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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpmul1000 | Structured version Visualization version GIF version |
Description: Multiply by 1000 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
Ref | Expression |
---|---|
dpmul1000.a | ⊢ 𝐴 ∈ ℕ0 |
dpmul1000.b | ⊢ 𝐵 ∈ ℕ0 |
dpmul1000.c | ⊢ 𝐶 ∈ ℕ0 |
dpmul1000.d | ⊢ 𝐷 ∈ ℝ |
Ref | Expression |
---|---|
dpmul1000 | ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpmul1000.a | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
2 | dpmul1000.b | . . . . . . . 8 ⊢ 𝐵 ∈ ℕ0 | |
3 | 2 | nn0rei 11303 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
4 | dpmul1000.c | . . . . . . . . 9 ⊢ 𝐶 ∈ ℕ0 | |
5 | 4 | nn0rei 11303 | . . . . . . . 8 ⊢ 𝐶 ∈ ℝ |
6 | dpmul1000.d | . . . . . . . 8 ⊢ 𝐷 ∈ ℝ | |
7 | dp2cl 29587 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ) | |
8 | 5, 6, 7 | mp2an 708 | . . . . . . 7 ⊢ _𝐶𝐷 ∈ ℝ |
9 | dp2cl 29587 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ _𝐶𝐷 ∈ ℝ) → _𝐵_𝐶𝐷 ∈ ℝ) | |
10 | 3, 8, 9 | mp2an 708 | . . . . . 6 ⊢ _𝐵_𝐶𝐷 ∈ ℝ |
11 | dpcl 29598 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ _𝐵_𝐶𝐷 ∈ ℝ) → (𝐴._𝐵_𝐶𝐷) ∈ ℝ) | |
12 | 1, 10, 11 | mp2an 708 | . . . . 5 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℝ |
13 | 12 | recni 10052 | . . . 4 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℂ |
14 | 10nn0 11516 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
15 | 0nn0 11307 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
16 | 14, 15 | deccl 11512 | . . . . 5 ⊢ ;;100 ∈ ℕ0 |
17 | 16 | nn0cni 11304 | . . . 4 ⊢ ;;100 ∈ ℂ |
18 | 14 | nn0cni 11304 | . . . 4 ⊢ ;10 ∈ ℂ |
19 | 13, 17, 18 | mulassi 10049 | . . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) |
20 | 1, 2, 8 | dpmul100 29605 | . . . 4 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;100) = ;;𝐴𝐵_𝐶𝐷 |
21 | 20 | oveq1i 6660 | . . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = (;;𝐴𝐵_𝐶𝐷 · ;10) |
22 | 16 | dec0u 11520 | . . . . 5 ⊢ (;10 · ;;100) = ;;;1000 |
23 | 18, 17, 22 | mulcomli 10047 | . . . 4 ⊢ (;;100 · ;10) = ;;;1000 |
24 | 23 | oveq2i 6661 | . . 3 ⊢ ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000) |
25 | 19, 21, 24 | 3eqtr3i 2652 | . 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000) |
26 | dfdec10 11497 | . . . 4 ⊢ ;;𝐴𝐵_𝐶𝐷 = ((;10 · ;𝐴𝐵) + _𝐶𝐷) | |
27 | 26 | oveq1i 6660 | . . 3 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) |
28 | 1, 2 | deccl 11512 | . . . . . 6 ⊢ ;𝐴𝐵 ∈ ℕ0 |
29 | 28 | nn0cni 11304 | . . . . 5 ⊢ ;𝐴𝐵 ∈ ℂ |
30 | 18, 29 | mulcli 10045 | . . . 4 ⊢ (;10 · ;𝐴𝐵) ∈ ℂ |
31 | 8 | recni 10052 | . . . 4 ⊢ _𝐶𝐷 ∈ ℂ |
32 | 30, 31, 18 | adddiri 10051 | . . 3 ⊢ (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) |
33 | 28, 4, 6 | dfdec100 29576 | . . . 4 ⊢ ;;;𝐴𝐵𝐶𝐷 = ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) |
34 | 14 | dec0u 11520 | . . . . . . 7 ⊢ (;10 · ;10) = ;;100 |
35 | 34 | oveq1i 6660 | . . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = (;;100 · ;𝐴𝐵) |
36 | 18, 18, 29 | mul32i 10232 | . . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10) |
37 | 35, 36 | eqtr3i 2646 | . . . . 5 ⊢ (;;100 · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10) |
38 | 4, 6 | dpmul10 29603 | . . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷 |
39 | dpval 29597 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷) | |
40 | 4, 6, 39 | mp2an 708 | . . . . . . 7 ⊢ (𝐶.𝐷) = _𝐶𝐷 |
41 | 40 | oveq1i 6660 | . . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = (_𝐶𝐷 · ;10) |
42 | 38, 41 | eqtr3i 2646 | . . . . 5 ⊢ ;𝐶𝐷 = (_𝐶𝐷 · ;10) |
43 | 37, 42 | oveq12i 6662 | . . . 4 ⊢ ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) |
44 | 33, 43 | eqtr2i 2645 | . . 3 ⊢ (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) = ;;;𝐴𝐵𝐶𝐷 |
45 | 27, 32, 44 | 3eqtri 2648 | . 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ;;;𝐴𝐵𝐶𝐷 |
46 | 25, 45 | eqtr3i 2646 | 1 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℕ0cn0 11292 ;cdc 11493 _cdp2 29577 .cdp 29595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-dec 11494 df-dp2 29578 df-dp 29596 |
This theorem is referenced by: dpmul4 29622 |
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