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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 11198 | . . . . . 6 ⊢ 0 ∈ ℝ | |
2 | 10pos 12676 | . . . . . 6 ⊢ 0 < ;10 | |
3 | 1, 2 | gtneii 11308 | . . . . 5 ⊢ ;10 ≠ 0 |
4 | dpexpp1.a | . . . . . . . . . 10 ⊢ 𝐴 ∈ ℕ0 | |
5 | dpexpp1.b | . . . . . . . . . 10 ⊢ 𝐵 ∈ ℝ+ | |
6 | 4, 5 | rpdp2cl 31919 | . . . . . . . . 9 ⊢ _𝐴𝐵 ∈ ℝ+ |
7 | rpre 12964 | . . . . . . . . 9 ⊢ (_𝐴𝐵 ∈ ℝ+ → _𝐴𝐵 ∈ ℝ) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ _𝐴𝐵 ∈ ℝ |
9 | 8 | recni 11210 | . . . . . . 7 ⊢ _𝐴𝐵 ∈ ℂ |
10 | 10re 12678 | . . . . . . . . . . 11 ⊢ ;10 ∈ ℝ | |
11 | 10, 2 | pm3.2i 471 | . . . . . . . . . 10 ⊢ (;10 ∈ ℝ ∧ 0 < ;10) |
12 | elrp 12958 | . . . . . . . . . 10 ⊢ (;10 ∈ ℝ+ ↔ (;10 ∈ ℝ ∧ 0 < ;10)) | |
13 | 11, 12 | mpbir 230 | . . . . . . . . 9 ⊢ ;10 ∈ ℝ+ |
14 | dpexpp1.p | . . . . . . . . 9 ⊢ 𝑃 ∈ ℤ | |
15 | rpexpcl 14028 | . . . . . . . . 9 ⊢ ((;10 ∈ ℝ+ ∧ 𝑃 ∈ ℤ) → (;10↑𝑃) ∈ ℝ+) | |
16 | 13, 14, 15 | mp2an 690 | . . . . . . . 8 ⊢ (;10↑𝑃) ∈ ℝ+ |
17 | rpcn 12966 | . . . . . . . 8 ⊢ ((;10↑𝑃) ∈ ℝ+ → (;10↑𝑃) ∈ ℂ) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (;10↑𝑃) ∈ ℂ |
19 | 9, 18 | mulcli 11203 | . . . . . 6 ⊢ (_𝐴𝐵 · (;10↑𝑃)) ∈ ℂ |
20 | 10nn0 12677 | . . . . . . 7 ⊢ ;10 ∈ ℕ0 | |
21 | 20 | nn0cni 12466 | . . . . . 6 ⊢ ;10 ∈ ℂ |
22 | 19, 21 | divcan1zi 11932 | . . . . 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 471 | . . . . . 6 ⊢ (;10 ∈ ℂ ∧ ;10 ≠ 0) |
25 | div23 11873 | . . . . . 6 ⊢ ((_𝐴𝐵 ∈ ℂ ∧ (;10↑𝑃) ∈ ℂ ∧ (;10 ∈ ℂ ∧ ;10 ≠ 0)) → ((_𝐴𝐵 · (;10↑𝑃)) / ;10) = ((_𝐴𝐵 / ;10) · (;10↑𝑃))) | |
26 | 9, 18, 24, 25 | mp3an 1461 | . . . . 5 ⊢ ((_𝐴𝐵 · (;10↑𝑃)) / ;10) = ((_𝐴𝐵 / ;10) · (;10↑𝑃)) |
27 | 26 | oveq1i 7403 | . . . 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 11938 | . . . 4 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ |
30 | 29, 18, 21 | mulassi 11207 | . . 3 ⊢ (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) = ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) |
31 | expp1z 14059 | . . . . . 6 ⊢ ((;10 ∈ ℂ ∧ ;10 ≠ 0 ∧ 𝑃 ∈ ℤ) → (;10↑(𝑃 + 1)) = ((;10↑𝑃) · ;10)) | |
32 | 21, 3, 14, 31 | mp3an 1461 | . . . . 5 ⊢ (;10↑(𝑃 + 1)) = ((;10↑𝑃) · ;10) |
33 | dpexpp1.1 | . . . . . 6 ⊢ (𝑃 + 1) = 𝑄 | |
34 | 33 | oveq2i 7404 | . . . . 5 ⊢ (;10↑(𝑃 + 1)) = (;10↑𝑄) |
35 | 32, 34 | eqtr3i 2761 | . . . 4 ⊢ ((;10↑𝑃) · ;10) = (;10↑𝑄) |
36 | 35 | oveq2i 7404 | . . 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 31937 | . . 3 ⊢ (𝐴.𝐵) = _𝐴𝐵 |
39 | 38 | oveq1i 7403 | . 2 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = (_𝐴𝐵 · (;10↑𝑃)) |
40 | 0nn0 12469 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
41 | 40, 6 | dpval3rp 31937 | . . . 4 ⊢ (0._𝐴𝐵) = _0_𝐴𝐵 |
42 | 6 | dp20h 31916 | . . . 4 ⊢ _0_𝐴𝐵 = (_𝐴𝐵 / ;10) |
43 | 41, 42 | eqtri 2759 | . . 3 ⊢ (0._𝐴𝐵) = (_𝐴𝐵 / ;10) |
44 | 43 | oveq1i 7403 | . 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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 class class class wbr 5141 (class class class)co 7393 ℂcc 11090 ℝcr 11091 0cc0 11092 1c1 11093 + caddc 11095 · cmul 11097 < clt 11230 / cdiv 11853 ℕ0cn0 12454 ℤcz 12540 ;cdc 12659 ℝ+crp 12956 ↑cexp 14009 _cdp2 31908 .cdp 31925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-rp 12957 df-seq 13949 df-exp 14010 df-dp2 31909 df-dp 31926 |
This theorem is referenced by: 0dp2dp 31946 hgt750lemd 33491 hgt750lem 33494 |
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