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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 30565 and dpcl 30567. (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 30548 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
2 | dpval 30566 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | |
3 | nn0cn 11908 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ) | |
4 | recn 10627 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
5 | dfdec10 12102 | . . . . 5 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
6 | 5 | oveq1i 7166 | . . . 4 ⊢ (;𝐴𝐵 / ;10) = (((;10 · 𝐴) + 𝐵) / ;10) |
7 | 10re 12118 | . . . . . . . . 9 ⊢ ;10 ∈ ℝ | |
8 | 7 | recni 10655 | . . . . . . . 8 ⊢ ;10 ∈ ℂ |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ;10 ∈ ℂ) |
10 | id 22 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
11 | 9, 10 | mulcld 10661 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (;10 · 𝐴) ∈ ℂ) |
12 | 10pos 12116 | . . . . . . . . 9 ⊢ 0 < ;10 | |
13 | 7, 12 | gt0ne0ii 11176 | . . . . . . . 8 ⊢ ;10 ≠ 0 |
14 | 8, 13 | pm3.2i 473 | . . . . . . 7 ⊢ (;10 ∈ ℂ ∧ ;10 ≠ 0) |
15 | divdir 11323 | . . . . . . 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 582 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) + 𝐵) / ;10) = (((;10 · 𝐴) / ;10) + (𝐵 / ;10))) |
18 | divcan3 11324 | . . . . . . . 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 7171 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (((;10 · 𝐴) / ;10) + (𝐵 / ;10)) = (𝐴 + (𝐵 / ;10))) |
21 | 20 | adantr 483 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) / ;10) + (𝐵 / ;10)) = (𝐴 + (𝐵 / ;10))) |
22 | 17, 21 | eqtrd 2856 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) + 𝐵) / ;10) = (𝐴 + (𝐵 / ;10))) |
23 | 6, 22 | syl5eq 2868 | . . 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 2882 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 / cdiv 11297 ℕ0cn0 11898 ;cdc 12099 _cdp2 30547 .cdp 30564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-dec 12100 df-dp2 30548 df-dp 30565 |
This theorem is referenced by: (None) |
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