dpmul1000 - Mathbox for Thierry Arnoux

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Theorem dpmul1000 32881

Description: Multiply by 1000 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)

**Hypotheses**
Ref	Expression
dpmul1000.a	⊢ 𝐴 ∈ ℕ₀
dpmul1000.b	⊢ 𝐵 ∈ ℕ₀
dpmul1000.c	⊢ 𝐶 ∈ ℕ₀
dpmul1000.d	⊢ 𝐷 ∈ ℝ

**Assertion**
Ref	Expression
dpmul1000	⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷

**Proof of Theorem dpmul1000**
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

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