dpmul10 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dpmul10		Structured version Visualization version GIF version

Theorem dpmul10 32563

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 11229	. . . 4 ⊢ 𝐵 ∈ ℂ
3		10nn 12694	. . . . 5 ⊢ ;10 ∈ ℕ
4	3	nncni 12223	. . . 4 ⊢ ;10 ∈ ℂ
5	3	nnne0i 12253	. . . 4 ⊢ ;10 ≠ 0
6	2, 4, 5	divcan2i 11958	. . 3 ⊢ (;10 · (𝐵 / ;10)) = 𝐵
7	6	oveq2i 7415	. 2 ⊢ ((;10 · 𝐴) + (;10 · (𝐵 / ;10))) = ((;10 · 𝐴) + 𝐵)
8		dpval2.a	. . . . 5 ⊢ 𝐴 ∈ ℕ₀
9	8, 1	dpval2 32561	. . . 4 ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10))
10	9	oveq2i 7415	. . 3 ⊢ (;10 · (𝐴.𝐵)) = (;10 · (𝐴 + (𝐵 / ;10)))
11		dpcl 32559	. . . . . 6 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
12	8, 1, 11	mp2an 689	. . . . 5 ⊢ (𝐴.𝐵) ∈ ℝ
13	12	recni 11229	. . . 4 ⊢ (𝐴.𝐵) ∈ ℂ
14	4, 13	mulcomi 11223	. . 3 ⊢ (;10 · (𝐴.𝐵)) = ((𝐴.𝐵) · ;10)
15	8	nn0cni 12485	. . . 4 ⊢ 𝐴 ∈ ℂ
16	2, 4, 5	divcli 11957	. . . 4 ⊢ (𝐵 / ;10) ∈ ℂ
17	4, 15, 16	adddii 11227	. . 3 ⊢ (;10 · (𝐴 + (𝐵 / ;10))) = ((;10 · 𝐴) + (;10 · (𝐵 / ;10)))
18	10, 14, 17	3eqtr3i 2762	. 2 ⊢ ((𝐴.𝐵) · ;10) = ((;10 · 𝐴) + (;10 · (𝐵 / ;10)))
19		dfdec10 12681	. 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
20	7, 18, 19	3eqtr4i 2764	1 ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7404 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 / cdiv 11872 ℕ₀cn0 12473 cdc 12678 .cdp 32556

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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-dec 12679 df-dp2 32540 df-dp 32557

This theorem is referenced by: decdiv10 32564 dpmul100 32565 dp3mul10 32566 dpmul1000 32567 dpmul 32581 dpmul4 32582

W3C validator