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 10637 | . . . . . 6 ⊢ 0 ∈ ℝ | |
2 | 10pos 12109 | . . . . . 6 ⊢ 0 < ;10 | |
3 | 1, 2 | gtneii 10746 | . . . . 5 ⊢ ;10 ≠ 0 |
4 | dpexpp1.a | . . . . . . . . . 10 ⊢ 𝐴 ∈ ℕ0 | |
5 | dpexpp1.b | . . . . . . . . . 10 ⊢ 𝐵 ∈ ℝ+ | |
6 | 4, 5 | rpdp2cl 30553 | . . . . . . . . 9 ⊢ _𝐴𝐵 ∈ ℝ+ |
7 | rpre 12391 | . . . . . . . . 9 ⊢ (_𝐴𝐵 ∈ ℝ+ → _𝐴𝐵 ∈ ℝ) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ _𝐴𝐵 ∈ ℝ |
9 | 8 | recni 10649 | . . . . . . 7 ⊢ _𝐴𝐵 ∈ ℂ |
10 | 10re 12111 | . . . . . . . . . . 11 ⊢ ;10 ∈ ℝ | |
11 | 10, 2 | pm3.2i 473 | . . . . . . . . . 10 ⊢ (;10 ∈ ℝ ∧ 0 < ;10) |
12 | elrp 12385 | . . . . . . . . . 10 ⊢ (;10 ∈ ℝ+ ↔ (;10 ∈ ℝ ∧ 0 < ;10)) | |
13 | 11, 12 | mpbir 233 | . . . . . . . . 9 ⊢ ;10 ∈ ℝ+ |
14 | dpexpp1.p | . . . . . . . . 9 ⊢ 𝑃 ∈ ℤ | |
15 | rpexpcl 13442 | . . . . . . . . 9 ⊢ ((;10 ∈ ℝ+ ∧ 𝑃 ∈ ℤ) → (;10↑𝑃) ∈ ℝ+) | |
16 | 13, 14, 15 | mp2an 690 | . . . . . . . 8 ⊢ (;10↑𝑃) ∈ ℝ+ |
17 | rpcn 12393 | . . . . . . . 8 ⊢ ((;10↑𝑃) ∈ ℝ+ → (;10↑𝑃) ∈ ℂ) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (;10↑𝑃) ∈ ℂ |
19 | 9, 18 | mulcli 10642 | . . . . . 6 ⊢ (_𝐴𝐵 · (;10↑𝑃)) ∈ ℂ |
20 | 10nn0 12110 | . . . . . . 7 ⊢ ;10 ∈ ℕ0 | |
21 | 20 | nn0cni 11903 | . . . . . 6 ⊢ ;10 ∈ ℂ |
22 | 19, 21 | divcan1zi 11370 | . . . . 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 473 | . . . . . 6 ⊢ (;10 ∈ ℂ ∧ ;10 ≠ 0) |
25 | div23 11311 | . . . . . 6 ⊢ ((_𝐴𝐵 ∈ ℂ ∧ (;10↑𝑃) ∈ ℂ ∧ (;10 ∈ ℂ ∧ ;10 ≠ 0)) → ((_𝐴𝐵 · (;10↑𝑃)) / ;10) = ((_𝐴𝐵 / ;10) · (;10↑𝑃))) | |
26 | 9, 18, 24, 25 | mp3an 1457 | . . . . 5 ⊢ ((_𝐴𝐵 · (;10↑𝑃)) / ;10) = ((_𝐴𝐵 / ;10) · (;10↑𝑃)) |
27 | 26 | oveq1i 7160 | . . . 4 ⊢ (((_𝐴𝐵 · (;10↑𝑃)) / ;10) · ;10) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) |
28 | 23, 27 | eqtr3i 2846 | . . 3 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) |
29 | 9, 21, 3 | divcli 11376 | . . . 4 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ |
30 | 29, 18, 21 | mulassi 10646 | . . 3 ⊢ (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) = ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) |
31 | expp1z 13472 | . . . . . 6 ⊢ ((;10 ∈ ℂ ∧ ;10 ≠ 0 ∧ 𝑃 ∈ ℤ) → (;10↑(𝑃 + 1)) = ((;10↑𝑃) · ;10)) | |
32 | 21, 3, 14, 31 | mp3an 1457 | . . . . 5 ⊢ (;10↑(𝑃 + 1)) = ((;10↑𝑃) · ;10) |
33 | dpexpp1.1 | . . . . . 6 ⊢ (𝑃 + 1) = 𝑄 | |
34 | 33 | oveq2i 7161 | . . . . 5 ⊢ (;10↑(𝑃 + 1)) = (;10↑𝑄) |
35 | 32, 34 | eqtr3i 2846 | . . . 4 ⊢ ((;10↑𝑃) · ;10) = (;10↑𝑄) |
36 | 35 | oveq2i 7161 | . . 3 ⊢ ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄)) |
37 | 28, 30, 36 | 3eqtri 2848 | . 2 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄)) |
38 | 4, 5 | dpval3rp 30571 | . . 3 ⊢ (𝐴.𝐵) = _𝐴𝐵 |
39 | 38 | oveq1i 7160 | . 2 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = (_𝐴𝐵 · (;10↑𝑃)) |
40 | 0nn0 11906 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
41 | 40, 6 | dpval3rp 30571 | . . . 4 ⊢ (0._𝐴𝐵) = _0_𝐴𝐵 |
42 | 6 | dp20h 30550 | . . . 4 ⊢ _0_𝐴𝐵 = (_𝐴𝐵 / ;10) |
43 | 41, 42 | eqtri 2844 | . . 3 ⊢ (0._𝐴𝐵) = (_𝐴𝐵 / ;10) |
44 | 43 | oveq1i 7160 | . 2 ⊢ ((0._𝐴𝐵) · (;10↑𝑄)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄)) |
45 | 37, 39, 44 | 3eqtr4i 2854 | 1 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5059 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 < clt 10669 / cdiv 11291 ℕ0cn0 11891 ℤcz 11975 ;cdc 12092 ℝ+crp 12383 ↑cexp 13423 _cdp2 30542 .cdp 30559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-dp2 30543 df-dp 30560 |
This theorem is referenced by: 0dp2dp 30580 hgt750lemd 31914 hgt750lem 31917 |
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