| 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 11263 | . . . . . 6 ⊢ 0 ∈ ℝ | |
| 2 | 10pos 12750 | . . . . . 6 ⊢ 0 < ;10 | |
| 3 | 1, 2 | gtneii 11373 | . . . . 5 ⊢ ;10 ≠ 0 |
| 4 | dpexpp1.a | . . . . . . . . . 10 ⊢ 𝐴 ∈ ℕ0 | |
| 5 | dpexpp1.b | . . . . . . . . . 10 ⊢ 𝐵 ∈ ℝ+ | |
| 6 | 4, 5 | rpdp2cl 32864 | . . . . . . . . 9 ⊢ _𝐴𝐵 ∈ ℝ+ |
| 7 | rpre 13043 | . . . . . . . . 9 ⊢ (_𝐴𝐵 ∈ ℝ+ → _𝐴𝐵 ∈ ℝ) | |
| 8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ _𝐴𝐵 ∈ ℝ |
| 9 | 8 | recni 11275 | . . . . . . 7 ⊢ _𝐴𝐵 ∈ ℂ |
| 10 | 10re 12752 | . . . . . . . . . . 11 ⊢ ;10 ∈ ℝ | |
| 11 | 10, 2 | pm3.2i 470 | . . . . . . . . . 10 ⊢ (;10 ∈ ℝ ∧ 0 < ;10) |
| 12 | elrp 13036 | . . . . . . . . . 10 ⊢ (;10 ∈ ℝ+ ↔ (;10 ∈ ℝ ∧ 0 < ;10)) | |
| 13 | 11, 12 | mpbir 231 | . . . . . . . . 9 ⊢ ;10 ∈ ℝ+ |
| 14 | dpexpp1.p | . . . . . . . . 9 ⊢ 𝑃 ∈ ℤ | |
| 15 | rpexpcl 14121 | . . . . . . . . 9 ⊢ ((;10 ∈ ℝ+ ∧ 𝑃 ∈ ℤ) → (;10↑𝑃) ∈ ℝ+) | |
| 16 | 13, 14, 15 | mp2an 692 | . . . . . . . 8 ⊢ (;10↑𝑃) ∈ ℝ+ |
| 17 | rpcn 13045 | . . . . . . . 8 ⊢ ((;10↑𝑃) ∈ ℝ+ → (;10↑𝑃) ∈ ℂ) | |
| 18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (;10↑𝑃) ∈ ℂ |
| 19 | 9, 18 | mulcli 11268 | . . . . . 6 ⊢ (_𝐴𝐵 · (;10↑𝑃)) ∈ ℂ |
| 20 | 10nn0 12751 | . . . . . . 7 ⊢ ;10 ∈ ℕ0 | |
| 21 | 20 | nn0cni 12538 | . . . . . 6 ⊢ ;10 ∈ ℂ |
| 22 | 19, 21 | divcan1zi 12003 | . . . . 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 11941 | . . . . . 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 7441 | . . . 4 ⊢ (((_𝐴𝐵 · (;10↑𝑃)) / ;10) · ;10) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) |
| 28 | 23, 27 | eqtr3i 2767 | . . 3 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) |
| 29 | 9, 21, 3 | divcli 12009 | . . . 4 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ |
| 30 | 29, 18, 21 | mulassi 11272 | . . 3 ⊢ (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) = ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) |
| 31 | expp1z 14152 | . . . . . 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 7442 | . . . . 5 ⊢ (;10↑(𝑃 + 1)) = (;10↑𝑄) |
| 35 | 32, 34 | eqtr3i 2767 | . . . 4 ⊢ ((;10↑𝑃) · ;10) = (;10↑𝑄) |
| 36 | 35 | oveq2i 7442 | . . 3 ⊢ ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄)) |
| 37 | 28, 30, 36 | 3eqtri 2769 | . 2 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄)) |
| 38 | 4, 5 | dpval3rp 32882 | . . 3 ⊢ (𝐴.𝐵) = _𝐴𝐵 |
| 39 | 38 | oveq1i 7441 | . 2 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = (_𝐴𝐵 · (;10↑𝑃)) |
| 40 | 0nn0 12541 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
| 41 | 40, 6 | dpval3rp 32882 | . . . 4 ⊢ (0._𝐴𝐵) = _0_𝐴𝐵 |
| 42 | 6 | dp20h 32861 | . . . 4 ⊢ _0_𝐴𝐵 = (_𝐴𝐵 / ;10) |
| 43 | 41, 42 | eqtri 2765 | . . 3 ⊢ (0._𝐴𝐵) = (_𝐴𝐵 / ;10) |
| 44 | 43 | oveq1i 7441 | . 2 ⊢ ((0._𝐴𝐵) · (;10↑𝑄)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄)) |
| 45 | 37, 39, 44 | 3eqtr4i 2775 | 1 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄)) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 / cdiv 11920 ℕ0cn0 12526 ℤcz 12613 ;cdc 12733 ℝ+crp 13034 ↑cexp 14102 _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-cnex 11211 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-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-dp2 32854 df-dp 32871 |
| This theorem is referenced by: 0dp2dp 32891 hgt750lemd 34663 hgt750lem 34666 |
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