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Mirrors > Home > MPE Home > Th. List > Mathboxes > decpmulnc | Structured version Visualization version GIF version |
Description: Partial products algorithm for two digit multiplication, no carry. Compare muladdi 10826. (Contributed by Steven Nguyen, 9-Dec-2022.) |
Ref | Expression |
---|---|
decpmulnc.a | ⊢ 𝐴 ∈ ℕ0 |
decpmulnc.b | ⊢ 𝐵 ∈ ℕ0 |
decpmulnc.c | ⊢ 𝐶 ∈ ℕ0 |
decpmulnc.d | ⊢ 𝐷 ∈ ℕ0 |
decpmulnc.1 | ⊢ (𝐴 · 𝐶) = 𝐸 |
decpmulnc.2 | ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹 |
decpmulnc.3 | ⊢ (𝐵 · 𝐷) = 𝐺 |
Ref | Expression |
---|---|
decpmulnc | ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐸𝐹𝐺 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decpmulnc.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
2 | decpmulnc.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
3 | 1, 2 | deccl 11860 | . 2 ⊢ ;𝐴𝐵 ∈ ℕ0 |
4 | decpmulnc.c | . 2 ⊢ 𝐶 ∈ ℕ0 | |
5 | decpmulnc.d | . 2 ⊢ 𝐷 ∈ ℕ0 | |
6 | eqid 2778 | . 2 ⊢ ;𝐶𝐷 = ;𝐶𝐷 | |
7 | decpmulnc.3 | . . 3 ⊢ (𝐵 · 𝐷) = 𝐺 | |
8 | 2, 5 | nn0mulcli 11682 | . . 3 ⊢ (𝐵 · 𝐷) ∈ ℕ0 |
9 | 7, 8 | eqeltrri 2856 | . 2 ⊢ 𝐺 ∈ ℕ0 |
10 | 1, 5 | nn0mulcli 11682 | . 2 ⊢ (𝐴 · 𝐷) ∈ ℕ0 |
11 | eqid 2778 | . . 3 ⊢ ;𝐴𝐵 = ;𝐴𝐵 | |
12 | decpmulnc.1 | . . 3 ⊢ (𝐴 · 𝐶) = 𝐸 | |
13 | 10 | nn0cni 11655 | . . . 4 ⊢ (𝐴 · 𝐷) ∈ ℂ |
14 | 2, 4 | nn0mulcli 11682 | . . . . 5 ⊢ (𝐵 · 𝐶) ∈ ℕ0 |
15 | 14 | nn0cni 11655 | . . . 4 ⊢ (𝐵 · 𝐶) ∈ ℂ |
16 | decpmulnc.2 | . . . 4 ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹 | |
17 | 13, 15, 16 | addcomli 10568 | . . 3 ⊢ ((𝐵 · 𝐶) + (𝐴 · 𝐷)) = 𝐹 |
18 | 1, 2, 10, 11, 4, 12, 17 | decrmanc 11903 | . 2 ⊢ ((;𝐴𝐵 · 𝐶) + (𝐴 · 𝐷)) = ;𝐸𝐹 |
19 | eqid 2778 | . . 3 ⊢ (𝐴 · 𝐷) = (𝐴 · 𝐷) | |
20 | 5, 1, 2, 11, 19, 7 | decmul1 11910 | . 2 ⊢ (;𝐴𝐵 · 𝐷) = ;(𝐴 · 𝐷)𝐺 |
21 | 3, 4, 5, 6, 9, 10, 18, 20 | decmul2c 11913 | 1 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐸𝐹𝐺 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 (class class class)co 6922 + caddc 10275 · cmul 10277 ℕ0cn0 11642 ;cdc 11845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-dec 11846 |
This theorem is referenced by: decpmul 38154 sqdeccom12 38155 |
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