dpmul1000 - Mathbox for Thierry Arnoux

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Theorem dpmul1000 32819

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 12510	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
4		dpmul1000.c	. . . . . . . . 9 ⊢ 𝐶 ∈ ℕ₀
5	4	nn0rei 12510	. . . . . . . 8 ⊢ 𝐶 ∈ ℝ
6		dpmul1000.d	. . . . . . . 8 ⊢ 𝐷 ∈ ℝ
7		dp2cl 32800	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ)
8	5, 6, 7	mp2an 692	. . . . . . 7 ⊢ _𝐶𝐷 ∈ ℝ
9		dp2cl 32800	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ _𝐶𝐷 ∈ ℝ) → _𝐵_𝐶𝐷 ∈ ℝ)
10	3, 8, 9	mp2an 692	. . . . . 6 ⊢ _𝐵_𝐶𝐷 ∈ ℝ
11		dpcl 32811	. . . . . 6 ⊢ ((𝐴 ∈ ℕ₀ ∧ _𝐵_𝐶𝐷 ∈ ℝ) → (𝐴._𝐵_𝐶𝐷) ∈ ℝ)
12	1, 10, 11	mp2an 692	. . . . 5 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℝ
13	12	recni 11247	. . . 4 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℂ
14		10nn0 12724	. . . . . 6 ⊢ ;10 ∈ ℕ₀
15		0nn0 12514	. . . . . 6 ⊢ 0 ∈ ℕ₀
16	14, 15	deccl 12721	. . . . 5 ⊢ ;;100 ∈ ℕ₀
17	16	nn0cni 12511	. . . 4 ⊢ ;;100 ∈ ℂ
18	14	nn0cni 12511	. . . 4 ⊢ ;10 ∈ ℂ
19	13, 17, 18	mulassi 11244	. . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10))
20	1, 2, 8	dpmul100 32817	. . . 4 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;100) = ;;𝐴𝐵_𝐶𝐷
21	20	oveq1i 7413	. . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = (;;𝐴𝐵_𝐶𝐷 · ;10)
22	16	dec0u 12727	. . . . 5 ⊢ (;10 · ;;100) = ;;;1000
23	18, 17, 22	mulcomli 11242	. . . 4 ⊢ (;;100 · ;10) = ;;;1000
24	23	oveq2i 7414	. . 3 ⊢ ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000)
25	19, 21, 24	3eqtr3i 2766	. 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000)
26		dfdec10 12709	. . . 4 ⊢ ;;𝐴𝐵_𝐶𝐷 = ((;10 · ;𝐴𝐵) + _𝐶𝐷)
27	26	oveq1i 7413	. . 3 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10)
28	1, 2	deccl 12721	. . . . . 6 ⊢ ;𝐴𝐵 ∈ ℕ₀
29	28	nn0cni 12511	. . . . 5 ⊢ ;𝐴𝐵 ∈ ℂ
30	18, 29	mulcli 11240	. . . 4 ⊢ (;10 · ;𝐴𝐵) ∈ ℂ
31	8	recni 11247	. . . 4 ⊢ _𝐶𝐷 ∈ ℂ
32	30, 31, 18	adddiri 11246	. . 3 ⊢ (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10))
33	28, 4, 6	dfdec100 32755	. . . 4 ⊢ ;;;𝐴𝐵𝐶𝐷 = ((;;100 · ;𝐴𝐵) + ;𝐶𝐷)
34	14	dec0u 12727	. . . . . . 7 ⊢ (;10 · ;10) = ;;100
35	34	oveq1i 7413	. . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = (;;100 · ;𝐴𝐵)
36	18, 18, 29	mul32i 11429	. . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10)
37	35, 36	eqtr3i 2760	. . . . 5 ⊢ (;;100 · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10)
38	4, 6	dpmul10 32815	. . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷
39		dpval 32810	. . . . . . . 8 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷)
40	4, 6, 39	mp2an 692	. . . . . . 7 ⊢ (𝐶.𝐷) = _𝐶𝐷
41	40	oveq1i 7413	. . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = (_𝐶𝐷 · ;10)
42	38, 41	eqtr3i 2760	. . . . 5 ⊢ ;𝐶𝐷 = (_𝐶𝐷 · ;10)
43	37, 42	oveq12i 7415	. . . 4 ⊢ ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10))
44	33, 43	eqtr2i 2759	. . 3 ⊢ (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) = ;;;𝐴𝐵𝐶𝐷
45	27, 32, 44	3eqtri 2762	. 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ;;;𝐴𝐵𝐶𝐷
46	25, 45	eqtr3i 2760	1 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7403 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 ℕ₀cn0 12499 cdc 12706 cdp2 32791 .cdp 32808

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 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-dec 12707 df-dp2 32792 df-dp 32809

This theorem is referenced by: dpmul4 32834

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