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 12101 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
4 | dp3mul10.c | . . . . . 6 ⊢ 𝐶 ∈ ℝ | |
5 | dp2cl 30874 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ) | |
6 | 3, 4, 5 | mp2an 692 | . . . . 5 ⊢ _𝐵𝐶 ∈ ℝ |
7 | 1, 6 | dpval2 30887 | . . . 4 ⊢ (𝐴._𝐵𝐶) = (𝐴 + (_𝐵𝐶 / ;10)) |
8 | 1 | nn0cni 12102 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
9 | 6 | recni 10847 | . . . . . 6 ⊢ _𝐵𝐶 ∈ ℂ |
10 | 10nn0 12311 | . . . . . . 7 ⊢ ;10 ∈ ℕ0 | |
11 | 10 | nn0cni 12102 | . . . . . 6 ⊢ ;10 ∈ ℂ |
12 | 10nn 12309 | . . . . . . 7 ⊢ ;10 ∈ ℕ | |
13 | 12 | nnne0i 11870 | . . . . . 6 ⊢ ;10 ≠ 0 |
14 | 9, 11, 13 | divcli 11574 | . . . . 5 ⊢ (_𝐵𝐶 / ;10) ∈ ℂ |
15 | 8, 14 | addcli 10839 | . . . 4 ⊢ (𝐴 + (_𝐵𝐶 / ;10)) ∈ ℂ |
16 | 7, 15 | eqeltri 2834 | . . 3 ⊢ (𝐴._𝐵𝐶) ∈ ℂ |
17 | 16, 11, 11 | mulassi 10844 | . 2 ⊢ (((𝐴._𝐵𝐶) · ;10) · ;10) = ((𝐴._𝐵𝐶) · (;10 · ;10)) |
18 | 1, 2, 4 | dfdec100 30864 | . . 3 ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶) |
19 | 11, 8, 11 | mul32i 11028 | . . . . 5 ⊢ ((;10 · 𝐴) · ;10) = ((;10 · ;10) · 𝐴) |
20 | 10 | dec0u 12314 | . . . . . 6 ⊢ (;10 · ;10) = ;;100 |
21 | 20 | oveq1i 7223 | . . . . 5 ⊢ ((;10 · ;10) · 𝐴) = (;;100 · 𝐴) |
22 | 19, 21 | eqtri 2765 | . . . 4 ⊢ ((;10 · 𝐴) · ;10) = (;;100 · 𝐴) |
23 | 2, 4 | dpval3 30888 | . . . . . 6 ⊢ (𝐵.𝐶) = _𝐵𝐶 |
24 | 23 | oveq1i 7223 | . . . . 5 ⊢ ((𝐵.𝐶) · ;10) = (_𝐵𝐶 · ;10) |
25 | 2, 4 | dpmul10 30889 | . . . . 5 ⊢ ((𝐵.𝐶) · ;10) = ;𝐵𝐶 |
26 | 24, 25 | eqtr3i 2767 | . . . 4 ⊢ (_𝐵𝐶 · ;10) = ;𝐵𝐶 |
27 | 22, 26 | oveq12i 7225 | . . 3 ⊢ (((;10 · 𝐴) · ;10) + (_𝐵𝐶 · ;10)) = ((;;100 · 𝐴) + ;𝐵𝐶) |
28 | 1, 6 | dpmul10 30889 | . . . . . 6 ⊢ ((𝐴._𝐵𝐶) · ;10) = ;𝐴_𝐵𝐶 |
29 | dfdec10 12296 | . . . . . 6 ⊢ ;𝐴_𝐵𝐶 = ((;10 · 𝐴) + _𝐵𝐶) | |
30 | 28, 29 | eqtri 2765 | . . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;10) = ((;10 · 𝐴) + _𝐵𝐶) |
31 | 30 | oveq1i 7223 | . . . 4 ⊢ (((𝐴._𝐵𝐶) · ;10) · ;10) = (((;10 · 𝐴) + _𝐵𝐶) · ;10) |
32 | 11, 8 | mulcli 10840 | . . . . 5 ⊢ (;10 · 𝐴) ∈ ℂ |
33 | 32, 9, 11 | adddiri 10846 | . . . 4 ⊢ (((;10 · 𝐴) + _𝐵𝐶) · ;10) = (((;10 · 𝐴) · ;10) + (_𝐵𝐶 · ;10)) |
34 | 31, 33 | eqtr2i 2766 | . . 3 ⊢ (((;10 · 𝐴) · ;10) + (_𝐵𝐶 · ;10)) = (((𝐴._𝐵𝐶) · ;10) · ;10) |
35 | 18, 27, 34 | 3eqtr2ri 2772 | . 2 ⊢ (((𝐴._𝐵𝐶) · ;10) · ;10) = ;;𝐴𝐵𝐶 |
36 | 20 | oveq2i 7224 | . 2 ⊢ ((𝐴._𝐵𝐶) · (;10 · ;10)) = ((𝐴._𝐵𝐶) · ;;100) |
37 | 17, 35, 36 | 3eqtr3ri 2774 | 1 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 / cdiv 11489 ℕ0cn0 12090 ;cdc 12293 _cdp2 30865 .cdp 30882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-dec 12294 df-dp2 30866 df-dp 30883 |
This theorem is referenced by: dpmul1000 30893 dpadd3 30906 dpmul 30907 dpmul4 30908 |
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