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Mirrors > Home > ILE Home > Th. List > nummac | GIF version |
Description: Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with 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) |
nummac.8 | ⊢ 𝑃 ∈ ℕ0 |
nummac.9 | ⊢ 𝐹 ∈ ℕ0 |
nummac.10 | ⊢ 𝐺 ∈ ℕ0 |
nummac.11 | ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 |
nummac.12 | ⊢ ((𝐵 · 𝑃) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) |
Ref | Expression |
---|---|
nummac | ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numma.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 | |
2 | 1 | nn0cni 8989 | . . . 4 ⊢ 𝑇 ∈ ℂ |
3 | numma.2 | . . . . . . . . 9 ⊢ 𝐴 ∈ ℕ0 | |
4 | 3 | nn0cni 8989 | . . . . . . . 8 ⊢ 𝐴 ∈ ℂ |
5 | nummac.8 | . . . . . . . . 9 ⊢ 𝑃 ∈ ℕ0 | |
6 | 5 | nn0cni 8989 | . . . . . . . 8 ⊢ 𝑃 ∈ ℂ |
7 | 4, 6 | mulcli 7771 | . . . . . . 7 ⊢ (𝐴 · 𝑃) ∈ ℂ |
8 | numma.4 | . . . . . . . 8 ⊢ 𝐶 ∈ ℕ0 | |
9 | 8 | nn0cni 8989 | . . . . . . 7 ⊢ 𝐶 ∈ ℂ |
10 | nummac.10 | . . . . . . . 8 ⊢ 𝐺 ∈ ℕ0 | |
11 | 10 | nn0cni 8989 | . . . . . . 7 ⊢ 𝐺 ∈ ℂ |
12 | 7, 9, 11 | addassi 7774 | . . . . . 6 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = ((𝐴 · 𝑃) + (𝐶 + 𝐺)) |
13 | nummac.11 | . . . . . 6 ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 | |
14 | 12, 13 | eqtri 2160 | . . . . 5 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = 𝐸 |
15 | 7, 9 | addcli 7770 | . . . . . 6 ⊢ ((𝐴 · 𝑃) + 𝐶) ∈ ℂ |
16 | 15, 11 | addcli 7770 | . . . . 5 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) ∈ ℂ |
17 | 14, 16 | eqeltrri 2213 | . . . 4 ⊢ 𝐸 ∈ ℂ |
18 | 2, 17, 11 | subdii 8169 | . . 3 ⊢ (𝑇 · (𝐸 − 𝐺)) = ((𝑇 · 𝐸) − (𝑇 · 𝐺)) |
19 | 18 | oveq1i 5784 | . 2 ⊢ ((𝑇 · (𝐸 − 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) = (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
20 | numma.3 | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
21 | numma.5 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
22 | numma.6 | . . 3 ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) | |
23 | numma.7 | . . 3 ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) | |
24 | 17, 11, 15 | subadd2i 8050 | . . . . 5 ⊢ ((𝐸 − 𝐺) = ((𝐴 · 𝑃) + 𝐶) ↔ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = 𝐸) |
25 | 14, 24 | mpbir 145 | . . . 4 ⊢ (𝐸 − 𝐺) = ((𝐴 · 𝑃) + 𝐶) |
26 | 25 | eqcomi 2143 | . . 3 ⊢ ((𝐴 · 𝑃) + 𝐶) = (𝐸 − 𝐺) |
27 | nummac.12 | . . 3 ⊢ ((𝐵 · 𝑃) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) | |
28 | 1, 3, 20, 8, 21, 22, 23, 5, 26, 27 | numma 9225 | . 2 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · (𝐸 − 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
29 | 2, 17 | mulcli 7771 | . . . . 5 ⊢ (𝑇 · 𝐸) ∈ ℂ |
30 | 2, 11 | mulcli 7771 | . . . . 5 ⊢ (𝑇 · 𝐺) ∈ ℂ |
31 | npcan 7971 | . . . . 5 ⊢ (((𝑇 · 𝐸) ∈ ℂ ∧ (𝑇 · 𝐺) ∈ ℂ) → (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) = (𝑇 · 𝐸)) | |
32 | 29, 30, 31 | mp2an 422 | . . . 4 ⊢ (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) = (𝑇 · 𝐸) |
33 | 32 | oveq1i 5784 | . . 3 ⊢ ((((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) + 𝐹) = ((𝑇 · 𝐸) + 𝐹) |
34 | 29, 30 | subcli 8038 | . . . 4 ⊢ ((𝑇 · 𝐸) − (𝑇 · 𝐺)) ∈ ℂ |
35 | nummac.9 | . . . . 5 ⊢ 𝐹 ∈ ℕ0 | |
36 | 35 | nn0cni 8989 | . . . 4 ⊢ 𝐹 ∈ ℂ |
37 | 34, 30, 36 | addassi 7774 | . . 3 ⊢ ((((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) + 𝐹) = (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
38 | 33, 37 | eqtr3i 2162 | . 2 ⊢ ((𝑇 · 𝐸) + 𝐹) = (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
39 | 19, 28, 38 | 3eqtr4i 2170 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7618 + caddc 7623 · cmul 7625 − cmin 7933 ℕ0cn0 8977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 df-inn 8721 df-n0 8978 |
This theorem is referenced by: numma2c 9227 numaddc 9229 nummul1c 9230 decmac 9233 |
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