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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 11756 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
4 | dpmul1000.c | . . . . . . . . 9 ⊢ 𝐶 ∈ ℕ0 | |
5 | 4 | nn0rei 11756 | . . . . . . . 8 ⊢ 𝐶 ∈ ℝ |
6 | dpmul1000.d | . . . . . . . 8 ⊢ 𝐷 ∈ ℝ | |
7 | dp2cl 30240 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ) | |
8 | 5, 6, 7 | mp2an 688 | . . . . . . 7 ⊢ _𝐶𝐷 ∈ ℝ |
9 | dp2cl 30240 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ _𝐶𝐷 ∈ ℝ) → _𝐵_𝐶𝐷 ∈ ℝ) | |
10 | 3, 8, 9 | mp2an 688 | . . . . . 6 ⊢ _𝐵_𝐶𝐷 ∈ ℝ |
11 | dpcl 30251 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ _𝐵_𝐶𝐷 ∈ ℝ) → (𝐴._𝐵_𝐶𝐷) ∈ ℝ) | |
12 | 1, 10, 11 | mp2an 688 | . . . . 5 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℝ |
13 | 12 | recni 10501 | . . . 4 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℂ |
14 | 10nn0 11965 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
15 | 0nn0 11760 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
16 | 14, 15 | deccl 11962 | . . . . 5 ⊢ ;;100 ∈ ℕ0 |
17 | 16 | nn0cni 11757 | . . . 4 ⊢ ;;100 ∈ ℂ |
18 | 14 | nn0cni 11757 | . . . 4 ⊢ ;10 ∈ ℂ |
19 | 13, 17, 18 | mulassi 10498 | . . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) |
20 | 1, 2, 8 | dpmul100 30257 | . . . 4 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;100) = ;;𝐴𝐵_𝐶𝐷 |
21 | 20 | oveq1i 7026 | . . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = (;;𝐴𝐵_𝐶𝐷 · ;10) |
22 | 16 | dec0u 11968 | . . . . 5 ⊢ (;10 · ;;100) = ;;;1000 |
23 | 18, 17, 22 | mulcomli 10496 | . . . 4 ⊢ (;;100 · ;10) = ;;;1000 |
24 | 23 | oveq2i 7027 | . . 3 ⊢ ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000) |
25 | 19, 21, 24 | 3eqtr3i 2827 | . 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000) |
26 | dfdec10 11950 | . . . 4 ⊢ ;;𝐴𝐵_𝐶𝐷 = ((;10 · ;𝐴𝐵) + _𝐶𝐷) | |
27 | 26 | oveq1i 7026 | . . 3 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) |
28 | 1, 2 | deccl 11962 | . . . . . 6 ⊢ ;𝐴𝐵 ∈ ℕ0 |
29 | 28 | nn0cni 11757 | . . . . 5 ⊢ ;𝐴𝐵 ∈ ℂ |
30 | 18, 29 | mulcli 10494 | . . . 4 ⊢ (;10 · ;𝐴𝐵) ∈ ℂ |
31 | 8 | recni 10501 | . . . 4 ⊢ _𝐶𝐷 ∈ ℂ |
32 | 30, 31, 18 | adddiri 10500 | . . 3 ⊢ (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) |
33 | 28, 4, 6 | dfdec100 30230 | . . . 4 ⊢ ;;;𝐴𝐵𝐶𝐷 = ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) |
34 | 14 | dec0u 11968 | . . . . . . 7 ⊢ (;10 · ;10) = ;;100 |
35 | 34 | oveq1i 7026 | . . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = (;;100 · ;𝐴𝐵) |
36 | 18, 18, 29 | mul32i 10683 | . . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10) |
37 | 35, 36 | eqtr3i 2821 | . . . . 5 ⊢ (;;100 · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10) |
38 | 4, 6 | dpmul10 30255 | . . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷 |
39 | dpval 30250 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷) | |
40 | 4, 6, 39 | mp2an 688 | . . . . . . 7 ⊢ (𝐶.𝐷) = _𝐶𝐷 |
41 | 40 | oveq1i 7026 | . . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = (_𝐶𝐷 · ;10) |
42 | 38, 41 | eqtr3i 2821 | . . . . 5 ⊢ ;𝐶𝐷 = (_𝐶𝐷 · ;10) |
43 | 37, 42 | oveq12i 7028 | . . . 4 ⊢ ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) |
44 | 33, 43 | eqtr2i 2820 | . . 3 ⊢ (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) = ;;;𝐴𝐵𝐶𝐷 |
45 | 27, 32, 44 | 3eqtri 2823 | . 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ;;;𝐴𝐵𝐶𝐷 |
46 | 25, 45 | eqtr3i 2821 | 1 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∈ wcel 2081 (class class class)co 7016 ℝcr 10382 0cc0 10383 1c1 10384 + caddc 10386 · cmul 10388 ℕ0cn0 11745 ;cdc 11947 _cdp2 30231 .cdp 30248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-dec 11948 df-dp2 30232 df-dp 30249 |
This theorem is referenced by: dpmul4 30274 |
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