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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 10216 | . . 3 ⊢ 0R ∈ R | |
2 | mulcnsr 10272 | . . . 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 652 | . . 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 698 | . 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 10235 | . . . . . . . 8 ⊢ (0R ∈ R → (0R ·R 0R) = 0R) | |
6 | 1, 5 | ax-mp 5 | . . . . . . 7 ⊢ (0R ·R 0R) = 0R |
7 | 6 | oveq2i 6915 | . . . . . 6 ⊢ (-1R ·R (0R ·R 0R)) = (-1R ·R 0R) |
8 | m1r 10218 | . . . . . . 7 ⊢ -1R ∈ R | |
9 | 00sr 10235 | . . . . . . 7 ⊢ (-1R ∈ R → (-1R ·R 0R) = 0R) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (-1R ·R 0R) = 0R |
11 | 7, 10 | eqtri 2848 | . . . . 5 ⊢ (-1R ·R (0R ·R 0R)) = 0R |
12 | 11 | oveq2i 6915 | . . . 4 ⊢ ((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))) = ((𝐴 ·R 𝐵) +R 0R) |
13 | mulclsr 10220 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) | |
14 | 0idsr 10233 | . . . . 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 | syl5eq 2872 | . . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))) = (𝐴 ·R 𝐵)) |
17 | mulcomsr 10225 | . . . . . 6 ⊢ (0R ·R 𝐵) = (𝐵 ·R 0R) | |
18 | 00sr 10235 | . . . . . 6 ⊢ (𝐵 ∈ R → (𝐵 ·R 0R) = 0R) | |
19 | 17, 18 | syl5eq 2872 | . . . . 5 ⊢ (𝐵 ∈ R → (0R ·R 𝐵) = 0R) |
20 | 00sr 10235 | . . . . 5 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | |
21 | 19, 20 | oveqan12rd 6924 | . . . 4 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R ·R 𝐵) +R (𝐴 ·R 0R)) = (0R +R 0R)) |
22 | 0idsr 10233 | . . . . 5 ⊢ (0R ∈ R → (0R +R 0R) = 0R) | |
23 | 1, 22 | ax-mp 5 | . . . 4 ⊢ (0R +R 0R) = 0R |
24 | 21, 23 | syl6eq 2876 | . . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R ·R 𝐵) +R (𝐴 ·R 0R)) = 0R) |
25 | 16, 24 | opeq12d 4630 | . 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 2860 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 · 〈𝐵, 0R〉) = 〈(𝐴 ·R 𝐵), 0R〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 〈cop 4402 (class class class)co 6904 Rcnr 10001 0Rc0r 10002 -1Rcm1r 10004 +R cplr 10005 ·R cmr 10006 · cmul 10256 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-omul 7830 df-er 8008 df-ec 8010 df-qs 8014 df-ni 10008 df-pli 10009 df-mi 10010 df-lti 10011 df-plpq 10044 df-mpq 10045 df-ltpq 10046 df-enq 10047 df-nq 10048 df-erq 10049 df-plq 10050 df-mq 10051 df-1nq 10052 df-rq 10053 df-ltnq 10054 df-np 10117 df-1p 10118 df-plp 10119 df-mp 10120 df-ltp 10121 df-enr 10191 df-nr 10192 df-plr 10193 df-mr 10194 df-0r 10196 df-m1r 10198 df-c 10257 df-mul 10263 |
This theorem is referenced by: axmulrcl 10290 ax1rid 10297 axrrecex 10299 axpre-mulgt0 10304 |
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