dpmul1000 - Mathbox for Thierry Arnoux

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Theorem dpmul1000 31075

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 12174	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
4		dpmul1000.c	. . . . . . . . 9 ⊢ 𝐶 ∈ ℕ₀
5	4	nn0rei 12174	. . . . . . . 8 ⊢ 𝐶 ∈ ℝ
6		dpmul1000.d	. . . . . . . 8 ⊢ 𝐷 ∈ ℝ
7		dp2cl 31056	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ)
8	5, 6, 7	mp2an 688	. . . . . . 7 ⊢ _𝐶𝐷 ∈ ℝ
9		dp2cl 31056	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ _𝐶𝐷 ∈ ℝ) → _𝐵_𝐶𝐷 ∈ ℝ)
10	3, 8, 9	mp2an 688	. . . . . 6 ⊢ _𝐵_𝐶𝐷 ∈ ℝ
11		dpcl 31067	. . . . . 6 ⊢ ((𝐴 ∈ ℕ₀ ∧ _𝐵_𝐶𝐷 ∈ ℝ) → (𝐴._𝐵_𝐶𝐷) ∈ ℝ)
12	1, 10, 11	mp2an 688	. . . . 5 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℝ
13	12	recni 10920	. . . 4 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℂ
14		10nn0 12384	. . . . . 6 ⊢ ;10 ∈ ℕ₀
15		0nn0 12178	. . . . . 6 ⊢ 0 ∈ ℕ₀
16	14, 15	deccl 12381	. . . . 5 ⊢ ;;100 ∈ ℕ₀
17	16	nn0cni 12175	. . . 4 ⊢ ;;100 ∈ ℂ
18	14	nn0cni 12175	. . . 4 ⊢ ;10 ∈ ℂ
19	13, 17, 18	mulassi 10917	. . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10))
20	1, 2, 8	dpmul100 31073	. . . 4 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;100) = ;;𝐴𝐵_𝐶𝐷
21	20	oveq1i 7265	. . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = (;;𝐴𝐵_𝐶𝐷 · ;10)
22	16	dec0u 12387	. . . . 5 ⊢ (;10 · ;;100) = ;;;1000
23	18, 17, 22	mulcomli 10915	. . . 4 ⊢ (;;100 · ;10) = ;;;1000
24	23	oveq2i 7266	. . 3 ⊢ ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000)
25	19, 21, 24	3eqtr3i 2774	. 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000)
26		dfdec10 12369	. . . 4 ⊢ ;;𝐴𝐵_𝐶𝐷 = ((;10 · ;𝐴𝐵) + _𝐶𝐷)
27	26	oveq1i 7265	. . 3 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10)
28	1, 2	deccl 12381	. . . . . 6 ⊢ ;𝐴𝐵 ∈ ℕ₀
29	28	nn0cni 12175	. . . . 5 ⊢ ;𝐴𝐵 ∈ ℂ
30	18, 29	mulcli 10913	. . . 4 ⊢ (;10 · ;𝐴𝐵) ∈ ℂ
31	8	recni 10920	. . . 4 ⊢ _𝐶𝐷 ∈ ℂ
32	30, 31, 18	adddiri 10919	. . 3 ⊢ (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10))
33	28, 4, 6	dfdec100 31046	. . . 4 ⊢ ;;;𝐴𝐵𝐶𝐷 = ((;;100 · ;𝐴𝐵) + ;𝐶𝐷)
34	14	dec0u 12387	. . . . . . 7 ⊢ (;10 · ;10) = ;;100
35	34	oveq1i 7265	. . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = (;;100 · ;𝐴𝐵)
36	18, 18, 29	mul32i 11101	. . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10)
37	35, 36	eqtr3i 2768	. . . . 5 ⊢ (;;100 · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10)
38	4, 6	dpmul10 31071	. . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷
39		dpval 31066	. . . . . . . 8 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷)
40	4, 6, 39	mp2an 688	. . . . . . 7 ⊢ (𝐶.𝐷) = _𝐶𝐷
41	40	oveq1i 7265	. . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = (_𝐶𝐷 · ;10)
42	38, 41	eqtr3i 2768	. . . . 5 ⊢ ;𝐶𝐷 = (_𝐶𝐷 · ;10)
43	37, 42	oveq12i 7267	. . . 4 ⊢ ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10))
44	33, 43	eqtr2i 2767	. . 3 ⊢ (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) = ;;;𝐴𝐵𝐶𝐷
45	27, 32, 44	3eqtri 2770	. 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ;;;𝐴𝐵𝐶𝐷
46	25, 45	eqtr3i 2768	1 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2108 (class class class)co 7255 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 ℕ₀cn0 12163 cdc 12366 cdp2 31047 .cdp 31064

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-dec 12367 df-dp2 31048 df-dp 31065

This theorem is referenced by: dpmul4 31090

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