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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpmul100 | Structured version Visualization version GIF version |
Description: Multiply by 100 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
Ref | Expression |
---|---|
dp3mul10.a | ⊢ 𝐴 ∈ ℕ0 |
dp3mul10.b | ⊢ 𝐵 ∈ ℕ0 |
dp3mul10.c | ⊢ 𝐶 ∈ ℝ |
Ref | Expression |
---|---|
dpmul100 | ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dp3mul10.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
2 | dp3mul10.b | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
3 | 2 | nn0rei 12516 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
4 | dp3mul10.c | . . . . . 6 ⊢ 𝐶 ∈ ℝ | |
5 | dp2cl 32688 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ) | |
6 | 3, 4, 5 | mp2an 690 | . . . . 5 ⊢ _𝐵𝐶 ∈ ℝ |
7 | 1, 6 | dpval2 32701 | . . . 4 ⊢ (𝐴._𝐵𝐶) = (𝐴 + (_𝐵𝐶 / ;10)) |
8 | 1 | nn0cni 12517 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
9 | 6 | recni 11260 | . . . . . 6 ⊢ _𝐵𝐶 ∈ ℂ |
10 | 10nn0 12728 | . . . . . . 7 ⊢ ;10 ∈ ℕ0 | |
11 | 10 | nn0cni 12517 | . . . . . 6 ⊢ ;10 ∈ ℂ |
12 | 10nn 12726 | . . . . . . 7 ⊢ ;10 ∈ ℕ | |
13 | 12 | nnne0i 12285 | . . . . . 6 ⊢ ;10 ≠ 0 |
14 | 9, 11, 13 | divcli 11989 | . . . . 5 ⊢ (_𝐵𝐶 / ;10) ∈ ℂ |
15 | 8, 14 | addcli 11252 | . . . 4 ⊢ (𝐴 + (_𝐵𝐶 / ;10)) ∈ ℂ |
16 | 7, 15 | eqeltri 2821 | . . 3 ⊢ (𝐴._𝐵𝐶) ∈ ℂ |
17 | 16, 11, 11 | mulassi 11257 | . 2 ⊢ (((𝐴._𝐵𝐶) · ;10) · ;10) = ((𝐴._𝐵𝐶) · (;10 · ;10)) |
18 | 1, 2, 4 | dfdec100 32678 | . . 3 ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶) |
19 | 11, 8, 11 | mul32i 11442 | . . . . 5 ⊢ ((;10 · 𝐴) · ;10) = ((;10 · ;10) · 𝐴) |
20 | 10 | dec0u 12731 | . . . . . 6 ⊢ (;10 · ;10) = ;;100 |
21 | 20 | oveq1i 7429 | . . . . 5 ⊢ ((;10 · ;10) · 𝐴) = (;;100 · 𝐴) |
22 | 19, 21 | eqtri 2753 | . . . 4 ⊢ ((;10 · 𝐴) · ;10) = (;;100 · 𝐴) |
23 | 2, 4 | dpval3 32702 | . . . . . 6 ⊢ (𝐵.𝐶) = _𝐵𝐶 |
24 | 23 | oveq1i 7429 | . . . . 5 ⊢ ((𝐵.𝐶) · ;10) = (_𝐵𝐶 · ;10) |
25 | 2, 4 | dpmul10 32703 | . . . . 5 ⊢ ((𝐵.𝐶) · ;10) = ;𝐵𝐶 |
26 | 24, 25 | eqtr3i 2755 | . . . 4 ⊢ (_𝐵𝐶 · ;10) = ;𝐵𝐶 |
27 | 22, 26 | oveq12i 7431 | . . 3 ⊢ (((;10 · 𝐴) · ;10) + (_𝐵𝐶 · ;10)) = ((;;100 · 𝐴) + ;𝐵𝐶) |
28 | 1, 6 | dpmul10 32703 | . . . . . 6 ⊢ ((𝐴._𝐵𝐶) · ;10) = ;𝐴_𝐵𝐶 |
29 | dfdec10 12713 | . . . . . 6 ⊢ ;𝐴_𝐵𝐶 = ((;10 · 𝐴) + _𝐵𝐶) | |
30 | 28, 29 | eqtri 2753 | . . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;10) = ((;10 · 𝐴) + _𝐵𝐶) |
31 | 30 | oveq1i 7429 | . . . 4 ⊢ (((𝐴._𝐵𝐶) · ;10) · ;10) = (((;10 · 𝐴) + _𝐵𝐶) · ;10) |
32 | 11, 8 | mulcli 11253 | . . . . 5 ⊢ (;10 · 𝐴) ∈ ℂ |
33 | 32, 9, 11 | adddiri 11259 | . . . 4 ⊢ (((;10 · 𝐴) + _𝐵𝐶) · ;10) = (((;10 · 𝐴) · ;10) + (_𝐵𝐶 · ;10)) |
34 | 31, 33 | eqtr2i 2754 | . . 3 ⊢ (((;10 · 𝐴) · ;10) + (_𝐵𝐶 · ;10)) = (((𝐴._𝐵𝐶) · ;10) · ;10) |
35 | 18, 27, 34 | 3eqtr2ri 2760 | . 2 ⊢ (((𝐴._𝐵𝐶) · ;10) · ;10) = ;;𝐴𝐵𝐶 |
36 | 20 | oveq2i 7430 | . 2 ⊢ ((𝐴._𝐵𝐶) · (;10 · ;10)) = ((𝐴._𝐵𝐶) · ;;100) |
37 | 17, 35, 36 | 3eqtr3ri 2762 | 1 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7419 ℂcc 11138 ℝcr 11139 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 / cdiv 11903 ℕ0cn0 12505 ;cdc 12710 _cdp2 32679 .cdp 32696 |
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 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
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 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-dec 12711 df-dp2 32680 df-dp 32697 |
This theorem is referenced by: dpmul1000 32707 dpadd3 32720 dpmul 32721 dpmul4 32722 |
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