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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 9147 | . . . 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 9355 | . . . . . 6 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
8 | 3, 7 | eqeltri 2243 | . . . . 5 ⊢ 𝑀 ∈ ℕ0 |
9 | 8 | nn0cni 9147 | . . . 4 ⊢ 𝑀 ∈ ℂ |
10 | 2, 9 | mulcomi 7926 | . . 3 ⊢ (𝑃 · 𝑀) = (𝑀 · 𝑃) |
11 | 10 | oveq1i 5863 | . 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 9147 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
18 | 17, 2 | mulcomi 7926 | . . . . 5 ⊢ (𝐴 · 𝑃) = (𝑃 · 𝐴) |
19 | 18 | oveq1i 5863 | . . . 4 ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = ((𝑃 · 𝐴) + (𝐶 + 𝐺)) |
20 | numma2c.11 | . . . 4 ⊢ ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸 | |
21 | 19, 20 | eqtri 2191 | . . 3 ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 |
22 | 6 | nn0cni 9147 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
23 | 22, 2 | mulcomi 7926 | . . . . 5 ⊢ (𝐵 · 𝑃) = (𝑃 · 𝐵) |
24 | 23 | oveq1i 5863 | . . . 4 ⊢ ((𝐵 · 𝑃) + 𝐷) = ((𝑃 · 𝐵) + 𝐷) |
25 | numma2c.12 | . . . 4 ⊢ ((𝑃 · 𝐵) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) | |
26 | 24, 25 | eqtri 2191 | . . 3 ⊢ ((𝐵 · 𝑃) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) |
27 | 4, 5, 6, 12, 13, 3, 14, 1, 15, 16, 21, 26 | nummac 9387 | . 2 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
28 | 11, 27 | eqtri 2191 | 1 ⊢ ((𝑃 · 𝑀) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 (class class class)co 5853 + caddc 7777 · cmul 7779 ℕ0cn0 9135 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-sub 8092 df-inn 8879 df-n0 9136 |
This theorem is referenced by: decma2c 9395 |
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