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 32056

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 11227	. . . 4 ⊢ 𝐵 ∈ ℂ
3		10nn 12692	. . . . 5 ⊢ ;10 ∈ ℕ
4	3	nncni 12221	. . . 4 ⊢ ;10 ∈ ℂ
5	3	nnne0i 12251	. . . 4 ⊢ ;10 ≠ 0
6	2, 4, 5	divcan2i 11956	. . 3 ⊢ (;10 · (𝐵 / ;10)) = 𝐵
7	6	oveq2i 7419	. 2 ⊢ ((;10 · 𝐴) + (;10 · (𝐵 / ;10))) = ((;10 · 𝐴) + 𝐵)
8		dpval2.a	. . . . 5 ⊢ 𝐴 ∈ ℕ₀
9	8, 1	dpval2 32054	. . . 4 ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10))
10	9	oveq2i 7419	. . 3 ⊢ (;10 · (𝐴.𝐵)) = (;10 · (𝐴 + (𝐵 / ;10)))
11		dpcl 32052	. . . . . 6 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
12	8, 1, 11	mp2an 690	. . . . 5 ⊢ (𝐴.𝐵) ∈ ℝ
13	12	recni 11227	. . . 4 ⊢ (𝐴.𝐵) ∈ ℂ
14	4, 13	mulcomi 11221	. . 3 ⊢ (;10 · (𝐴.𝐵)) = ((𝐴.𝐵) · ;10)
15	8	nn0cni 12483	. . . 4 ⊢ 𝐴 ∈ ℂ
16	2, 4, 5	divcli 11955	. . . 4 ⊢ (𝐵 / ;10) ∈ ℂ
17	4, 15, 16	adddii 11225	. . 3 ⊢ (;10 · (𝐴 + (𝐵 / ;10))) = ((;10 · 𝐴) + (;10 · (𝐵 / ;10)))
18	10, 14, 17	3eqtr3i 2768	. 2 ⊢ ((𝐴.𝐵) · ;10) = ((;10 · 𝐴) + (;10 · (𝐵 / ;10)))
19		dfdec10 12679	. 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
20	7, 18, 19	3eqtr4i 2770	1 ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2106 (class class class)co 7408 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 / cdiv 11870 ℕ₀cn0 12471 cdc 12676 .cdp 32049

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-dec 12677 df-dp2 32033 df-dp 32050

This theorem is referenced by: decdiv10 32057 dpmul100 32058 dp3mul10 32059 dpmul1000 32060 dpmul 32074 dpmul4 32075

W3C validator