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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 12534 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
4 | dpmul1000.c | . . . . . . . . 9 ⊢ 𝐶 ∈ ℕ0 | |
5 | 4 | nn0rei 12534 | . . . . . . . 8 ⊢ 𝐶 ∈ ℝ |
6 | dpmul1000.d | . . . . . . . 8 ⊢ 𝐷 ∈ ℝ | |
7 | dp2cl 32846 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ) | |
8 | 5, 6, 7 | mp2an 692 | . . . . . . 7 ⊢ _𝐶𝐷 ∈ ℝ |
9 | dp2cl 32846 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ _𝐶𝐷 ∈ ℝ) → _𝐵_𝐶𝐷 ∈ ℝ) | |
10 | 3, 8, 9 | mp2an 692 | . . . . . 6 ⊢ _𝐵_𝐶𝐷 ∈ ℝ |
11 | dpcl 32857 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ _𝐵_𝐶𝐷 ∈ ℝ) → (𝐴._𝐵_𝐶𝐷) ∈ ℝ) | |
12 | 1, 10, 11 | mp2an 692 | . . . . 5 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℝ |
13 | 12 | recni 11272 | . . . 4 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℂ |
14 | 10nn0 12748 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
15 | 0nn0 12538 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
16 | 14, 15 | deccl 12745 | . . . . 5 ⊢ ;;100 ∈ ℕ0 |
17 | 16 | nn0cni 12535 | . . . 4 ⊢ ;;100 ∈ ℂ |
18 | 14 | nn0cni 12535 | . . . 4 ⊢ ;10 ∈ ℂ |
19 | 13, 17, 18 | mulassi 11269 | . . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) |
20 | 1, 2, 8 | dpmul100 32863 | . . . 4 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;100) = ;;𝐴𝐵_𝐶𝐷 |
21 | 20 | oveq1i 7440 | . . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = (;;𝐴𝐵_𝐶𝐷 · ;10) |
22 | 16 | dec0u 12751 | . . . . 5 ⊢ (;10 · ;;100) = ;;;1000 |
23 | 18, 17, 22 | mulcomli 11267 | . . . 4 ⊢ (;;100 · ;10) = ;;;1000 |
24 | 23 | oveq2i 7441 | . . 3 ⊢ ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000) |
25 | 19, 21, 24 | 3eqtr3i 2770 | . 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000) |
26 | dfdec10 12733 | . . . 4 ⊢ ;;𝐴𝐵_𝐶𝐷 = ((;10 · ;𝐴𝐵) + _𝐶𝐷) | |
27 | 26 | oveq1i 7440 | . . 3 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) |
28 | 1, 2 | deccl 12745 | . . . . . 6 ⊢ ;𝐴𝐵 ∈ ℕ0 |
29 | 28 | nn0cni 12535 | . . . . 5 ⊢ ;𝐴𝐵 ∈ ℂ |
30 | 18, 29 | mulcli 11265 | . . . 4 ⊢ (;10 · ;𝐴𝐵) ∈ ℂ |
31 | 8 | recni 11272 | . . . 4 ⊢ _𝐶𝐷 ∈ ℂ |
32 | 30, 31, 18 | adddiri 11271 | . . 3 ⊢ (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) |
33 | 28, 4, 6 | dfdec100 32836 | . . . 4 ⊢ ;;;𝐴𝐵𝐶𝐷 = ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) |
34 | 14 | dec0u 12751 | . . . . . . 7 ⊢ (;10 · ;10) = ;;100 |
35 | 34 | oveq1i 7440 | . . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = (;;100 · ;𝐴𝐵) |
36 | 18, 18, 29 | mul32i 11454 | . . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10) |
37 | 35, 36 | eqtr3i 2764 | . . . . 5 ⊢ (;;100 · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10) |
38 | 4, 6 | dpmul10 32861 | . . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷 |
39 | dpval 32856 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷) | |
40 | 4, 6, 39 | mp2an 692 | . . . . . . 7 ⊢ (𝐶.𝐷) = _𝐶𝐷 |
41 | 40 | oveq1i 7440 | . . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = (_𝐶𝐷 · ;10) |
42 | 38, 41 | eqtr3i 2764 | . . . . 5 ⊢ ;𝐶𝐷 = (_𝐶𝐷 · ;10) |
43 | 37, 42 | oveq12i 7442 | . . . 4 ⊢ ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) |
44 | 33, 43 | eqtr2i 2763 | . . 3 ⊢ (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) = ;;;𝐴𝐵𝐶𝐷 |
45 | 27, 32, 44 | 3eqtri 2766 | . 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ;;;𝐴𝐵𝐶𝐷 |
46 | 25, 45 | eqtr3i 2764 | 1 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 (class class class)co 7430 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 ℕ0cn0 12523 ;cdc 12730 _cdp2 32837 .cdp 32854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-dec 12731 df-dp2 32838 df-dp 32855 |
This theorem is referenced by: dpmul4 32880 |
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