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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 41087 | . 2 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐸𝐹;𝐺𝐻 |
9 | dfdec10 12667 | . 2 ⊢ ;;𝐸𝐹;𝐺𝐻 = ((;10 · ;𝐸𝐹) + ;𝐺𝐻) | |
10 | 1, 3 | nn0mulcli 12497 | . . . . 5 ⊢ (𝐴 · 𝐶) ∈ ℕ0 |
11 | 5, 10 | eqeltrri 2831 | . . . 4 ⊢ 𝐸 ∈ ℕ0 |
12 | 2, 3 | nn0mulcli 12497 | . . . . . 6 ⊢ (𝐵 · 𝐶) ∈ ℕ0 |
13 | 1, 4, 12 | numcl 12677 | . . . . 5 ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) ∈ ℕ0 |
14 | 6, 13 | eqeltrri 2831 | . . . 4 ⊢ 𝐹 ∈ ℕ0 |
15 | 11, 14 | deccl 12679 | . . 3 ⊢ ;𝐸𝐹 ∈ ℕ0 |
16 | 0nn0 12474 | . . 3 ⊢ 0 ∈ ℕ0 | |
17 | decpmul.g | . . 3 ⊢ 𝐺 ∈ ℕ0 | |
18 | decpmul.h | . . 3 ⊢ 𝐻 ∈ ℕ0 | |
19 | 15 | dec0u 12685 | . . 3 ⊢ (;10 · ;𝐸𝐹) = ;;𝐸𝐹0 |
20 | eqid 2733 | . . 3 ⊢ ;𝐺𝐻 = ;𝐺𝐻 | |
21 | 11, 14, 17 | decaddcom 41084 | . . . 4 ⊢ (;𝐸𝐹 + 𝐺) = (;𝐸𝐺 + 𝐹) |
22 | decpmul.4 | . . . 4 ⊢ (;𝐸𝐺 + 𝐹) = 𝐼 | |
23 | 21, 22 | eqtri 2761 | . . 3 ⊢ (;𝐸𝐹 + 𝐺) = 𝐼 |
24 | 18 | nn0cni 12471 | . . . 4 ⊢ 𝐻 ∈ ℂ |
25 | 24 | addlidi 11389 | . . 3 ⊢ (0 + 𝐻) = 𝐻 |
26 | 15, 16, 17, 18, 19, 20, 23, 25 | decadd 12718 | . 2 ⊢ ((;10 · ;𝐸𝐹) + ;𝐺𝐻) = ;𝐼𝐻 |
27 | 8, 9, 26 | 3eqtri 2765 | 1 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;𝐼𝐻 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7396 0cc0 11097 1c1 11098 + caddc 11100 · cmul 11102 ℕ0cn0 12459 ;cdc 12664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-ltxr 11240 df-sub 11433 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-dec 12665 |
This theorem is referenced by: ex-decpmul 41092 |
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