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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 32855 and dpcl 32857. (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 32838 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
2 | dpval 32856 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | |
3 | nn0cn 12533 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ) | |
4 | recn 11242 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
5 | dfdec10 12733 | . . . . 5 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
6 | 5 | oveq1i 7440 | . . . 4 ⊢ (;𝐴𝐵 / ;10) = (((;10 · 𝐴) + 𝐵) / ;10) |
7 | 10re 12749 | . . . . . . . . 9 ⊢ ;10 ∈ ℝ | |
8 | 7 | recni 11272 | . . . . . . . 8 ⊢ ;10 ∈ ℂ |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ;10 ∈ ℂ) |
10 | id 22 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
11 | 9, 10 | mulcld 11278 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (;10 · 𝐴) ∈ ℂ) |
12 | 10pos 12747 | . . . . . . . . 9 ⊢ 0 < ;10 | |
13 | 7, 12 | gt0ne0ii 11796 | . . . . . . . 8 ⊢ ;10 ≠ 0 |
14 | 8, 13 | pm3.2i 470 | . . . . . . 7 ⊢ (;10 ∈ ℂ ∧ ;10 ≠ 0) |
15 | divdir 11944 | . . . . . . 7 ⊢ (((;10 · 𝐴) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (;10 ∈ ℂ ∧ ;10 ≠ 0)) → (((;10 · 𝐴) + 𝐵) / ;10) = (((;10 · 𝐴) / ;10) + (𝐵 / ;10))) | |
16 | 14, 15 | mp3an3 1449 | . . . . . 6 ⊢ (((;10 · 𝐴) ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) + 𝐵) / ;10) = (((;10 · 𝐴) / ;10) + (𝐵 / ;10))) |
17 | 11, 16 | sylan 580 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) + 𝐵) / ;10) = (((;10 · 𝐴) / ;10) + (𝐵 / ;10))) |
18 | divcan3 11945 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ ;10 ∈ ℂ ∧ ;10 ≠ 0) → ((;10 · 𝐴) / ;10) = 𝐴) | |
19 | 8, 13, 18 | mp3an23 1452 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((;10 · 𝐴) / ;10) = 𝐴) |
20 | 19 | oveq1d 7445 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (((;10 · 𝐴) / ;10) + (𝐵 / ;10)) = (𝐴 + (𝐵 / ;10))) |
21 | 20 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) / ;10) + (𝐵 / ;10)) = (𝐴 + (𝐵 / ;10))) |
22 | 17, 21 | eqtrd 2774 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) + 𝐵) / ;10) = (𝐴 + (𝐵 / ;10))) |
23 | 6, 22 | eqtrid 2786 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (;𝐴𝐵 / ;10) = (𝐴 + (𝐵 / ;10))) |
24 | 3, 4, 23 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (;𝐴𝐵 / ;10) = (𝐴 + (𝐵 / ;10))) |
25 | 1, 2, 24 | 3eqtr4a 2800 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 / cdiv 11917 ℕ0cn0 12523 ;cdc 12730 _cdp2 32837 .cdp 32854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-dec 12731 df-dp2 32838 df-dp 32855 |
This theorem is referenced by: (None) |
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