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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 11091. (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 12114 | . 2 ⊢ ;𝐴𝐵 ∈ ℕ0 |
4 | decpmulnc.c | . 2 ⊢ 𝐶 ∈ ℕ0 | |
5 | decpmulnc.d | . 2 ⊢ 𝐷 ∈ ℕ0 | |
6 | eqid 2821 | . 2 ⊢ ;𝐶𝐷 = ;𝐶𝐷 | |
7 | decpmulnc.3 | . . 3 ⊢ (𝐵 · 𝐷) = 𝐺 | |
8 | 2, 5 | nn0mulcli 11936 | . . 3 ⊢ (𝐵 · 𝐷) ∈ ℕ0 |
9 | 7, 8 | eqeltrri 2910 | . 2 ⊢ 𝐺 ∈ ℕ0 |
10 | 1, 5 | nn0mulcli 11936 | . 2 ⊢ (𝐴 · 𝐷) ∈ ℕ0 |
11 | eqid 2821 | . . 3 ⊢ ;𝐴𝐵 = ;𝐴𝐵 | |
12 | decpmulnc.1 | . . 3 ⊢ (𝐴 · 𝐶) = 𝐸 | |
13 | 10 | nn0cni 11910 | . . . 4 ⊢ (𝐴 · 𝐷) ∈ ℂ |
14 | 2, 4 | nn0mulcli 11936 | . . . . 5 ⊢ (𝐵 · 𝐶) ∈ ℕ0 |
15 | 14 | nn0cni 11910 | . . . 4 ⊢ (𝐵 · 𝐶) ∈ ℂ |
16 | decpmulnc.2 | . . . 4 ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹 | |
17 | 13, 15, 16 | addcomli 10832 | . . 3 ⊢ ((𝐵 · 𝐶) + (𝐴 · 𝐷)) = 𝐹 |
18 | 1, 2, 10, 11, 4, 12, 17 | decrmanc 12156 | . 2 ⊢ ((;𝐴𝐵 · 𝐶) + (𝐴 · 𝐷)) = ;𝐸𝐹 |
19 | eqid 2821 | . . 3 ⊢ (𝐴 · 𝐷) = (𝐴 · 𝐷) | |
20 | 5, 1, 2, 11, 19, 7 | decmul1 12163 | . 2 ⊢ (;𝐴𝐵 · 𝐷) = ;(𝐴 · 𝐷)𝐺 |
21 | 3, 4, 5, 6, 9, 10, 18, 20 | decmul2c 12165 | 1 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐸𝐹𝐺 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 + caddc 10540 · cmul 10542 ℕ0cn0 11898 ;cdc 12099 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-dec 12100 |
This theorem is referenced by: decpmul 39223 sqdeccom12 39224 |
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