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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpfrac1 | Structured version Visualization version GIF version |
Description: Prove a simple equivalence involving the decimal point. See df-dp 31570 and dpcl 31572. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.) |
Ref | Expression |
---|---|
dpfrac1 | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dp2 31553 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
2 | dpval 31571 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | |
3 | nn0cn 12382 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ) | |
4 | recn 11100 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
5 | dfdec10 12580 | . . . . 5 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
6 | 5 | oveq1i 7362 | . . . 4 ⊢ (;𝐴𝐵 / ;10) = (((;10 · 𝐴) + 𝐵) / ;10) |
7 | 10re 12596 | . . . . . . . . 9 ⊢ ;10 ∈ ℝ | |
8 | 7 | recni 11128 | . . . . . . . 8 ⊢ ;10 ∈ ℂ |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ;10 ∈ ℂ) |
10 | id 22 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
11 | 9, 10 | mulcld 11134 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (;10 · 𝐴) ∈ ℂ) |
12 | 10pos 12594 | . . . . . . . . 9 ⊢ 0 < ;10 | |
13 | 7, 12 | gt0ne0ii 11650 | . . . . . . . 8 ⊢ ;10 ≠ 0 |
14 | 8, 13 | pm3.2i 472 | . . . . . . 7 ⊢ (;10 ∈ ℂ ∧ ;10 ≠ 0) |
15 | divdir 11797 | . . . . . . 7 ⊢ (((;10 · 𝐴) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (;10 ∈ ℂ ∧ ;10 ≠ 0)) → (((;10 · 𝐴) + 𝐵) / ;10) = (((;10 · 𝐴) / ;10) + (𝐵 / ;10))) | |
16 | 14, 15 | mp3an3 1451 | . . . . . 6 ⊢ (((;10 · 𝐴) ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) + 𝐵) / ;10) = (((;10 · 𝐴) / ;10) + (𝐵 / ;10))) |
17 | 11, 16 | sylan 581 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) + 𝐵) / ;10) = (((;10 · 𝐴) / ;10) + (𝐵 / ;10))) |
18 | divcan3 11798 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ ;10 ∈ ℂ ∧ ;10 ≠ 0) → ((;10 · 𝐴) / ;10) = 𝐴) | |
19 | 8, 13, 18 | mp3an23 1454 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((;10 · 𝐴) / ;10) = 𝐴) |
20 | 19 | oveq1d 7367 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (((;10 · 𝐴) / ;10) + (𝐵 / ;10)) = (𝐴 + (𝐵 / ;10))) |
21 | 20 | adantr 482 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) / ;10) + (𝐵 / ;10)) = (𝐴 + (𝐵 / ;10))) |
22 | 17, 21 | eqtrd 2778 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) + 𝐵) / ;10) = (𝐴 + (𝐵 / ;10))) |
23 | 6, 22 | eqtrid 2790 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (;𝐴𝐵 / ;10) = (𝐴 + (𝐵 / ;10))) |
24 | 3, 4, 23 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (;𝐴𝐵 / ;10) = (𝐴 + (𝐵 / ;10))) |
25 | 1, 2, 24 | 3eqtr4a 2804 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 (class class class)co 7352 ℂcc 11008 ℝcr 11009 0cc0 11010 1c1 11011 + caddc 11013 · cmul 11015 / cdiv 11771 ℕ0cn0 12372 ;cdc 12577 _cdp2 31552 .cdp 31569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-dec 12578 df-dp2 31553 df-dp 31570 |
This theorem is referenced by: (None) |
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