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Mirrors > Home > MPE Home > Th. List > nummul2c | Structured version Visualization version 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 12685 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2830 | . . 3 ⊢ 𝑁 ∈ ℕ0 |
7 | 6 | nn0cni 12479 | . 2 ⊢ 𝑁 ∈ ℂ |
8 | nummul1c.2 | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
9 | 8 | nn0cni 12479 | . 2 ⊢ 𝑃 ∈ ℂ |
10 | nummul1c.6 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
11 | nummul1c.7 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
12 | 3 | nn0cni 12479 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
13 | 12, 9 | mulcomi 11217 | . . . . 5 ⊢ (𝐴 · 𝑃) = (𝑃 · 𝐴) |
14 | 13 | oveq1i 7413 | . . . 4 ⊢ ((𝐴 · 𝑃) + 𝐸) = ((𝑃 · 𝐴) + 𝐸) |
15 | nummul2c.7 | . . . 4 ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 | |
16 | 14, 15 | eqtri 2761 | . . 3 ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 |
17 | 4 | nn0cni 12479 | . . . 4 ⊢ 𝐵 ∈ ℂ |
18 | nummul2c.8 | . . . 4 ⊢ (𝑃 · 𝐵) = ((𝑇 · 𝐸) + 𝐷) | |
19 | 9, 17, 18 | mulcomli 11218 | . . 3 ⊢ (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷) |
20 | 2, 8, 3, 4, 1, 10, 11, 16, 19 | nummul1c 12721 | . 2 ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) |
21 | 7, 9, 20 | mulcomli 11218 | 1 ⊢ (𝑃 · 𝑁) = ((𝑇 · 𝐶) + 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7403 + caddc 11108 · cmul 11110 ℕ0cn0 12467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-ltxr 11248 df-sub 11441 df-nn 12208 df-n0 12468 |
This theorem is referenced by: decmul2c 12738 |
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