0R) = 0R |
11 | 7, 10 | eqtri 2198 | . . . . 5 ⊢ (-1R ·R (0R ·R 0R)) = 0R |
12 | 11 | oveq2i 5880 | . . . 4 ⊢ ((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))) = ((𝐴 ·R 𝐵) +R 0R) |
13 | mulclsr 7744 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) | |
14 | 0idsr 7757 | . . . . 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 | eqtrid 2222 | . . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))) = (𝐴 ·R 𝐵)) |
17 | mulcomsrg 7747 | . . . . . . 7 ⊢ ((0R ∈ R ∧ 𝐵 ∈ R) → (0R ·R 𝐵) = (𝐵 ·R 0R)) | |
18 | 1, 17 | mpan 424 | . . . . . 6 ⊢ (𝐵 ∈ R → (0R ·R 𝐵) = (𝐵 ·R 0R)) |
19 | 00sr 7759 | . . . . . 6 ⊢ (𝐵 ∈ R → (𝐵 ·R 0R) = 0R) | |
20 | 18, 19 | eqtrd 2210 | . . . . 5 ⊢ (𝐵 ∈ R → (0R ·R 𝐵) = 0R) |
21 | 00sr 7759 | . . . . 5 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | |
22 | 20, 21 | oveqan12rd 5889 | . . . 4 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R ·R 𝐵) +R (𝐴 ·R 0R)) = (0R +R 0R)) |
23 | 0idsr 7757 | . . . . 5 ⊢ (0R ∈ R → (0R +R 0R) = 0R) | |
24 | 1, 23 | ax-mp 5 | . . . 4 ⊢ (0R +R 0R) = 0R |
25 | 22, 24 | eqtrdi 2226 | . . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R ·R 𝐵) +R (𝐴 ·R 0R)) = 0R) |
26 | 16, 25 | opeq12d 3784 | . 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 2210 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 · 〈𝐵, 0R〉) = 〈(𝐴 ·R 𝐵), 0R〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 〈cop 3594 (class class class)co 5869 Rcnr 7287 0Rc0r 7288 -1Rcm1r 7290 +R cplr 7291 ·R cmr 7292 · cmul 7807 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-eprel 4286 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-1o 6411 df-2o 6412 df-oadd 6415 df-omul 6416 df-er 6529 df-ec 6531 df-qs 6535 df-ni 7294 df-pli 7295 df-mi 7296 df-lti 7297 df-plpq 7334 df-mpq 7335 df-enq 7337 df-nqqs 7338 df-plqqs 7339 df-mqqs 7340 df-1nqqs 7341 df-rq 7342 df-ltnqqs 7343 df-enq0 7414 df-nq0 7415 df-0nq0 7416 df-plq0 7417 df-mq0 7418 df-inp 7456 df-i1p 7457 df-iplp 7458 df-imp 7459 df-enr 7716 df-nr 7717 df-plr 7718 df-mr 7719 df-0r 7721 df-m1r 7723 df-c 7808 df-mul 7814 |
This theorem is referenced by: recidpirq 7848 axmulrcl 7857 ax1rid 7867 axprecex 7870 axpre-mulgt0 7877 axpre-mulext 7878 |
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