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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 32871 and dpcl 32873. (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 32854 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
| 2 | dpval 32872 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | |
| 3 | nn0cn 12536 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ) | |
| 4 | recn 11245 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
| 5 | dfdec10 12736 | . . . . 5 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
| 6 | 5 | oveq1i 7441 | . . . 4 ⊢ (;𝐴𝐵 / ;10) = (((;10 · 𝐴) + 𝐵) / ;10) |
| 7 | 10re 12752 | . . . . . . . . 9 ⊢ ;10 ∈ ℝ | |
| 8 | 7 | recni 11275 | . . . . . . . 8 ⊢ ;10 ∈ ℂ |
| 9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ;10 ∈ ℂ) |
| 10 | id 22 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
| 11 | 9, 10 | mulcld 11281 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (;10 · 𝐴) ∈ ℂ) |
| 12 | 10pos 12750 | . . . . . . . . 9 ⊢ 0 < ;10 | |
| 13 | 7, 12 | gt0ne0ii 11799 | . . . . . . . 8 ⊢ ;10 ≠ 0 |
| 14 | 8, 13 | pm3.2i 470 | . . . . . . 7 ⊢ (;10 ∈ ℂ ∧ ;10 ≠ 0) |
| 15 | divdir 11947 | . . . . . . 7 ⊢ (((;10 · 𝐴) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (;10 ∈ ℂ ∧ ;10 ≠ 0)) → (((;10 · 𝐴) + 𝐵) / ;10) = (((;10 · 𝐴) / ;10) + (𝐵 / ;10))) | |
| 16 | 14, 15 | mp3an3 1452 | . . . . . 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 11948 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ ;10 ∈ ℂ ∧ ;10 ≠ 0) → ((;10 · 𝐴) / ;10) = 𝐴) | |
| 19 | 8, 13, 18 | mp3an23 1455 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((;10 · 𝐴) / ;10) = 𝐴) |
| 20 | 19 | oveq1d 7446 | . . . . . 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 2777 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((;10 · 𝐴) + 𝐵) / ;10) = (𝐴 + (𝐵 / ;10))) |
| 23 | 6, 22 | eqtrid 2789 | . . 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 2803 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 / cdiv 11920 ℕ0cn0 12526 ;cdc 12733 _cdp2 32853 .cdp 32870 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12734 df-dp2 32854 df-dp 32871 |
| This theorem is referenced by: (None) |
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