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Mirrors > Home > MPE Home > Th. List > mulresr | Structured version Visualization version GIF version |
Description: Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulresr | ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 · 〈𝐵, 0R〉) = 〈(𝐴 ·R 𝐵), 0R〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0r 11117 | . . 3 ⊢ 0R ∈ R | |
2 | mulcnsr 11173 | . . . 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 660 | . . 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 705 | . 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 11136 | . . . . . . . 8 ⊢ (0R ∈ R → (0R ·R 0R) = 0R) | |
6 | 1, 5 | ax-mp 5 | . . . . . . 7 ⊢ (0R ·R 0R) = 0R |
7 | 6 | oveq2i 7441 | . . . . . 6 ⊢ (-1R ·R (0R ·R 0R)) = (-1R ·R 0R) |
8 | m1r 11119 | . . . . . . 7 ⊢ -1R ∈ R | |
9 | 00sr 11136 | . . . . . . 7 ⊢ (-1R ∈ R → (-1R ·R 0R) = 0R) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (-1R ·R 0R) = 0R |
11 | 7, 10 | eqtri 2762 | . . . . 5 ⊢ (-1R ·R (0R ·R 0R)) = 0R |
12 | 11 | oveq2i 7441 | . . . 4 ⊢ ((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))) = ((𝐴 ·R 𝐵) +R 0R) |
13 | mulclsr 11121 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) | |
14 | 0idsr 11134 | . . . . 5 ⊢ ((𝐴 ·R 𝐵) ∈ R → ((𝐴 ·R 𝐵) +R 0R) = (𝐴 ·R 𝐵)) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((𝐴 ·R 𝐵) +R 0R) = (𝐴 ·R 𝐵)) |
16 | 12, 15 | eqtrid 2786 | . . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))) = (𝐴 ·R 𝐵)) |
17 | mulcomsr 11126 | . . . . . 6 ⊢ (0R ·R 𝐵) = (𝐵 ·R 0R) | |
18 | 00sr 11136 | . . . . . 6 ⊢ (𝐵 ∈ R → (𝐵 ·R 0R) = 0R) | |
19 | 17, 18 | eqtrid 2786 | . . . . 5 ⊢ (𝐵 ∈ R → (0R ·R 𝐵) = 0R) |
20 | 00sr 11136 | . . . . 5 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | |
21 | 19, 20 | oveqan12rd 7450 | . . . 4 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R ·R 𝐵) +R (𝐴 ·R 0R)) = (0R +R 0R)) |
22 | 0idsr 11134 | . . . . 5 ⊢ (0R ∈ R → (0R +R 0R) = 0R) | |
23 | 1, 22 | ax-mp 5 | . . . 4 ⊢ (0R +R 0R) = 0R |
24 | 21, 23 | eqtrdi 2790 | . . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R ·R 𝐵) +R (𝐴 ·R 0R)) = 0R) |
25 | 16, 24 | opeq12d 4885 | . 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〉) |
26 | 4, 25 | eqtrd 2774 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 · 〈𝐵, 0R〉) = 〈(𝐴 ·R 𝐵), 0R〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 〈cop 4636 (class class class)co 7430 Rcnr 10902 0Rc0r 10903 -1Rcm1r 10905 +R cplr 10906 ·R cmr 10907 · cmul 11157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-omul 8509 df-er 8743 df-ec 8745 df-qs 8749 df-ni 10909 df-pli 10910 df-mi 10911 df-lti 10912 df-plpq 10945 df-mpq 10946 df-ltpq 10947 df-enq 10948 df-nq 10949 df-erq 10950 df-plq 10951 df-mq 10952 df-1nq 10953 df-rq 10954 df-ltnq 10955 df-np 11018 df-1p 11019 df-plp 11020 df-mp 11021 df-ltp 11022 df-enr 11092 df-nr 11093 df-plr 11094 df-mr 11095 df-0r 11097 df-m1r 11099 df-c 11158 df-mul 11164 |
This theorem is referenced by: axmulrcl 11191 ax1rid 11198 axrrecex 11200 axpre-mulgt0 11205 |
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