![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
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 42301 | . 2 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐸𝐹;𝐺𝐻 |
9 | dfdec10 12734 | . 2 ⊢ ;;𝐸𝐹;𝐺𝐻 = ((;10 · ;𝐸𝐹) + ;𝐺𝐻) | |
10 | 1, 3 | nn0mulcli 12562 | . . . . 5 ⊢ (𝐴 · 𝐶) ∈ ℕ0 |
11 | 5, 10 | eqeltrri 2836 | . . . 4 ⊢ 𝐸 ∈ ℕ0 |
12 | 2, 3 | nn0mulcli 12562 | . . . . . 6 ⊢ (𝐵 · 𝐶) ∈ ℕ0 |
13 | 1, 4, 12 | numcl 12744 | . . . . 5 ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) ∈ ℕ0 |
14 | 6, 13 | eqeltrri 2836 | . . . 4 ⊢ 𝐹 ∈ ℕ0 |
15 | 11, 14 | deccl 12746 | . . 3 ⊢ ;𝐸𝐹 ∈ ℕ0 |
16 | 0nn0 12539 | . . 3 ⊢ 0 ∈ ℕ0 | |
17 | decpmul.g | . . 3 ⊢ 𝐺 ∈ ℕ0 | |
18 | decpmul.h | . . 3 ⊢ 𝐻 ∈ ℕ0 | |
19 | 15 | dec0u 12752 | . . 3 ⊢ (;10 · ;𝐸𝐹) = ;;𝐸𝐹0 |
20 | eqid 2735 | . . 3 ⊢ ;𝐺𝐻 = ;𝐺𝐻 | |
21 | 11, 14, 17 | decaddcom 42298 | . . . 4 ⊢ (;𝐸𝐹 + 𝐺) = (;𝐸𝐺 + 𝐹) |
22 | decpmul.4 | . . . 4 ⊢ (;𝐸𝐺 + 𝐹) = 𝐼 | |
23 | 21, 22 | eqtri 2763 | . . 3 ⊢ (;𝐸𝐹 + 𝐺) = 𝐼 |
24 | 18 | nn0cni 12536 | . . . 4 ⊢ 𝐻 ∈ ℂ |
25 | 24 | addlidi 11447 | . . 3 ⊢ (0 + 𝐻) = 𝐻 |
26 | 15, 16, 17, 18, 19, 20, 23, 25 | decadd 12785 | . 2 ⊢ ((;10 · ;𝐸𝐹) + ;𝐺𝐻) = ;𝐼𝐻 |
27 | 8, 9, 26 | 3eqtri 2767 | 1 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;𝐼𝐻 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ℕ0cn0 12524 ;cdc 12731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-dec 12732 |
This theorem is referenced by: ex-decpmul 42319 |
Copyright terms: Public domain | W3C validator |