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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 11911 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
4 | dpmul1000.c | . . . . . . . . 9 ⊢ 𝐶 ∈ ℕ0 | |
5 | 4 | nn0rei 11911 | . . . . . . . 8 ⊢ 𝐶 ∈ ℝ |
6 | dpmul1000.d | . . . . . . . 8 ⊢ 𝐷 ∈ ℝ | |
7 | dp2cl 30558 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ) | |
8 | 5, 6, 7 | mp2an 690 | . . . . . . 7 ⊢ _𝐶𝐷 ∈ ℝ |
9 | dp2cl 30558 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ _𝐶𝐷 ∈ ℝ) → _𝐵_𝐶𝐷 ∈ ℝ) | |
10 | 3, 8, 9 | mp2an 690 | . . . . . 6 ⊢ _𝐵_𝐶𝐷 ∈ ℝ |
11 | dpcl 30569 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ _𝐵_𝐶𝐷 ∈ ℝ) → (𝐴._𝐵_𝐶𝐷) ∈ ℝ) | |
12 | 1, 10, 11 | mp2an 690 | . . . . 5 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℝ |
13 | 12 | recni 10657 | . . . 4 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℂ |
14 | 10nn0 12119 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
15 | 0nn0 11915 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
16 | 14, 15 | deccl 12116 | . . . . 5 ⊢ ;;100 ∈ ℕ0 |
17 | 16 | nn0cni 11912 | . . . 4 ⊢ ;;100 ∈ ℂ |
18 | 14 | nn0cni 11912 | . . . 4 ⊢ ;10 ∈ ℂ |
19 | 13, 17, 18 | mulassi 10654 | . . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) |
20 | 1, 2, 8 | dpmul100 30575 | . . . 4 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;100) = ;;𝐴𝐵_𝐶𝐷 |
21 | 20 | oveq1i 7168 | . . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = (;;𝐴𝐵_𝐶𝐷 · ;10) |
22 | 16 | dec0u 12122 | . . . . 5 ⊢ (;10 · ;;100) = ;;;1000 |
23 | 18, 17, 22 | mulcomli 10652 | . . . 4 ⊢ (;;100 · ;10) = ;;;1000 |
24 | 23 | oveq2i 7169 | . . 3 ⊢ ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000) |
25 | 19, 21, 24 | 3eqtr3i 2854 | . 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000) |
26 | dfdec10 12104 | . . . 4 ⊢ ;;𝐴𝐵_𝐶𝐷 = ((;10 · ;𝐴𝐵) + _𝐶𝐷) | |
27 | 26 | oveq1i 7168 | . . 3 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) |
28 | 1, 2 | deccl 12116 | . . . . . 6 ⊢ ;𝐴𝐵 ∈ ℕ0 |
29 | 28 | nn0cni 11912 | . . . . 5 ⊢ ;𝐴𝐵 ∈ ℂ |
30 | 18, 29 | mulcli 10650 | . . . 4 ⊢ (;10 · ;𝐴𝐵) ∈ ℂ |
31 | 8 | recni 10657 | . . . 4 ⊢ _𝐶𝐷 ∈ ℂ |
32 | 30, 31, 18 | adddiri 10656 | . . 3 ⊢ (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) |
33 | 28, 4, 6 | dfdec100 30548 | . . . 4 ⊢ ;;;𝐴𝐵𝐶𝐷 = ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) |
34 | 14 | dec0u 12122 | . . . . . . 7 ⊢ (;10 · ;10) = ;;100 |
35 | 34 | oveq1i 7168 | . . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = (;;100 · ;𝐴𝐵) |
36 | 18, 18, 29 | mul32i 10838 | . . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10) |
37 | 35, 36 | eqtr3i 2848 | . . . . 5 ⊢ (;;100 · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10) |
38 | 4, 6 | dpmul10 30573 | . . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷 |
39 | dpval 30568 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷) | |
40 | 4, 6, 39 | mp2an 690 | . . . . . . 7 ⊢ (𝐶.𝐷) = _𝐶𝐷 |
41 | 40 | oveq1i 7168 | . . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = (_𝐶𝐷 · ;10) |
42 | 38, 41 | eqtr3i 2848 | . . . . 5 ⊢ ;𝐶𝐷 = (_𝐶𝐷 · ;10) |
43 | 37, 42 | oveq12i 7170 | . . . 4 ⊢ ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) |
44 | 33, 43 | eqtr2i 2847 | . . 3 ⊢ (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) = ;;;𝐴𝐵𝐶𝐷 |
45 | 27, 32, 44 | 3eqtri 2850 | . 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ;;;𝐴𝐵𝐶𝐷 |
46 | 25, 45 | eqtr3i 2848 | 1 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 ℕ0cn0 11900 ;cdc 12101 _cdp2 30549 .cdp 30566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-dec 12102 df-dp2 30550 df-dp 30567 |
This theorem is referenced by: dpmul4 30592 |
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