Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpexpp1 | Structured version Visualization version GIF version |
Description: Add one zero to the mantisse, and a one to the exponent in a scientific notation. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
Ref | Expression |
---|---|
dpexpp1.a | ⊢ 𝐴 ∈ ℕ0 |
dpexpp1.b | ⊢ 𝐵 ∈ ℝ+ |
dpexpp1.1 | ⊢ (𝑃 + 1) = 𝑄 |
dpexpp1.p | ⊢ 𝑃 ∈ ℤ |
dpexpp1.q | ⊢ 𝑄 ∈ ℤ |
Ref | Expression |
---|---|
dpexpp1 | ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10800 | . . . . . 6 ⊢ 0 ∈ ℝ | |
2 | 10pos 12275 | . . . . . 6 ⊢ 0 < ;10 | |
3 | 1, 2 | gtneii 10909 | . . . . 5 ⊢ ;10 ≠ 0 |
4 | dpexpp1.a | . . . . . . . . . 10 ⊢ 𝐴 ∈ ℕ0 | |
5 | dpexpp1.b | . . . . . . . . . 10 ⊢ 𝐵 ∈ ℝ+ | |
6 | 4, 5 | rpdp2cl 30830 | . . . . . . . . 9 ⊢ _𝐴𝐵 ∈ ℝ+ |
7 | rpre 12559 | . . . . . . . . 9 ⊢ (_𝐴𝐵 ∈ ℝ+ → _𝐴𝐵 ∈ ℝ) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ _𝐴𝐵 ∈ ℝ |
9 | 8 | recni 10812 | . . . . . . 7 ⊢ _𝐴𝐵 ∈ ℂ |
10 | 10re 12277 | . . . . . . . . . . 11 ⊢ ;10 ∈ ℝ | |
11 | 10, 2 | pm3.2i 474 | . . . . . . . . . 10 ⊢ (;10 ∈ ℝ ∧ 0 < ;10) |
12 | elrp 12553 | . . . . . . . . . 10 ⊢ (;10 ∈ ℝ+ ↔ (;10 ∈ ℝ ∧ 0 < ;10)) | |
13 | 11, 12 | mpbir 234 | . . . . . . . . 9 ⊢ ;10 ∈ ℝ+ |
14 | dpexpp1.p | . . . . . . . . 9 ⊢ 𝑃 ∈ ℤ | |
15 | rpexpcl 13619 | . . . . . . . . 9 ⊢ ((;10 ∈ ℝ+ ∧ 𝑃 ∈ ℤ) → (;10↑𝑃) ∈ ℝ+) | |
16 | 13, 14, 15 | mp2an 692 | . . . . . . . 8 ⊢ (;10↑𝑃) ∈ ℝ+ |
17 | rpcn 12561 | . . . . . . . 8 ⊢ ((;10↑𝑃) ∈ ℝ+ → (;10↑𝑃) ∈ ℂ) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (;10↑𝑃) ∈ ℂ |
19 | 9, 18 | mulcli 10805 | . . . . . 6 ⊢ (_𝐴𝐵 · (;10↑𝑃)) ∈ ℂ |
20 | 10nn0 12276 | . . . . . . 7 ⊢ ;10 ∈ ℕ0 | |
21 | 20 | nn0cni 12067 | . . . . . 6 ⊢ ;10 ∈ ℂ |
22 | 19, 21 | divcan1zi 11533 | . . . . 5 ⊢ (;10 ≠ 0 → (((_𝐴𝐵 · (;10↑𝑃)) / ;10) · ;10) = (_𝐴𝐵 · (;10↑𝑃))) |
23 | 3, 22 | ax-mp 5 | . . . 4 ⊢ (((_𝐴𝐵 · (;10↑𝑃)) / ;10) · ;10) = (_𝐴𝐵 · (;10↑𝑃)) |
24 | 21, 3 | pm3.2i 474 | . . . . . 6 ⊢ (;10 ∈ ℂ ∧ ;10 ≠ 0) |
25 | div23 11474 | . . . . . 6 ⊢ ((_𝐴𝐵 ∈ ℂ ∧ (;10↑𝑃) ∈ ℂ ∧ (;10 ∈ ℂ ∧ ;10 ≠ 0)) → ((_𝐴𝐵 · (;10↑𝑃)) / ;10) = ((_𝐴𝐵 / ;10) · (;10↑𝑃))) | |
26 | 9, 18, 24, 25 | mp3an 1463 | . . . . 5 ⊢ ((_𝐴𝐵 · (;10↑𝑃)) / ;10) = ((_𝐴𝐵 / ;10) · (;10↑𝑃)) |
27 | 26 | oveq1i 7201 | . . . 4 ⊢ (((_𝐴𝐵 · (;10↑𝑃)) / ;10) · ;10) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) |
28 | 23, 27 | eqtr3i 2761 | . . 3 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) |
29 | 9, 21, 3 | divcli 11539 | . . . 4 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ |
30 | 29, 18, 21 | mulassi 10809 | . . 3 ⊢ (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) = ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) |
31 | expp1z 13649 | . . . . . 6 ⊢ ((;10 ∈ ℂ ∧ ;10 ≠ 0 ∧ 𝑃 ∈ ℤ) → (;10↑(𝑃 + 1)) = ((;10↑𝑃) · ;10)) | |
32 | 21, 3, 14, 31 | mp3an 1463 | . . . . 5 ⊢ (;10↑(𝑃 + 1)) = ((;10↑𝑃) · ;10) |
33 | dpexpp1.1 | . . . . . 6 ⊢ (𝑃 + 1) = 𝑄 | |
34 | 33 | oveq2i 7202 | . . . . 5 ⊢ (;10↑(𝑃 + 1)) = (;10↑𝑄) |
35 | 32, 34 | eqtr3i 2761 | . . . 4 ⊢ ((;10↑𝑃) · ;10) = (;10↑𝑄) |
36 | 35 | oveq2i 7202 | . . 3 ⊢ ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄)) |
37 | 28, 30, 36 | 3eqtri 2763 | . 2 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄)) |
38 | 4, 5 | dpval3rp 30848 | . . 3 ⊢ (𝐴.𝐵) = _𝐴𝐵 |
39 | 38 | oveq1i 7201 | . 2 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = (_𝐴𝐵 · (;10↑𝑃)) |
40 | 0nn0 12070 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
41 | 40, 6 | dpval3rp 30848 | . . . 4 ⊢ (0._𝐴𝐵) = _0_𝐴𝐵 |
42 | 6 | dp20h 30827 | . . . 4 ⊢ _0_𝐴𝐵 = (_𝐴𝐵 / ;10) |
43 | 41, 42 | eqtri 2759 | . . 3 ⊢ (0._𝐴𝐵) = (_𝐴𝐵 / ;10) |
44 | 43 | oveq1i 7201 | . 2 ⊢ ((0._𝐴𝐵) · (;10↑𝑄)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄)) |
45 | 37, 39, 44 | 3eqtr4i 2769 | 1 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 class class class wbr 5039 (class class class)co 7191 ℂcc 10692 ℝcr 10693 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 < clt 10832 / cdiv 11454 ℕ0cn0 12055 ℤcz 12141 ;cdc 12258 ℝ+crp 12551 ↑cexp 13600 _cdp2 30819 .cdp 30836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-rp 12552 df-seq 13540 df-exp 13601 df-dp2 30820 df-dp 30837 |
This theorem is referenced by: 0dp2dp 30857 hgt750lemd 32294 hgt750lem 32297 |
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