dpfrac1 - Mathbox for Thierry Arnoux

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Theorem dpfrac1 32636

Description: Prove a simple equivalence involving the decimal point. See df-dp 32633 and dpcl 32635. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.)

**Assertion**
Ref	Expression
dpfrac1	⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10))

**Proof of Theorem dpfrac1**
Step	Hyp	Ref	Expression
1		df-dp2 32616	. 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
2		dpval 32634	. 2 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)
3		nn0cn 12520	. . 3 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ ℂ)
4		recn 11236	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
5		dfdec10 12718	. . . . 5 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
6	5	oveq1i 7436	. . . 4 ⊢ (;𝐴𝐵 / ;10) = (((;10 · 𝐴) + 𝐵) / ;10)
7		10re 12734	. . . . . . . . 9 ⊢ ;10 ∈ ℝ
8	7	recni 11266	. . . . . . . 8 ⊢ ;10 ∈ ℂ
9	8	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ;10 ∈ ℂ)
10		id 22	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
11	9, 10	mulcld 11272	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (;10 · 𝐴) ∈ ℂ)
12		10pos 12732	. . . . . . . . 9 ⊢ 0 < ;10
13	7, 12	gt0ne0ii 11788	. . . . . . . 8 ⊢ ;10 ≠ 0
14	8, 13	pm3.2i 469	. . . . . . 7 ⊢ (;10 ∈ ℂ ∧ ;10 ≠ 0)
15		divdir 11935	. . . . . . 7 ⊢ (((;10 · 𝐴) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (;10 ∈ ℂ ∧ ;10 ≠ 0)) → (((;10 · 𝐴) + 𝐵) / ;10) = (((;10 · 𝐴) / ;10) + (𝐵 / ;10)))
16	14, 15	mp3an3 1446	. . . . . 6 ⊢ (((;10 · 𝐴) ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) + 𝐵) / ;10) = (((;10 · 𝐴) / ;10) + (𝐵 / ;10)))
17	11, 16	sylan 578	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) + 𝐵) / ;10) = (((;10 · 𝐴) / ;10) + (𝐵 / ;10)))
18		divcan3 11936	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ ;10 ∈ ℂ ∧ ;10 ≠ 0) → ((;10 · 𝐴) / ;10) = 𝐴)
19	8, 13, 18	mp3an23 1449	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((;10 · 𝐴) / ;10) = 𝐴)
20	19	oveq1d 7441	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (((;10 · 𝐴) / ;10) + (𝐵 / ;10)) = (𝐴 + (𝐵 / ;10)))
21	20	adantr 479	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) / ;10) + (𝐵 / ;10)) = (𝐴 + (𝐵 / ;10)))
22	17, 21	eqtrd 2768	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) + 𝐵) / ;10) = (𝐴 + (𝐵 / ;10)))
23	6, 22	eqtrid 2780	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (;𝐴𝐵 / ;10) = (𝐴 + (𝐵 / ;10)))
24	3, 4, 23	syl2an 594	. 2 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (;𝐴𝐵 / ;10) = (𝐴 + (𝐵 / ;10)))
25	1, 2, 24	3eqtr4a 2794	1 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 / cdiv 11909 ℕ₀cn0 12510 cdc 12715 cdp2 32615 .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: (None)

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