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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 12099 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2906 | . . 3 ⊢ 𝑁 ∈ ℕ0 |
7 | 6 | nn0cni 11897 | . 2 ⊢ 𝑁 ∈ ℂ |
8 | nummul1c.2 | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
9 | 8 | nn0cni 11897 | . 2 ⊢ 𝑃 ∈ ℂ |
10 | nummul1c.6 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
11 | nummul1c.7 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
12 | 3 | nn0cni 11897 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
13 | 12, 9 | mulcomi 10637 | . . . . 5 ⊢ (𝐴 · 𝑃) = (𝑃 · 𝐴) |
14 | 13 | oveq1i 7155 | . . . 4 ⊢ ((𝐴 · 𝑃) + 𝐸) = ((𝑃 · 𝐴) + 𝐸) |
15 | nummul2c.7 | . . . 4 ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 | |
16 | 14, 15 | eqtri 2841 | . . 3 ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 |
17 | 4 | nn0cni 11897 | . . . 4 ⊢ 𝐵 ∈ ℂ |
18 | nummul2c.8 | . . . 4 ⊢ (𝑃 · 𝐵) = ((𝑇 · 𝐸) + 𝐷) | |
19 | 9, 17, 18 | mulcomli 10638 | . . 3 ⊢ (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷) |
20 | 2, 8, 3, 4, 1, 10, 11, 16, 19 | nummul1c 12135 | . 2 ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) |
21 | 7, 9, 20 | mulcomli 10638 | 1 ⊢ (𝑃 · 𝑁) = ((𝑇 · 𝐶) + 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 + caddc 10528 · cmul 10530 ℕ0cn0 11885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-sub 10860 df-nn 11627 df-n0 11886 |
This theorem is referenced by: decmul2c 12152 |
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