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Mirrors > Home > MPE Home > Th. List > numma | Structured version Visualization version GIF version |
Description: Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numma.1 | ⊢ 𝑇 ∈ ℕ0 |
numma.2 | ⊢ 𝐴 ∈ ℕ0 |
numma.3 | ⊢ 𝐵 ∈ ℕ0 |
numma.4 | ⊢ 𝐶 ∈ ℕ0 |
numma.5 | ⊢ 𝐷 ∈ ℕ0 |
numma.6 | ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) |
numma.7 | ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) |
numma.8 | ⊢ 𝑃 ∈ ℕ0 |
numma.9 | ⊢ ((𝐴 · 𝑃) + 𝐶) = 𝐸 |
numma.10 | ⊢ ((𝐵 · 𝑃) + 𝐷) = 𝐹 |
Ref | Expression |
---|---|
numma | ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numma.6 | . . . 4 ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) | |
2 | 1 | oveq1i 6660 | . . 3 ⊢ (𝑀 · 𝑃) = (((𝑇 · 𝐴) + 𝐵) · 𝑃) |
3 | numma.7 | . . 3 ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) | |
4 | 2, 3 | oveq12i 6662 | . 2 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) |
5 | numma.1 | . . . . . . 7 ⊢ 𝑇 ∈ ℕ0 | |
6 | 5 | nn0cni 11304 | . . . . . 6 ⊢ 𝑇 ∈ ℂ |
7 | numma.2 | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
8 | 7 | nn0cni 11304 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
9 | numma.8 | . . . . . . . 8 ⊢ 𝑃 ∈ ℕ0 | |
10 | 9 | nn0cni 11304 | . . . . . . 7 ⊢ 𝑃 ∈ ℂ |
11 | 8, 10 | mulcli 10045 | . . . . . 6 ⊢ (𝐴 · 𝑃) ∈ ℂ |
12 | numma.4 | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 | |
13 | 12 | nn0cni 11304 | . . . . . 6 ⊢ 𝐶 ∈ ℂ |
14 | 6, 11, 13 | adddii 10050 | . . . . 5 ⊢ (𝑇 · ((𝐴 · 𝑃) + 𝐶)) = ((𝑇 · (𝐴 · 𝑃)) + (𝑇 · 𝐶)) |
15 | 6, 8, 10 | mulassi 10049 | . . . . . 6 ⊢ ((𝑇 · 𝐴) · 𝑃) = (𝑇 · (𝐴 · 𝑃)) |
16 | 15 | oveq1i 6660 | . . . . 5 ⊢ (((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) = ((𝑇 · (𝐴 · 𝑃)) + (𝑇 · 𝐶)) |
17 | 14, 16 | eqtr4i 2647 | . . . 4 ⊢ (𝑇 · ((𝐴 · 𝑃) + 𝐶)) = (((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) |
18 | 17 | oveq1i 6660 | . . 3 ⊢ ((𝑇 · ((𝐴 · 𝑃) + 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) |
19 | 6, 8 | mulcli 10045 | . . . . . 6 ⊢ (𝑇 · 𝐴) ∈ ℂ |
20 | numma.3 | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
21 | 20 | nn0cni 11304 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
22 | 19, 21, 10 | adddiri 10051 | . . . . 5 ⊢ (((𝑇 · 𝐴) + 𝐵) · 𝑃) = (((𝑇 · 𝐴) · 𝑃) + (𝐵 · 𝑃)) |
23 | 22 | oveq1i 6660 | . . . 4 ⊢ ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝐵 · 𝑃)) + ((𝑇 · 𝐶) + 𝐷)) |
24 | 19, 10 | mulcli 10045 | . . . . 5 ⊢ ((𝑇 · 𝐴) · 𝑃) ∈ ℂ |
25 | 6, 13 | mulcli 10045 | . . . . 5 ⊢ (𝑇 · 𝐶) ∈ ℂ |
26 | 21, 10 | mulcli 10045 | . . . . 5 ⊢ (𝐵 · 𝑃) ∈ ℂ |
27 | numma.5 | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
28 | 27 | nn0cni 11304 | . . . . 5 ⊢ 𝐷 ∈ ℂ |
29 | 24, 25, 26, 28 | add4i 10260 | . . . 4 ⊢ ((((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝐵 · 𝑃)) + ((𝑇 · 𝐶) + 𝐷)) |
30 | 23, 29 | eqtr4i 2647 | . . 3 ⊢ ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) |
31 | 18, 30 | eqtr4i 2647 | . 2 ⊢ ((𝑇 · ((𝐴 · 𝑃) + 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) |
32 | numma.9 | . . . 4 ⊢ ((𝐴 · 𝑃) + 𝐶) = 𝐸 | |
33 | 32 | oveq2i 6661 | . . 3 ⊢ (𝑇 · ((𝐴 · 𝑃) + 𝐶)) = (𝑇 · 𝐸) |
34 | numma.10 | . . 3 ⊢ ((𝐵 · 𝑃) + 𝐷) = 𝐹 | |
35 | 33, 34 | oveq12i 6662 | . 2 ⊢ ((𝑇 · ((𝐴 · 𝑃) + 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((𝑇 · 𝐸) + 𝐹) |
36 | 4, 31, 35 | 3eqtr2i 2650 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 (class class class)co 6650 + caddc 9939 · cmul 9941 ℕ0cn0 11292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-nn 11021 df-n0 11293 |
This theorem is referenced by: nummac 11558 numadd 11560 decma 11564 decmaOLD 11565 |
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