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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 10502 | . . 3 ⊢ 0R ∈ R | |
2 | mulcnsr 10558 | . . . 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 658 | . . 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 703 | . 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 10521 | . . . . . . . 8 ⊢ (0R ∈ R → (0R ·R 0R) = 0R) | |
6 | 1, 5 | ax-mp 5 | . . . . . . 7 ⊢ (0R ·R 0R) = 0R |
7 | 6 | oveq2i 7167 | . . . . . 6 ⊢ (-1R ·R (0R ·R 0R)) = (-1R ·R 0R) |
8 | m1r 10504 | . . . . . . 7 ⊢ -1R ∈ R | |
9 | 00sr 10521 | . . . . . . 7 ⊢ (-1R ∈ R → (-1R ·R 0R) = 0R) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (-1R ·R 0R) = 0R |
11 | 7, 10 | eqtri 2844 | . . . . 5 ⊢ (-1R ·R (0R ·R 0R)) = 0R |
12 | 11 | oveq2i 7167 | . . . 4 ⊢ ((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))) = ((𝐴 ·R 𝐵) +R 0R) |
13 | mulclsr 10506 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) | |
14 | 0idsr 10519 | . . . . 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 2868 | . . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((𝐴 ·R 𝐵) +R (-1R ·R (0R ·R 0R))) = (𝐴 ·R 𝐵)) |
17 | mulcomsr 10511 | . . . . . 6 ⊢ (0R ·R 𝐵) = (𝐵 ·R 0R) | |
18 | 00sr 10521 | . . . . . 6 ⊢ (𝐵 ∈ R → (𝐵 ·R 0R) = 0R) | |
19 | 17, 18 | syl5eq 2868 | . . . . 5 ⊢ (𝐵 ∈ R → (0R ·R 𝐵) = 0R) |
20 | 00sr 10521 | . . . . 5 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | |
21 | 19, 20 | oveqan12rd 7176 | . . . 4 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R ·R 𝐵) +R (𝐴 ·R 0R)) = (0R +R 0R)) |
22 | 0idsr 10519 | . . . . 5 ⊢ (0R ∈ R → (0R +R 0R) = 0R) | |
23 | 1, 22 | ax-mp 5 | . . . 4 ⊢ (0R +R 0R) = 0R |
24 | 21, 23 | syl6eq 2872 | . . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R ·R 𝐵) +R (𝐴 ·R 0R)) = 0R) |
25 | 16, 24 | opeq12d 4811 | . 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 2856 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 · 〈𝐵, 0R〉) = 〈(𝐴 ·R 𝐵), 0R〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 (class class class)co 7156 Rcnr 10287 0Rc0r 10288 -1Rcm1r 10290 +R cplr 10291 ·R cmr 10292 · cmul 10542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-ec 8291 df-qs 8295 df-ni 10294 df-pli 10295 df-mi 10296 df-lti 10297 df-plpq 10330 df-mpq 10331 df-ltpq 10332 df-enq 10333 df-nq 10334 df-erq 10335 df-plq 10336 df-mq 10337 df-1nq 10338 df-rq 10339 df-ltnq 10340 df-np 10403 df-1p 10404 df-plp 10405 df-mp 10406 df-ltp 10407 df-enr 10477 df-nr 10478 df-plr 10479 df-mr 10480 df-0r 10482 df-m1r 10484 df-c 10543 df-mul 10549 |
This theorem is referenced by: axmulrcl 10576 ax1rid 10583 axrrecex 10585 axpre-mulgt0 10590 |
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