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Mirrors > Home > ILE Home > Th. List > numma2c | 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) |
numma2c.8 | ⊢ 𝑃 ∈ ℕ0 |
numma2c.9 | ⊢ 𝐹 ∈ ℕ0 |
numma2c.10 | ⊢ 𝐺 ∈ ℕ0 |
numma2c.11 | ⊢ ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸 |
numma2c.12 | ⊢ ((𝑃 · 𝐵) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) |
Ref | Expression |
---|---|
numma2c | ⊢ ((𝑃 · 𝑀) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numma2c.8 | . . . . 5 ⊢ 𝑃 ∈ ℕ0 | |
2 | 1 | nn0cni 8746 | . . . 4 ⊢ 𝑃 ∈ ℂ |
3 | numma.6 | . . . . . 6 ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) | |
4 | numma.1 | . . . . . . 7 ⊢ 𝑇 ∈ ℕ0 | |
5 | numma.2 | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
6 | numma.3 | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
7 | 4, 5, 6 | numcl 8950 | . . . . . 6 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
8 | 3, 7 | eqeltri 2161 | . . . . 5 ⊢ 𝑀 ∈ ℕ0 |
9 | 8 | nn0cni 8746 | . . . 4 ⊢ 𝑀 ∈ ℂ |
10 | 2, 9 | mulcomi 7555 | . . 3 ⊢ (𝑃 · 𝑀) = (𝑀 · 𝑃) |
11 | 10 | oveq1i 5676 | . 2 ⊢ ((𝑃 · 𝑀) + 𝑁) = ((𝑀 · 𝑃) + 𝑁) |
12 | numma.4 | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
13 | numma.5 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
14 | numma.7 | . . 3 ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) | |
15 | numma2c.9 | . . 3 ⊢ 𝐹 ∈ ℕ0 | |
16 | numma2c.10 | . . 3 ⊢ 𝐺 ∈ ℕ0 | |
17 | 5 | nn0cni 8746 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
18 | 17, 2 | mulcomi 7555 | . . . . 5 ⊢ (𝐴 · 𝑃) = (𝑃 · 𝐴) |
19 | 18 | oveq1i 5676 | . . . 4 ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = ((𝑃 · 𝐴) + (𝐶 + 𝐺)) |
20 | numma2c.11 | . . . 4 ⊢ ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸 | |
21 | 19, 20 | eqtri 2109 | . . 3 ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 |
22 | 6 | nn0cni 8746 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
23 | 22, 2 | mulcomi 7555 | . . . . 5 ⊢ (𝐵 · 𝑃) = (𝑃 · 𝐵) |
24 | 23 | oveq1i 5676 | . . . 4 ⊢ ((𝐵 · 𝑃) + 𝐷) = ((𝑃 · 𝐵) + 𝐷) |
25 | numma2c.12 | . . . 4 ⊢ ((𝑃 · 𝐵) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) | |
26 | 24, 25 | eqtri 2109 | . . 3 ⊢ ((𝐵 · 𝑃) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) |
27 | 4, 5, 6, 12, 13, 3, 14, 1, 15, 16, 21, 26 | nummac 8982 | . 2 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
28 | 11, 27 | eqtri 2109 | 1 ⊢ ((𝑃 · 𝑀) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 ∈ wcel 1439 (class class class)co 5666 + caddc 7414 · cmul 7416 ℕ0cn0 8734 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-cnre 7517 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-sub 7716 df-inn 8484 df-n0 8735 |
This theorem is referenced by: decma2c 8990 |
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