Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpmul10 | Structured version Visualization version GIF version |
Description: Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
Ref | Expression |
---|---|
dpval2.a | ⊢ 𝐴 ∈ ℕ0 |
dpval2.b | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
dpmul10 | ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpval2.b | . . . . 5 ⊢ 𝐵 ∈ ℝ | |
2 | 1 | recni 10643 | . . . 4 ⊢ 𝐵 ∈ ℂ |
3 | 10nn 12102 | . . . . 5 ⊢ ;10 ∈ ℕ | |
4 | 3 | nncni 11636 | . . . 4 ⊢ ;10 ∈ ℂ |
5 | 3 | nnne0i 11665 | . . . 4 ⊢ ;10 ≠ 0 |
6 | 2, 4, 5 | divcan2i 11371 | . . 3 ⊢ (;10 · (𝐵 / ;10)) = 𝐵 |
7 | 6 | oveq2i 7156 | . 2 ⊢ ((;10 · 𝐴) + (;10 · (𝐵 / ;10))) = ((;10 · 𝐴) + 𝐵) |
8 | dpval2.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
9 | 8, 1 | dpval2 30496 | . . . 4 ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10)) |
10 | 9 | oveq2i 7156 | . . 3 ⊢ (;10 · (𝐴.𝐵)) = (;10 · (𝐴 + (𝐵 / ;10))) |
11 | dpcl 30494 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ) | |
12 | 8, 1, 11 | mp2an 688 | . . . . 5 ⊢ (𝐴.𝐵) ∈ ℝ |
13 | 12 | recni 10643 | . . . 4 ⊢ (𝐴.𝐵) ∈ ℂ |
14 | 4, 13 | mulcomi 10637 | . . 3 ⊢ (;10 · (𝐴.𝐵)) = ((𝐴.𝐵) · ;10) |
15 | 8 | nn0cni 11897 | . . . 4 ⊢ 𝐴 ∈ ℂ |
16 | 2, 4, 5 | divcli 11370 | . . . 4 ⊢ (𝐵 / ;10) ∈ ℂ |
17 | 4, 15, 16 | adddii 10641 | . . 3 ⊢ (;10 · (𝐴 + (𝐵 / ;10))) = ((;10 · 𝐴) + (;10 · (𝐵 / ;10))) |
18 | 10, 14, 17 | 3eqtr3i 2849 | . 2 ⊢ ((𝐴.𝐵) · ;10) = ((;10 · 𝐴) + (;10 · (𝐵 / ;10))) |
19 | dfdec10 12089 | . 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
20 | 7, 18, 19 | 3eqtr4i 2851 | 1 ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 / cdiv 11285 ℕ0cn0 11885 ;cdc 12086 .cdp 30491 |
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 ax-pre-mulgt0 10602 |
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-rmo 3143 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-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 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 30475 df-dp 30492 |
This theorem is referenced by: decdiv10 30499 dpmul100 30500 dp3mul10 30501 dpmul1000 30502 dpmul 30516 dpmul4 30517 |
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