|   | 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 12537 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ | 
| 4 | dpmul1000.c | . . . . . . . . 9 ⊢ 𝐶 ∈ ℕ0 | |
| 5 | 4 | nn0rei 12537 | . . . . . . . 8 ⊢ 𝐶 ∈ ℝ | 
| 6 | dpmul1000.d | . . . . . . . 8 ⊢ 𝐷 ∈ ℝ | |
| 7 | dp2cl 32862 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ) | |
| 8 | 5, 6, 7 | mp2an 692 | . . . . . . 7 ⊢ _𝐶𝐷 ∈ ℝ | 
| 9 | dp2cl 32862 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ _𝐶𝐷 ∈ ℝ) → _𝐵_𝐶𝐷 ∈ ℝ) | |
| 10 | 3, 8, 9 | mp2an 692 | . . . . . 6 ⊢ _𝐵_𝐶𝐷 ∈ ℝ | 
| 11 | dpcl 32873 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ _𝐵_𝐶𝐷 ∈ ℝ) → (𝐴._𝐵_𝐶𝐷) ∈ ℝ) | |
| 12 | 1, 10, 11 | mp2an 692 | . . . . 5 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℝ | 
| 13 | 12 | recni 11275 | . . . 4 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℂ | 
| 14 | 10nn0 12751 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
| 15 | 0nn0 12541 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
| 16 | 14, 15 | deccl 12748 | . . . . 5 ⊢ ;;100 ∈ ℕ0 | 
| 17 | 16 | nn0cni 12538 | . . . 4 ⊢ ;;100 ∈ ℂ | 
| 18 | 14 | nn0cni 12538 | . . . 4 ⊢ ;10 ∈ ℂ | 
| 19 | 13, 17, 18 | mulassi 11272 | . . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) | 
| 20 | 1, 2, 8 | dpmul100 32879 | . . . 4 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;100) = ;;𝐴𝐵_𝐶𝐷 | 
| 21 | 20 | oveq1i 7441 | . . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = (;;𝐴𝐵_𝐶𝐷 · ;10) | 
| 22 | 16 | dec0u 12754 | . . . . 5 ⊢ (;10 · ;;100) = ;;;1000 | 
| 23 | 18, 17, 22 | mulcomli 11270 | . . . 4 ⊢ (;;100 · ;10) = ;;;1000 | 
| 24 | 23 | oveq2i 7442 | . . 3 ⊢ ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000) | 
| 25 | 19, 21, 24 | 3eqtr3i 2773 | . 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000) | 
| 26 | dfdec10 12736 | . . . 4 ⊢ ;;𝐴𝐵_𝐶𝐷 = ((;10 · ;𝐴𝐵) + _𝐶𝐷) | |
| 27 | 26 | oveq1i 7441 | . . 3 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) | 
| 28 | 1, 2 | deccl 12748 | . . . . . 6 ⊢ ;𝐴𝐵 ∈ ℕ0 | 
| 29 | 28 | nn0cni 12538 | . . . . 5 ⊢ ;𝐴𝐵 ∈ ℂ | 
| 30 | 18, 29 | mulcli 11268 | . . . 4 ⊢ (;10 · ;𝐴𝐵) ∈ ℂ | 
| 31 | 8 | recni 11275 | . . . 4 ⊢ _𝐶𝐷 ∈ ℂ | 
| 32 | 30, 31, 18 | adddiri 11274 | . . 3 ⊢ (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) | 
| 33 | 28, 4, 6 | dfdec100 32832 | . . . 4 ⊢ ;;;𝐴𝐵𝐶𝐷 = ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) | 
| 34 | 14 | dec0u 12754 | . . . . . . 7 ⊢ (;10 · ;10) = ;;100 | 
| 35 | 34 | oveq1i 7441 | . . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = (;;100 · ;𝐴𝐵) | 
| 36 | 18, 18, 29 | mul32i 11457 | . . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10) | 
| 37 | 35, 36 | eqtr3i 2767 | . . . . 5 ⊢ (;;100 · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10) | 
| 38 | 4, 6 | dpmul10 32877 | . . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷 | 
| 39 | dpval 32872 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷) | |
| 40 | 4, 6, 39 | mp2an 692 | . . . . . . 7 ⊢ (𝐶.𝐷) = _𝐶𝐷 | 
| 41 | 40 | oveq1i 7441 | . . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = (_𝐶𝐷 · ;10) | 
| 42 | 38, 41 | eqtr3i 2767 | . . . . 5 ⊢ ;𝐶𝐷 = (_𝐶𝐷 · ;10) | 
| 43 | 37, 42 | oveq12i 7443 | . . . 4 ⊢ ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) | 
| 44 | 33, 43 | eqtr2i 2766 | . . 3 ⊢ (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) = ;;;𝐴𝐵𝐶𝐷 | 
| 45 | 27, 32, 44 | 3eqtri 2769 | . 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ;;;𝐴𝐵𝐶𝐷 | 
| 46 | 25, 45 | eqtr3i 2767 | 1 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℕ0cn0 12526 ;cdc 12733 _cdp2 32853 .cdp 32870 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12734 df-dp2 32854 df-dp 32871 | 
| This theorem is referenced by: dpmul4 32896 | 
| Copyright terms: Public domain | W3C validator |