Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
||
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 11079. (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 12101 | . 2 ⊢ ;𝐴𝐵 ∈ ℕ0 |
4 | decpmulnc.c | . 2 ⊢ 𝐶 ∈ ℕ0 | |
5 | decpmulnc.d | . 2 ⊢ 𝐷 ∈ ℕ0 | |
6 | eqid 2818 | . 2 ⊢ ;𝐶𝐷 = ;𝐶𝐷 | |
7 | decpmulnc.3 | . . 3 ⊢ (𝐵 · 𝐷) = 𝐺 | |
8 | 2, 5 | nn0mulcli 11923 | . . 3 ⊢ (𝐵 · 𝐷) ∈ ℕ0 |
9 | 7, 8 | eqeltrri 2907 | . 2 ⊢ 𝐺 ∈ ℕ0 |
10 | 1, 5 | nn0mulcli 11923 | . 2 ⊢ (𝐴 · 𝐷) ∈ ℕ0 |
11 | eqid 2818 | . . 3 ⊢ ;𝐴𝐵 = ;𝐴𝐵 | |
12 | decpmulnc.1 | . . 3 ⊢ (𝐴 · 𝐶) = 𝐸 | |
13 | 10 | nn0cni 11897 | . . . 4 ⊢ (𝐴 · 𝐷) ∈ ℂ |
14 | 2, 4 | nn0mulcli 11923 | . . . . 5 ⊢ (𝐵 · 𝐶) ∈ ℕ0 |
15 | 14 | nn0cni 11897 | . . . 4 ⊢ (𝐵 · 𝐶) ∈ ℂ |
16 | decpmulnc.2 | . . . 4 ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹 | |
17 | 13, 15, 16 | addcomli 10820 | . . 3 ⊢ ((𝐵 · 𝐶) + (𝐴 · 𝐷)) = 𝐹 |
18 | 1, 2, 10, 11, 4, 12, 17 | decrmanc 12143 | . 2 ⊢ ((;𝐴𝐵 · 𝐶) + (𝐴 · 𝐷)) = ;𝐸𝐹 |
19 | eqid 2818 | . . 3 ⊢ (𝐴 · 𝐷) = (𝐴 · 𝐷) | |
20 | 5, 1, 2, 11, 19, 7 | decmul1 12150 | . 2 ⊢ (;𝐴𝐵 · 𝐷) = ;(𝐴 · 𝐷)𝐺 |
21 | 3, 4, 5, 6, 9, 10, 18, 20 | decmul2c 12152 | 1 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐸𝐹𝐺 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 + caddc 10528 · cmul 10530 ℕ0cn0 11885 ;cdc 12086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-sub 10860 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-dec 12087 |
This theorem is referenced by: decpmul 39052 sqdeccom12 39053 |
Copyright terms: Public domain | W3C validator |