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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 11260 | . . . . . 6 ⊢ 0 ∈ ℝ | |
2 | 10pos 12747 | . . . . . 6 ⊢ 0 < ;10 | |
3 | 1, 2 | gtneii 11370 | . . . . 5 ⊢ ;10 ≠ 0 |
4 | dpexpp1.a | . . . . . . . . . 10 ⊢ 𝐴 ∈ ℕ0 | |
5 | dpexpp1.b | . . . . . . . . . 10 ⊢ 𝐵 ∈ ℝ+ | |
6 | 4, 5 | rpdp2cl 32848 | . . . . . . . . 9 ⊢ _𝐴𝐵 ∈ ℝ+ |
7 | rpre 13040 | . . . . . . . . 9 ⊢ (_𝐴𝐵 ∈ ℝ+ → _𝐴𝐵 ∈ ℝ) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ _𝐴𝐵 ∈ ℝ |
9 | 8 | recni 11272 | . . . . . . 7 ⊢ _𝐴𝐵 ∈ ℂ |
10 | 10re 12749 | . . . . . . . . . . 11 ⊢ ;10 ∈ ℝ | |
11 | 10, 2 | pm3.2i 470 | . . . . . . . . . 10 ⊢ (;10 ∈ ℝ ∧ 0 < ;10) |
12 | elrp 13033 | . . . . . . . . . 10 ⊢ (;10 ∈ ℝ+ ↔ (;10 ∈ ℝ ∧ 0 < ;10)) | |
13 | 11, 12 | mpbir 231 | . . . . . . . . 9 ⊢ ;10 ∈ ℝ+ |
14 | dpexpp1.p | . . . . . . . . 9 ⊢ 𝑃 ∈ ℤ | |
15 | rpexpcl 14117 | . . . . . . . . 9 ⊢ ((;10 ∈ ℝ+ ∧ 𝑃 ∈ ℤ) → (;10↑𝑃) ∈ ℝ+) | |
16 | 13, 14, 15 | mp2an 692 | . . . . . . . 8 ⊢ (;10↑𝑃) ∈ ℝ+ |
17 | rpcn 13042 | . . . . . . . 8 ⊢ ((;10↑𝑃) ∈ ℝ+ → (;10↑𝑃) ∈ ℂ) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (;10↑𝑃) ∈ ℂ |
19 | 9, 18 | mulcli 11265 | . . . . . 6 ⊢ (_𝐴𝐵 · (;10↑𝑃)) ∈ ℂ |
20 | 10nn0 12748 | . . . . . . 7 ⊢ ;10 ∈ ℕ0 | |
21 | 20 | nn0cni 12535 | . . . . . 6 ⊢ ;10 ∈ ℂ |
22 | 19, 21 | divcan1zi 12000 | . . . . 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 470 | . . . . . 6 ⊢ (;10 ∈ ℂ ∧ ;10 ≠ 0) |
25 | div23 11938 | . . . . . 6 ⊢ ((_𝐴𝐵 ∈ ℂ ∧ (;10↑𝑃) ∈ ℂ ∧ (;10 ∈ ℂ ∧ ;10 ≠ 0)) → ((_𝐴𝐵 · (;10↑𝑃)) / ;10) = ((_𝐴𝐵 / ;10) · (;10↑𝑃))) | |
26 | 9, 18, 24, 25 | mp3an 1460 | . . . . 5 ⊢ ((_𝐴𝐵 · (;10↑𝑃)) / ;10) = ((_𝐴𝐵 / ;10) · (;10↑𝑃)) |
27 | 26 | oveq1i 7440 | . . . 4 ⊢ (((_𝐴𝐵 · (;10↑𝑃)) / ;10) · ;10) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) |
28 | 23, 27 | eqtr3i 2764 | . . 3 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) |
29 | 9, 21, 3 | divcli 12006 | . . . 4 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ |
30 | 29, 18, 21 | mulassi 11269 | . . 3 ⊢ (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) = ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) |
31 | expp1z 14148 | . . . . . 6 ⊢ ((;10 ∈ ℂ ∧ ;10 ≠ 0 ∧ 𝑃 ∈ ℤ) → (;10↑(𝑃 + 1)) = ((;10↑𝑃) · ;10)) | |
32 | 21, 3, 14, 31 | mp3an 1460 | . . . . 5 ⊢ (;10↑(𝑃 + 1)) = ((;10↑𝑃) · ;10) |
33 | dpexpp1.1 | . . . . . 6 ⊢ (𝑃 + 1) = 𝑄 | |
34 | 33 | oveq2i 7441 | . . . . 5 ⊢ (;10↑(𝑃 + 1)) = (;10↑𝑄) |
35 | 32, 34 | eqtr3i 2764 | . . . 4 ⊢ ((;10↑𝑃) · ;10) = (;10↑𝑄) |
36 | 35 | oveq2i 7441 | . . 3 ⊢ ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄)) |
37 | 28, 30, 36 | 3eqtri 2766 | . 2 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄)) |
38 | 4, 5 | dpval3rp 32866 | . . 3 ⊢ (𝐴.𝐵) = _𝐴𝐵 |
39 | 38 | oveq1i 7440 | . 2 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = (_𝐴𝐵 · (;10↑𝑃)) |
40 | 0nn0 12538 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
41 | 40, 6 | dpval3rp 32866 | . . . 4 ⊢ (0._𝐴𝐵) = _0_𝐴𝐵 |
42 | 6 | dp20h 32845 | . . . 4 ⊢ _0_𝐴𝐵 = (_𝐴𝐵 / ;10) |
43 | 41, 42 | eqtri 2762 | . . 3 ⊢ (0._𝐴𝐵) = (_𝐴𝐵 / ;10) |
44 | 43 | oveq1i 7440 | . 2 ⊢ ((0._𝐴𝐵) · (;10↑𝑄)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄)) |
45 | 37, 39, 44 | 3eqtr4i 2772 | 1 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 class class class wbr 5147 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 / cdiv 11917 ℕ0cn0 12523 ℤcz 12610 ;cdc 12730 ℝ+crp 13031 ↑cexp 14098 _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-cnex 11208 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-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-seq 14039 df-exp 14099 df-dp2 32838 df-dp 32855 |
This theorem is referenced by: 0dp2dp 32875 hgt750lemd 34641 hgt750lem 34644 |
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