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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 12530 | . . . 4 ⊢ 𝐵 ∈ ℝ |
4 | dp3mul10.c | . . . 4 ⊢ 𝐶 ∈ ℝ | |
5 | dp2cl 32730 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ) | |
6 | 3, 4, 5 | mp2an 690 | . . 3 ⊢ _𝐵𝐶 ∈ ℝ |
7 | 1, 6 | dpmul10 32745 | . 2 ⊢ ((𝐴._𝐵𝐶) · ;10) = ;𝐴_𝐵𝐶 |
8 | dfdec10 12727 | . 2 ⊢ ;𝐴_𝐵𝐶 = ((;10 · 𝐴) + _𝐵𝐶) | |
9 | 10nn 12740 | . . . . . . 7 ⊢ ;10 ∈ ℕ | |
10 | 9 | nncni 12269 | . . . . . 6 ⊢ ;10 ∈ ℂ |
11 | 1 | nn0cni 12531 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
12 | 10, 11 | mulcli 11267 | . . . . 5 ⊢ (;10 · 𝐴) ∈ ℂ |
13 | 3 | recni 11274 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
14 | 4 | recni 11274 | . . . . . 6 ⊢ 𝐶 ∈ ℂ |
15 | 9 | nnne0i 12299 | . . . . . 6 ⊢ ;10 ≠ 0 |
16 | 14, 10, 15 | divcli 12003 | . . . . 5 ⊢ (𝐶 / ;10) ∈ ℂ |
17 | 12, 13, 16 | addassi 11270 | . . . 4 ⊢ (((;10 · 𝐴) + 𝐵) + (𝐶 / ;10)) = ((;10 · 𝐴) + (𝐵 + (𝐶 / ;10))) |
18 | dfdec10 12727 | . . . . 5 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
19 | 18 | oveq1i 7433 | . . . 4 ⊢ (;𝐴𝐵 + (𝐶 / ;10)) = (((;10 · 𝐴) + 𝐵) + (𝐶 / ;10)) |
20 | df-dp2 32722 | . . . . 5 ⊢ _𝐵𝐶 = (𝐵 + (𝐶 / ;10)) | |
21 | 20 | oveq2i 7434 | . . . 4 ⊢ ((;10 · 𝐴) + _𝐵𝐶) = ((;10 · 𝐴) + (𝐵 + (𝐶 / ;10))) |
22 | 17, 19, 21 | 3eqtr4ri 2764 | . . 3 ⊢ ((;10 · 𝐴) + _𝐵𝐶) = (;𝐴𝐵 + (𝐶 / ;10)) |
23 | 1, 2 | deccl 12739 | . . . 4 ⊢ ;𝐴𝐵 ∈ ℕ0 |
24 | 23, 4 | dpval2 32743 | . . 3 ⊢ (;𝐴𝐵.𝐶) = (;𝐴𝐵 + (𝐶 / ;10)) |
25 | 22, 24 | eqtr4i 2756 | . 2 ⊢ ((;10 · 𝐴) + _𝐵𝐶) = (;𝐴𝐵.𝐶) |
26 | 7, 8, 25 | 3eqtri 2757 | 1 ⊢ ((𝐴._𝐵𝐶) · ;10) = (;𝐴𝐵.𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7423 ℝcr 11153 0cc0 11154 1c1 11155 + caddc 11157 · cmul 11159 / cdiv 11917 ℕ0cn0 12519 ;cdc 12724 _cdp2 32721 .cdp 32738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-dec 12725 df-dp2 32722 df-dp 32739 |
This theorem is referenced by: dpmul4 32764 |
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