dpmul10 - Mathbox for Thierry Arnoux

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Theorem dpmul10 32639

Description: Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.)

**Hypotheses**
Ref	Expression
dpval2.a	⊢ 𝐴 ∈ ℕ₀
dpval2.b	⊢ 𝐵 ∈ ℝ

**Assertion**
Ref	Expression
dpmul10	⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵

**Proof of Theorem dpmul10**
Step	Hyp	Ref	Expression
1		dpval2.b	. . . . 5 ⊢ 𝐵 ∈ ℝ
2	1	recni 11266	. . . 4 ⊢ 𝐵 ∈ ℂ
3		10nn 12731	. . . . 5 ⊢ ;10 ∈ ℕ
4	3	nncni 12260	. . . 4 ⊢ ;10 ∈ ℂ
5	3	nnne0i 12290	. . . 4 ⊢ ;10 ≠ 0
6	2, 4, 5	divcan2i 11995	. . 3 ⊢ (;10 · (𝐵 / ;10)) = 𝐵
7	6	oveq2i 7437	. 2 ⊢ ((;10 · 𝐴) + (;10 · (𝐵 / ;10))) = ((;10 · 𝐴) + 𝐵)
8		dpval2.a	. . . . 5 ⊢ 𝐴 ∈ ℕ₀
9	8, 1	dpval2 32637	. . . 4 ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10))
10	9	oveq2i 7437	. . 3 ⊢ (;10 · (𝐴.𝐵)) = (;10 · (𝐴 + (𝐵 / ;10)))
11		dpcl 32635	. . . . . 6 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
12	8, 1, 11	mp2an 690	. . . . 5 ⊢ (𝐴.𝐵) ∈ ℝ
13	12	recni 11266	. . . 4 ⊢ (𝐴.𝐵) ∈ ℂ
14	4, 13	mulcomi 11260	. . 3 ⊢ (;10 · (𝐴.𝐵)) = ((𝐴.𝐵) · ;10)
15	8	nn0cni 12522	. . . 4 ⊢ 𝐴 ∈ ℂ
16	2, 4, 5	divcli 11994	. . . 4 ⊢ (𝐵 / ;10) ∈ ℂ
17	4, 15, 16	adddii 11264	. . 3 ⊢ (;10 · (𝐴 + (𝐵 / ;10))) = ((;10 · 𝐴) + (;10 · (𝐵 / ;10)))
18	10, 14, 17	3eqtr3i 2764	. 2 ⊢ ((𝐴.𝐵) · ;10) = ((;10 · 𝐴) + (;10 · (𝐵 / ;10)))
19		dfdec10 12718	. 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
20	7, 18, 19	3eqtr4i 2766	1 ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7426 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 / cdiv 11909 ℕ₀cn0 12510 cdc 12715 .cdp 32632

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-dec 12716 df-dp2 32616 df-dp 32633

This theorem is referenced by: decdiv10 32640 dpmul100 32641 dp3mul10 32642 dpmul1000 32643 dpmul 32657 dpmul4 32658

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