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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 12271 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2827 | . . 3 ⊢ 𝑁 ∈ ℕ0 |
7 | 6 | nn0cni 12067 | . 2 ⊢ 𝑁 ∈ ℂ |
8 | nummul1c.2 | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
9 | 8 | nn0cni 12067 | . 2 ⊢ 𝑃 ∈ ℂ |
10 | nummul1c.6 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
11 | nummul1c.7 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
12 | 3 | nn0cni 12067 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
13 | 12, 9 | mulcomi 10806 | . . . . 5 ⊢ (𝐴 · 𝑃) = (𝑃 · 𝐴) |
14 | 13 | oveq1i 7201 | . . . 4 ⊢ ((𝐴 · 𝑃) + 𝐸) = ((𝑃 · 𝐴) + 𝐸) |
15 | nummul2c.7 | . . . 4 ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 | |
16 | 14, 15 | eqtri 2759 | . . 3 ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 |
17 | 4 | nn0cni 12067 | . . . 4 ⊢ 𝐵 ∈ ℂ |
18 | nummul2c.8 | . . . 4 ⊢ (𝑃 · 𝐵) = ((𝑇 · 𝐸) + 𝐷) | |
19 | 9, 17, 18 | mulcomli 10807 | . . 3 ⊢ (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷) |
20 | 2, 8, 3, 4, 1, 10, 11, 16, 19 | nummul1c 12307 | . 2 ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) |
21 | 7, 9, 20 | mulcomli 10807 | 1 ⊢ (𝑃 · 𝑁) = ((𝑇 · 𝐶) + 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 (class class class)co 7191 + caddc 10697 · cmul 10699 ℕ0cn0 12055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-sub 11029 df-nn 11796 df-n0 12056 |
This theorem is referenced by: decmul2c 12324 |
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