![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 12535 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
4 | dpmul1000.c | . . . . . . . . 9 ⊢ 𝐶 ∈ ℕ0 | |
5 | 4 | nn0rei 12535 | . . . . . . . 8 ⊢ 𝐶 ∈ ℝ |
6 | dpmul1000.d | . . . . . . . 8 ⊢ 𝐷 ∈ ℝ | |
7 | dp2cl 32741 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ) | |
8 | 5, 6, 7 | mp2an 690 | . . . . . . 7 ⊢ _𝐶𝐷 ∈ ℝ |
9 | dp2cl 32741 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ _𝐶𝐷 ∈ ℝ) → _𝐵_𝐶𝐷 ∈ ℝ) | |
10 | 3, 8, 9 | mp2an 690 | . . . . . 6 ⊢ _𝐵_𝐶𝐷 ∈ ℝ |
11 | dpcl 32752 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ _𝐵_𝐶𝐷 ∈ ℝ) → (𝐴._𝐵_𝐶𝐷) ∈ ℝ) | |
12 | 1, 10, 11 | mp2an 690 | . . . . 5 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℝ |
13 | 12 | recni 11278 | . . . 4 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℂ |
14 | 10nn0 12747 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
15 | 0nn0 12539 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
16 | 14, 15 | deccl 12744 | . . . . 5 ⊢ ;;100 ∈ ℕ0 |
17 | 16 | nn0cni 12536 | . . . 4 ⊢ ;;100 ∈ ℂ |
18 | 14 | nn0cni 12536 | . . . 4 ⊢ ;10 ∈ ℂ |
19 | 13, 17, 18 | mulassi 11275 | . . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) |
20 | 1, 2, 8 | dpmul100 32758 | . . . 4 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;100) = ;;𝐴𝐵_𝐶𝐷 |
21 | 20 | oveq1i 7434 | . . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = (;;𝐴𝐵_𝐶𝐷 · ;10) |
22 | 16 | dec0u 12750 | . . . . 5 ⊢ (;10 · ;;100) = ;;;1000 |
23 | 18, 17, 22 | mulcomli 11273 | . . . 4 ⊢ (;;100 · ;10) = ;;;1000 |
24 | 23 | oveq2i 7435 | . . 3 ⊢ ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000) |
25 | 19, 21, 24 | 3eqtr3i 2762 | . 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000) |
26 | dfdec10 12732 | . . . 4 ⊢ ;;𝐴𝐵_𝐶𝐷 = ((;10 · ;𝐴𝐵) + _𝐶𝐷) | |
27 | 26 | oveq1i 7434 | . . 3 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) |
28 | 1, 2 | deccl 12744 | . . . . . 6 ⊢ ;𝐴𝐵 ∈ ℕ0 |
29 | 28 | nn0cni 12536 | . . . . 5 ⊢ ;𝐴𝐵 ∈ ℂ |
30 | 18, 29 | mulcli 11271 | . . . 4 ⊢ (;10 · ;𝐴𝐵) ∈ ℂ |
31 | 8 | recni 11278 | . . . 4 ⊢ _𝐶𝐷 ∈ ℂ |
32 | 30, 31, 18 | adddiri 11277 | . . 3 ⊢ (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) |
33 | 28, 4, 6 | dfdec100 32731 | . . . 4 ⊢ ;;;𝐴𝐵𝐶𝐷 = ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) |
34 | 14 | dec0u 12750 | . . . . . . 7 ⊢ (;10 · ;10) = ;;100 |
35 | 34 | oveq1i 7434 | . . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = (;;100 · ;𝐴𝐵) |
36 | 18, 18, 29 | mul32i 11460 | . . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10) |
37 | 35, 36 | eqtr3i 2756 | . . . . 5 ⊢ (;;100 · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10) |
38 | 4, 6 | dpmul10 32756 | . . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷 |
39 | dpval 32751 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷) | |
40 | 4, 6, 39 | mp2an 690 | . . . . . . 7 ⊢ (𝐶.𝐷) = _𝐶𝐷 |
41 | 40 | oveq1i 7434 | . . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = (_𝐶𝐷 · ;10) |
42 | 38, 41 | eqtr3i 2756 | . . . . 5 ⊢ ;𝐶𝐷 = (_𝐶𝐷 · ;10) |
43 | 37, 42 | oveq12i 7436 | . . . 4 ⊢ ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) |
44 | 33, 43 | eqtr2i 2755 | . . 3 ⊢ (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) = ;;;𝐴𝐵𝐶𝐷 |
45 | 27, 32, 44 | 3eqtri 2758 | . 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ;;;𝐴𝐵𝐶𝐷 |
46 | 25, 45 | eqtr3i 2756 | 1 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7424 ℝcr 11157 0cc0 11158 1c1 11159 + caddc 11161 · cmul 11163 ℕ0cn0 12524 ;cdc 12729 _cdp2 32732 .cdp 32749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-dec 12730 df-dp2 32733 df-dp 32750 |
This theorem is referenced by: dpmul4 32775 |
Copyright terms: Public domain | W3C validator |