dpmul1000 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dpmul1000		Structured version Visualization version GIF version

Theorem dpmul1000 33159

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 12515	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
4		dpmul1000.c	. . . . . . . . 9 ⊢ 𝐶 ∈ ℕ₀
5	4	nn0rei 12515	. . . . . . . 8 ⊢ 𝐶 ∈ ℝ
6		dpmul1000.d	. . . . . . . 8 ⊢ 𝐷 ∈ ℝ
7		dp2cl 33140	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ)
8	5, 6, 7	mp2an 704	. . . . . . 7 ⊢ _𝐶𝐷 ∈ ℝ
9		dp2cl 33140	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ _𝐶𝐷 ∈ ℝ) → _𝐵_𝐶𝐷 ∈ ℝ)
10	3, 8, 9	mp2an 704	. . . . . 6 ⊢ _𝐵_𝐶𝐷 ∈ ℝ
11		dpcl 33151	. . . . . 6 ⊢ ((𝐴 ∈ ℕ₀ ∧ _𝐵_𝐶𝐷 ∈ ℝ) → (𝐴._𝐵_𝐶𝐷) ∈ ℝ)
12	1, 10, 11	mp2an 704	. . . . 5 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℝ
13	12	recni 11223	. . . 4 ⊢ (𝐴._𝐵_𝐶𝐷) ∈ ℂ
14		10nn0 12733	. . . . . 6 ⊢ ;10 ∈ ℕ₀
15		0nn0 12519	. . . . . 6 ⊢ 0 ∈ ℕ₀
16	14, 15	deccl 12726	. . . . 5 ⊢ ;;100 ∈ ℕ₀
17	16	nn0cni 12516	. . . 4 ⊢ ;;100 ∈ ℂ
18	14	nn0cni 12516	. . . 4 ⊢ ;10 ∈ ℂ
19	13, 17, 18	mulassi 11220	. . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10))
20	1, 2, 8	dpmul100 33157	. . . 4 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;100) = ;;𝐴𝐵_𝐶𝐷
21	20	oveq1i 7421	. . 3 ⊢ (((𝐴._𝐵_𝐶𝐷) · ;;100) · ;10) = (;;𝐴𝐵_𝐶𝐷 · ;10)
22	16	dec0u 12737	. . . . 5 ⊢ (;10 · ;;100) = ;;;1000
23	18, 17, 22	mulcomli 11218	. . . 4 ⊢ (;;100 · ;10) = ;;;1000
24	23	oveq2i 7422	. . 3 ⊢ ((𝐴._𝐵_𝐶𝐷) · (;;100 · ;10)) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000)
25	19, 21, 24	3eqtr3i 2800	. 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ((𝐴._𝐵_𝐶𝐷) · ;;;1000)
26		dfdec10 12714	. . . 4 ⊢ ;;𝐴𝐵_𝐶𝐷 = ((;10 · ;𝐴𝐵) + _𝐶𝐷)
27	26	oveq1i 7421	. . 3 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10)
28	1, 2	deccl 12726	. . . . . 6 ⊢ ;𝐴𝐵 ∈ ℕ₀
29	28	nn0cni 12516	. . . . 5 ⊢ ;𝐴𝐵 ∈ ℂ
30	18, 29	mulcli 11216	. . . 4 ⊢ (;10 · ;𝐴𝐵) ∈ ℂ
31	8	recni 11223	. . . 4 ⊢ _𝐶𝐷 ∈ ℂ
32	30, 31, 18	adddiri 11222	. . 3 ⊢ (((;10 · ;𝐴𝐵) + _𝐶𝐷) · ;10) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10))
33	28, 4, 6	dfdec100 33115	. . . 4 ⊢ ;;;𝐴𝐵𝐶𝐷 = ((;;100 · ;𝐴𝐵) + ;𝐶𝐷)
34	14	dec0u 12737	. . . . . . 7 ⊢ (;10 · ;10) = ;;100
35	34	oveq1i 7421	. . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = (;;100 · ;𝐴𝐵)
36	18, 18, 29	mul32i 11406	. . . . . 6 ⊢ ((;10 · ;10) · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10)
37	35, 36	eqtr3i 2794	. . . . 5 ⊢ (;;100 · ;𝐴𝐵) = ((;10 · ;𝐴𝐵) · ;10)
38	4, 6	dpmul10 33155	. . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷
39		dpval 33150	. . . . . . . 8 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷)
40	4, 6, 39	mp2an 704	. . . . . . 7 ⊢ (𝐶.𝐷) = _𝐶𝐷
41	40	oveq1i 7421	. . . . . 6 ⊢ ((𝐶.𝐷) · ;10) = (_𝐶𝐷 · ;10)
42	38, 41	eqtr3i 2794	. . . . 5 ⊢ ;𝐶𝐷 = (_𝐶𝐷 · ;10)
43	37, 42	oveq12i 7423	. . . 4 ⊢ ((;;100 · ;𝐴𝐵) + ;𝐶𝐷) = (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10))
44	33, 43	eqtr2i 2793	. . 3 ⊢ (((;10 · ;𝐴𝐵) · ;10) + (_𝐶𝐷 · ;10)) = ;;;𝐴𝐵𝐶𝐷
45	27, 32, 44	3eqtri 2796	. 2 ⊢ (;;𝐴𝐵_𝐶𝐷 · ;10) = ;;;𝐴𝐵𝐶𝐷
46	25, 45	eqtr3i 2794	1 ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷

Colors of variables: wff setvar class

Syntax hints: = wceq 1567 ∈ wcel 2149 (class class class)co 7411 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 · cmul 11105 ℕ₀cn0 12504 cdc 12711 cdp2 33131 .cdp 33148

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-dec 12712 df-dp2 33132 df-dp 33149

This theorem is referenced by: dpmul4 33174

W3C validator