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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dp3mul10 | Structured version Visualization version GIF version |
Description: Multiply by 10 a decimal expansion with 3 digits. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
Ref | Expression |
---|---|
dp3mul10.a | ⊢ 𝐴 ∈ ℕ0 |
dp3mul10.b | ⊢ 𝐵 ∈ ℕ0 |
dp3mul10.c | ⊢ 𝐶 ∈ ℝ |
Ref | Expression |
---|---|
dp3mul10 | ⊢ ((𝐴._𝐵𝐶) · ;10) = (;𝐴𝐵.𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dp3mul10.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
2 | dp3mul10.b | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
3 | 2 | nn0rei 11896 | . . . 4 ⊢ 𝐵 ∈ ℝ |
4 | dp3mul10.c | . . . 4 ⊢ 𝐶 ∈ ℝ | |
5 | dp2cl 30582 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ) | |
6 | 3, 4, 5 | mp2an 691 | . . 3 ⊢ _𝐵𝐶 ∈ ℝ |
7 | 1, 6 | dpmul10 30597 | . 2 ⊢ ((𝐴._𝐵𝐶) · ;10) = ;𝐴_𝐵𝐶 |
8 | dfdec10 12089 | . 2 ⊢ ;𝐴_𝐵𝐶 = ((;10 · 𝐴) + _𝐵𝐶) | |
9 | 10nn 12102 | . . . . . . 7 ⊢ ;10 ∈ ℕ | |
10 | 9 | nncni 11635 | . . . . . 6 ⊢ ;10 ∈ ℂ |
11 | 1 | nn0cni 11897 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
12 | 10, 11 | mulcli 10637 | . . . . 5 ⊢ (;10 · 𝐴) ∈ ℂ |
13 | 3 | recni 10644 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
14 | 4 | recni 10644 | . . . . . 6 ⊢ 𝐶 ∈ ℂ |
15 | 9 | nnne0i 11665 | . . . . . 6 ⊢ ;10 ≠ 0 |
16 | 14, 10, 15 | divcli 11371 | . . . . 5 ⊢ (𝐶 / ;10) ∈ ℂ |
17 | 12, 13, 16 | addassi 10640 | . . . 4 ⊢ (((;10 · 𝐴) + 𝐵) + (𝐶 / ;10)) = ((;10 · 𝐴) + (𝐵 + (𝐶 / ;10))) |
18 | dfdec10 12089 | . . . . 5 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
19 | 18 | oveq1i 7145 | . . . 4 ⊢ (;𝐴𝐵 + (𝐶 / ;10)) = (((;10 · 𝐴) + 𝐵) + (𝐶 / ;10)) |
20 | df-dp2 30574 | . . . . 5 ⊢ _𝐵𝐶 = (𝐵 + (𝐶 / ;10)) | |
21 | 20 | oveq2i 7146 | . . . 4 ⊢ ((;10 · 𝐴) + _𝐵𝐶) = ((;10 · 𝐴) + (𝐵 + (𝐶 / ;10))) |
22 | 17, 19, 21 | 3eqtr4ri 2832 | . . 3 ⊢ ((;10 · 𝐴) + _𝐵𝐶) = (;𝐴𝐵 + (𝐶 / ;10)) |
23 | 1, 2 | deccl 12101 | . . . 4 ⊢ ;𝐴𝐵 ∈ ℕ0 |
24 | 23, 4 | dpval2 30595 | . . 3 ⊢ (;𝐴𝐵.𝐶) = (;𝐴𝐵 + (𝐶 / ;10)) |
25 | 22, 24 | eqtr4i 2824 | . 2 ⊢ ((;10 · 𝐴) + _𝐵𝐶) = (;𝐴𝐵.𝐶) |
26 | 7, 8, 25 | 3eqtri 2825 | 1 ⊢ ((𝐴._𝐵𝐶) · ;10) = (;𝐴𝐵.𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 / cdiv 11286 ℕ0cn0 11885 ;cdc 12086 _cdp2 30573 .cdp 30590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-dec 12087 df-dp2 30574 df-dp 30591 |
This theorem is referenced by: dpmul4 30616 |
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