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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 11248. (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 12273 | . 2 ⊢ ;𝐴𝐵 ∈ ℕ0 |
4 | decpmulnc.c | . 2 ⊢ 𝐶 ∈ ℕ0 | |
5 | decpmulnc.d | . 2 ⊢ 𝐷 ∈ ℕ0 | |
6 | eqid 2736 | . 2 ⊢ ;𝐶𝐷 = ;𝐶𝐷 | |
7 | decpmulnc.3 | . . 3 ⊢ (𝐵 · 𝐷) = 𝐺 | |
8 | 2, 5 | nn0mulcli 12093 | . . 3 ⊢ (𝐵 · 𝐷) ∈ ℕ0 |
9 | 7, 8 | eqeltrri 2828 | . 2 ⊢ 𝐺 ∈ ℕ0 |
10 | 1, 5 | nn0mulcli 12093 | . 2 ⊢ (𝐴 · 𝐷) ∈ ℕ0 |
11 | eqid 2736 | . . 3 ⊢ ;𝐴𝐵 = ;𝐴𝐵 | |
12 | decpmulnc.1 | . . 3 ⊢ (𝐴 · 𝐶) = 𝐸 | |
13 | 10 | nn0cni 12067 | . . . 4 ⊢ (𝐴 · 𝐷) ∈ ℂ |
14 | 2, 4 | nn0mulcli 12093 | . . . . 5 ⊢ (𝐵 · 𝐶) ∈ ℕ0 |
15 | 14 | nn0cni 12067 | . . . 4 ⊢ (𝐵 · 𝐶) ∈ ℂ |
16 | decpmulnc.2 | . . . 4 ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹 | |
17 | 13, 15, 16 | addcomli 10989 | . . 3 ⊢ ((𝐵 · 𝐶) + (𝐴 · 𝐷)) = 𝐹 |
18 | 1, 2, 10, 11, 4, 12, 17 | decrmanc 12315 | . 2 ⊢ ((;𝐴𝐵 · 𝐶) + (𝐴 · 𝐷)) = ;𝐸𝐹 |
19 | eqid 2736 | . . 3 ⊢ (𝐴 · 𝐷) = (𝐴 · 𝐷) | |
20 | 5, 1, 2, 11, 19, 7 | decmul1 12322 | . 2 ⊢ (;𝐴𝐵 · 𝐷) = ;(𝐴 · 𝐷)𝐺 |
21 | 3, 4, 5, 6, 9, 10, 18, 20 | decmul2c 12324 | 1 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐸𝐹𝐺 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 (class class class)co 7191 + caddc 10697 · cmul 10699 ℕ0cn0 12055 ;cdc 12258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-sub 11029 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-dec 12259 |
This theorem is referenced by: decpmul 39964 sqdeccom12 39965 |
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