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Mirrors > Home > ILE Home > Th. List > nummul2c | GIF version |
Description: The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
nummul1c.1 | ⊢ 𝑇 ∈ ℕ0 |
nummul1c.2 | ⊢ 𝑃 ∈ ℕ0 |
nummul1c.3 | ⊢ 𝐴 ∈ ℕ0 |
nummul1c.4 | ⊢ 𝐵 ∈ ℕ0 |
nummul1c.5 | ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) |
nummul1c.6 | ⊢ 𝐷 ∈ ℕ0 |
nummul1c.7 | ⊢ 𝐸 ∈ ℕ0 |
nummul2c.7 | ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 |
nummul2c.8 | ⊢ (𝑃 · 𝐵) = ((𝑇 · 𝐸) + 𝐷) |
Ref | Expression |
---|---|
nummul2c | ⊢ (𝑃 · 𝑁) = ((𝑇 · 𝐶) + 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nummul1c.5 | . . . 4 ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) | |
2 | nummul1c.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 | |
3 | nummul1c.3 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
4 | nummul1c.4 | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
5 | 2, 3, 4 | numcl 9410 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2260 | . . 3 ⊢ 𝑁 ∈ ℕ0 |
7 | 6 | nn0cni 9202 | . 2 ⊢ 𝑁 ∈ ℂ |
8 | nummul1c.2 | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
9 | 8 | nn0cni 9202 | . 2 ⊢ 𝑃 ∈ ℂ |
10 | nummul1c.6 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
11 | nummul1c.7 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
12 | 3 | nn0cni 9202 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
13 | 12, 9 | mulcomi 7977 | . . . . 5 ⊢ (𝐴 · 𝑃) = (𝑃 · 𝐴) |
14 | 13 | oveq1i 5898 | . . . 4 ⊢ ((𝐴 · 𝑃) + 𝐸) = ((𝑃 · 𝐴) + 𝐸) |
15 | nummul2c.7 | . . . 4 ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 | |
16 | 14, 15 | eqtri 2208 | . . 3 ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 |
17 | 4 | nn0cni 9202 | . . . 4 ⊢ 𝐵 ∈ ℂ |
18 | nummul2c.8 | . . . 4 ⊢ (𝑃 · 𝐵) = ((𝑇 · 𝐸) + 𝐷) | |
19 | 9, 17, 18 | mulcomli 7978 | . . 3 ⊢ (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷) |
20 | 2, 8, 3, 4, 1, 10, 11, 16, 19 | nummul1c 9446 | . 2 ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) |
21 | 7, 9, 20 | mulcomli 7978 | 1 ⊢ (𝑃 · 𝑁) = ((𝑇 · 𝐶) + 𝐷) |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 ∈ wcel 2158 (class class class)co 5888 + caddc 7828 · cmul 7830 ℕ0cn0 9190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-sub 8144 df-inn 8934 df-n0 9191 |
This theorem is referenced by: decmul2c 9463 |
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