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Mirrors > Home > MPE Home > Th. List > Mathboxes > decpmul | Structured version Visualization version GIF version |
Description: Partial products algorithm for two digit multiplication. (Contributed by Steven Nguyen, 10-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 |
decpmul.3 | ⊢ (𝐵 · 𝐷) = ;𝐺𝐻 |
decpmul.4 | ⊢ (;𝐸𝐺 + 𝐹) = 𝐼 |
decpmul.g | ⊢ 𝐺 ∈ ℕ0 |
decpmul.h | ⊢ 𝐻 ∈ ℕ0 |
Ref | Expression |
---|---|
decpmul | ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;𝐼𝐻 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decpmulnc.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
2 | decpmulnc.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
3 | decpmulnc.c | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
4 | decpmulnc.d | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
5 | decpmulnc.1 | . . 3 ⊢ (𝐴 · 𝐶) = 𝐸 | |
6 | decpmulnc.2 | . . 3 ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹 | |
7 | decpmul.3 | . . 3 ⊢ (𝐵 · 𝐷) = ;𝐺𝐻 | |
8 | 1, 2, 3, 4, 5, 6, 7 | decpmulnc 39180 | . 2 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐸𝐹;𝐺𝐻 |
9 | dfdec10 12104 | . 2 ⊢ ;;𝐸𝐹;𝐺𝐻 = ((;10 · ;𝐸𝐹) + ;𝐺𝐻) | |
10 | 1, 3 | nn0mulcli 11938 | . . . . 5 ⊢ (𝐴 · 𝐶) ∈ ℕ0 |
11 | 5, 10 | eqeltrri 2912 | . . . 4 ⊢ 𝐸 ∈ ℕ0 |
12 | 2, 3 | nn0mulcli 11938 | . . . . . 6 ⊢ (𝐵 · 𝐶) ∈ ℕ0 |
13 | 1, 4, 12 | numcl 12114 | . . . . 5 ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) ∈ ℕ0 |
14 | 6, 13 | eqeltrri 2912 | . . . 4 ⊢ 𝐹 ∈ ℕ0 |
15 | 11, 14 | deccl 12116 | . . 3 ⊢ ;𝐸𝐹 ∈ ℕ0 |
16 | 0nn0 11915 | . . 3 ⊢ 0 ∈ ℕ0 | |
17 | decpmul.g | . . 3 ⊢ 𝐺 ∈ ℕ0 | |
18 | decpmul.h | . . 3 ⊢ 𝐻 ∈ ℕ0 | |
19 | 15 | dec0u 12122 | . . 3 ⊢ (;10 · ;𝐸𝐹) = ;;𝐸𝐹0 |
20 | eqid 2823 | . . 3 ⊢ ;𝐺𝐻 = ;𝐺𝐻 | |
21 | 11, 14, 17 | decaddcom 39177 | . . . 4 ⊢ (;𝐸𝐹 + 𝐺) = (;𝐸𝐺 + 𝐹) |
22 | decpmul.4 | . . . 4 ⊢ (;𝐸𝐺 + 𝐹) = 𝐼 | |
23 | 21, 22 | eqtri 2846 | . . 3 ⊢ (;𝐸𝐹 + 𝐺) = 𝐼 |
24 | 18 | nn0cni 11912 | . . . 4 ⊢ 𝐻 ∈ ℂ |
25 | 24 | addid2i 10830 | . . 3 ⊢ (0 + 𝐻) = 𝐻 |
26 | 15, 16, 17, 18, 19, 20, 23, 25 | decadd 12155 | . 2 ⊢ ((;10 · ;𝐸𝐹) + ;𝐺𝐻) = ;𝐼𝐻 |
27 | 8, 9, 26 | 3eqtri 2850 | 1 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;𝐼𝐻 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 ℕ0cn0 11900 ;cdc 12101 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-dec 12102 |
This theorem is referenced by: ex-decpmul 39185 |
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