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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 9013 | . . . 4 ⊢ 𝑇 ∈ ℂ |
3 | numma.2 | . . . . . . . . 9 ⊢ 𝐴 ∈ ℕ0 | |
4 | 3 | nn0cni 9013 | . . . . . . . 8 ⊢ 𝐴 ∈ ℂ |
5 | nummac.8 | . . . . . . . . 9 ⊢ 𝑃 ∈ ℕ0 | |
6 | 5 | nn0cni 9013 | . . . . . . . 8 ⊢ 𝑃 ∈ ℂ |
7 | 4, 6 | mulcli 7795 | . . . . . . 7 ⊢ (𝐴 · 𝑃) ∈ ℂ |
8 | numma.4 | . . . . . . . 8 ⊢ 𝐶 ∈ ℕ0 | |
9 | 8 | nn0cni 9013 | . . . . . . 7 ⊢ 𝐶 ∈ ℂ |
10 | nummac.10 | . . . . . . . 8 ⊢ 𝐺 ∈ ℕ0 | |
11 | 10 | nn0cni 9013 | . . . . . . 7 ⊢ 𝐺 ∈ ℂ |
12 | 7, 9, 11 | addassi 7798 | . . . . . 6 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = ((𝐴 · 𝑃) + (𝐶 + 𝐺)) |
13 | nummac.11 | . . . . . 6 ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 | |
14 | 12, 13 | eqtri 2161 | . . . . 5 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = 𝐸 |
15 | 7, 9 | addcli 7794 | . . . . . 6 ⊢ ((𝐴 · 𝑃) + 𝐶) ∈ ℂ |
16 | 15, 11 | addcli 7794 | . . . . 5 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) ∈ ℂ |
17 | 14, 16 | eqeltrri 2214 | . . . 4 ⊢ 𝐸 ∈ ℂ |
18 | 2, 17, 11 | subdii 8193 | . . 3 ⊢ (𝑇 · (𝐸 − 𝐺)) = ((𝑇 · 𝐸) − (𝑇 · 𝐺)) |
19 | 18 | oveq1i 5792 | . 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 8074 | . . . . 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 2144 | . . 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 9249 | . 2 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · (𝐸 − 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
29 | 2, 17 | mulcli 7795 | . . . . 5 ⊢ (𝑇 · 𝐸) ∈ ℂ |
30 | 2, 11 | mulcli 7795 | . . . . 5 ⊢ (𝑇 · 𝐺) ∈ ℂ |
31 | npcan 7995 | . . . . 5 ⊢ (((𝑇 · 𝐸) ∈ ℂ ∧ (𝑇 · 𝐺) ∈ ℂ) → (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) = (𝑇 · 𝐸)) | |
32 | 29, 30, 31 | mp2an 423 | . . . 4 ⊢ (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) = (𝑇 · 𝐸) |
33 | 32 | oveq1i 5792 | . . 3 ⊢ ((((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) + 𝐹) = ((𝑇 · 𝐸) + 𝐹) |
34 | 29, 30 | subcli 8062 | . . . 4 ⊢ ((𝑇 · 𝐸) − (𝑇 · 𝐺)) ∈ ℂ |
35 | nummac.9 | . . . . 5 ⊢ 𝐹 ∈ ℕ0 | |
36 | 35 | nn0cni 9013 | . . . 4 ⊢ 𝐹 ∈ ℂ |
37 | 34, 30, 36 | addassi 7798 | . . 3 ⊢ ((((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) + 𝐹) = (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
38 | 33, 37 | eqtr3i 2163 | . 2 ⊢ ((𝑇 · 𝐸) + 𝐹) = (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
39 | 19, 28, 38 | 3eqtr4i 2171 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 (class class class)co 5782 ℂcc 7642 + caddc 7647 · cmul 7649 − cmin 7957 ℕ0cn0 9001 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-inn 8745 df-n0 9002 |
This theorem is referenced by: numma2c 9251 numaddc 9253 nummul1c 9254 decmac 9257 |
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