dpmul1000 - Mathbox for Thierry Arnoux

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Theorem dpmul1000 32976

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 12442	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
4		dpmul1000.c	. . . . . . . . 9 ⊢ 𝐶 ∈ ℕ₀
5	4	nn0rei 12442	. . . . . . . 8 ⊢ 𝐶 ∈ ℝ
6		dpmul1000.d	. . . . . . . 8 ⊢ 𝐷 ∈ ℝ
7		dp2cl 32957	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ)
8	5, 6, 7	mp2an 693	. . . . . . 7 ⊢ _𝐶𝐷 ∈ ℝ
9		dp2cl 32957	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ _𝐶𝐷 ∈ ℝ) → _𝐵_𝐶𝐷 ∈ ℝ)
10	3, 8, 9	mp2an 693	. . . . . 6 ⊢ _𝐵_𝐶𝐷 ∈ ℝ
11		dpcl 32968	. . . . . 6 ⊢ ((𝐴 ∈ ℕ₀ ∧ _𝐵_𝐶𝐷 ∈ ℝ) → (𝐴._𝐵_𝐶𝐷) ∈ ℝ)
12	1, 10, 11	mp2an 693	. . . . 5 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℝ
13	12	recni 11153	. . . 4 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℂ
14		10nn0 12656	. . . . . 6 ⊢ ;10 ∈ ℕ₀
15		0nn0 12446	. . . . . 6 ⊢ 0 ∈ ℕ₀
16	14, 15	deccl 12653	. . . . 5 ⊢ ;;100 ∈ ℕ₀
17	16	nn0cni 12443	. . . 4 ⊢ ;;100 ∈ ℂ
18	14	nn0cni 12443	. . . 4 ⊢ ;10 ∈ ℂ
19	13, 17, 18	mulassi 11150	. . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10))
20	1, 2, 8	dpmul100 32974	. . . 4 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;100) = ;;𝐴𝐵_𝐶𝐷
21	20	oveq1i 7371	. . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = (;;𝐴𝐵_𝐶𝐷 · ;10)
22	16	dec0u 12659	. . . . 5 ⊢ (;10 · ;;100) = ;;;1000
23	18, 17, 22	mulcomli 11148	. . . 4 ⊢ (;;100 · ;10) = ;;;1000
24	23	oveq2i 7372	. . 3 ⊢ ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000)
25	19, 21, 24	3eqtr3i 2768	. 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000)
26		dfdec10 12641	. . . 4 ⊢ ;;𝐴𝐵_𝐶𝐷 = ((;10 · ;𝐴𝐵) + _𝐶𝐷)
27	26	oveq1i 7371	. . 3 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10)
28	1, 2	deccl 12653	. . . . . 6 ⊢ ;𝐴𝐵 ∈ ℕ₀
29	28	nn0cni 12443	. . . . 5 ⊢ ;𝐴𝐵 ∈ ℂ
30	18, 29	mulcli 11146	. . . 4 ⊢ (;10 · ;𝐴𝐵) ∈ ℂ
31	8	recni 11153	. . . 4 ⊢ _𝐶𝐷 ∈ ℂ
32	30, 31, 18	adddiri 11152	. . 3 ⊢ (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10))
33	28, 4, 6	dfdec100 32921	. . . 4 ⊢ ;;;𝐴𝐵𝐶𝐷 = ((;;100 · ;𝐴𝐵) + ;𝐶𝐷)
34	14	dec0u 12659	. . . . . . 7 ⊢ (;10 · ;10) = ;;100
35	34	oveq1i 7371	. . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = (;;100 · ;𝐴𝐵)
36	18, 18, 29	mul32i 11336	. . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10)
37	35, 36	eqtr3i 2762	. . . . 5 ⊢ (;;100 · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10)
38	4, 6	dpmul10 32972	. . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷
39		dpval 32967	. . . . . . . 8 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷)
40	4, 6, 39	mp2an 693	. . . . . . 7 ⊢ (𝐶.𝐷) = _𝐶𝐷
41	40	oveq1i 7371	. . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = (_𝐶𝐷 · ;10)
42	38, 41	eqtr3i 2762	. . . . 5 ⊢ ;𝐶𝐷 = (_𝐶𝐷 · ;10)
43	37, 42	oveq12i 7373	. . . 4 ⊢ ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10))
44	33, 43	eqtr2i 2761	. . 3 ⊢ (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) = ;;;𝐴𝐵𝐶𝐷
45	27, 32, 44	3eqtri 2764	. 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ;;;𝐴𝐵𝐶𝐷
46	25, 45	eqtr3i 2762	1 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 (class class class)co 7361 ℝcr 11031 0cc0 11032 1c1 11033 + caddc 11035 · cmul 11037 ℕ₀cn0 12431 cdc 12638 cdp2 32948 .cdp 32965

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-dec 12639 df-dp2 32949 df-dp 32966

This theorem is referenced by: dpmul4 32991

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