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Mathbox for Thierry Arnoux |
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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 11745 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
4 | dp3mul10.c | . . . . . 6 ⊢ 𝐶 ∈ ℝ | |
5 | dp2cl 30211 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ) | |
6 | 3, 4, 5 | mp2an 688 | . . . . 5 ⊢ _𝐵𝐶 ∈ ℝ |
7 | 1, 6 | dpval2 30224 | . . . 4 ⊢ (𝐴._𝐵𝐶) = (𝐴 + (_𝐵𝐶 / ;10)) |
8 | 1 | nn0cni 11746 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
9 | 6 | recni 10490 | . . . . . 6 ⊢ _𝐵𝐶 ∈ ℂ |
10 | 10nn0 11954 | . . . . . . 7 ⊢ ;10 ∈ ℕ0 | |
11 | 10 | nn0cni 11746 | . . . . . 6 ⊢ ;10 ∈ ℂ |
12 | 10nn 11952 | . . . . . . 7 ⊢ ;10 ∈ ℕ | |
13 | 12 | nnne0i 11514 | . . . . . 6 ⊢ ;10 ≠ 0 |
14 | 9, 11, 13 | divcli 11219 | . . . . 5 ⊢ (_𝐵𝐶 / ;10) ∈ ℂ |
15 | 8, 14 | addcli 10482 | . . . 4 ⊢ (𝐴 + (_𝐵𝐶 / ;10)) ∈ ℂ |
16 | 7, 15 | eqeltri 2877 | . . 3 ⊢ (𝐴._𝐵𝐶) ∈ ℂ |
17 | 16, 11, 11 | mulassi 10487 | . 2 ⊢ (((𝐴._𝐵𝐶) · ;10) · ;10) = ((𝐴._𝐵𝐶) · (;10 · ;10)) |
18 | 1, 2, 4 | dfdec100 30201 | . . 3 ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶) |
19 | 11, 8, 11 | mul32i 10672 | . . . . 5 ⊢ ((;10 · 𝐴) · ;10) = ((;10 · ;10) · 𝐴) |
20 | 10 | dec0u 11957 | . . . . . 6 ⊢ (;10 · ;10) = ;;100 |
21 | 20 | oveq1i 7017 | . . . . 5 ⊢ ((;10 · ;10) · 𝐴) = (;;100 · 𝐴) |
22 | 19, 21 | eqtri 2817 | . . . 4 ⊢ ((;10 · 𝐴) · ;10) = (;;100 · 𝐴) |
23 | 2, 4 | dpval3 30225 | . . . . . 6 ⊢ (𝐵.𝐶) = _𝐵𝐶 |
24 | 23 | oveq1i 7017 | . . . . 5 ⊢ ((𝐵.𝐶) · ;10) = (_𝐵𝐶 · ;10) |
25 | 2, 4 | dpmul10 30226 | . . . . 5 ⊢ ((𝐵.𝐶) · ;10) = ;𝐵𝐶 |
26 | 24, 25 | eqtr3i 2819 | . . . 4 ⊢ (_𝐵𝐶 · ;10) = ;𝐵𝐶 |
27 | 22, 26 | oveq12i 7019 | . . 3 ⊢ (((;10 · 𝐴) · ;10) + (_𝐵𝐶 · ;10)) = ((;;100 · 𝐴) + ;𝐵𝐶) |
28 | 1, 6 | dpmul10 30226 | . . . . . 6 ⊢ ((𝐴._𝐵𝐶) · ;10) = ;𝐴_𝐵𝐶 |
29 | dfdec10 11939 | . . . . . 6 ⊢ ;𝐴_𝐵𝐶 = ((;10 · 𝐴) + _𝐵𝐶) | |
30 | 28, 29 | eqtri 2817 | . . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;10) = ((;10 · 𝐴) + _𝐵𝐶) |
31 | 30 | oveq1i 7017 | . . . 4 ⊢ (((𝐴._𝐵𝐶) · ;10) · ;10) = (((;10 · 𝐴) + _𝐵𝐶) · ;10) |
32 | 11, 8 | mulcli 10483 | . . . . 5 ⊢ (;10 · 𝐴) ∈ ℂ |
33 | 32, 9, 11 | adddiri 10489 | . . . 4 ⊢ (((;10 · 𝐴) + _𝐵𝐶) · ;10) = (((;10 · 𝐴) · ;10) + (_𝐵𝐶 · ;10)) |
34 | 31, 33 | eqtr2i 2818 | . . 3 ⊢ (((;10 · 𝐴) · ;10) + (_𝐵𝐶 · ;10)) = (((𝐴._𝐵𝐶) · ;10) · ;10) |
35 | 18, 27, 34 | 3eqtr2ri 2824 | . 2 ⊢ (((𝐴._𝐵𝐶) · ;10) · ;10) = ;;𝐴𝐵𝐶 |
36 | 20 | oveq2i 7018 | . 2 ⊢ ((𝐴._𝐵𝐶) · (;10 · ;10)) = ((𝐴._𝐵𝐶) · ;;100) |
37 | 17, 35, 36 | 3eqtr3ri 2826 | 1 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1520 ∈ wcel 2079 (class class class)co 7007 ℂcc 10370 ℝcr 10371 0cc0 10372 1c1 10373 + caddc 10375 · cmul 10377 / cdiv 11134 ℕ0cn0 11734 ;cdc 11936 _cdp2 30202 .cdp 30219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-dec 11937 df-dp2 30203 df-dp 30220 |
This theorem is referenced by: dpmul1000 30230 dpadd3 30243 dpmul 30244 dpmul4 30245 |
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