dpexpp1 - Mathbox for Thierry Arnoux

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Theorem dpexpp1 32895

Description: Add one zero to the mantisse, and a one to the exponent in a scientific notation. (Contributed by Thierry Arnoux, 16-Dec-2021.)

**Hypotheses**
Ref	Expression
dpexpp1.a	⊢ 𝐴 ∈ ℕ₀
dpexpp1.b	⊢ 𝐵 ∈ ℝ⁺
dpexpp1.1	⊢ (𝑃 + 1) = 𝑄
dpexpp1.p	⊢ 𝑃 ∈ ℤ
dpexpp1.q	⊢ 𝑄 ∈ ℤ

**Assertion**
Ref	Expression
dpexpp1	⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄))

**Proof of Theorem dpexpp1**
Step	Hyp	Ref	Expression
1		0re 11121	. . . . . 6 ⊢ 0 ∈ ℝ
2		10pos 12611	. . . . . 6 ⊢ 0 < ;10
3	1, 2	gtneii 11232	. . . . 5 ⊢ ;10 ≠ 0
4		dpexpp1.a	. . . . . . . . . 10 ⊢ 𝐴 ∈ ℕ₀
5		dpexpp1.b	. . . . . . . . . 10 ⊢ 𝐵 ∈ ℝ⁺
6	4, 5	rpdp2cl 32869	. . . . . . . . 9 ⊢ _𝐴𝐵 ∈ ℝ⁺
7		rpre 12901	. . . . . . . . 9 ⊢ (_𝐴𝐵 ∈ ℝ⁺ → _𝐴𝐵 ∈ ℝ)
8	6, 7	ax-mp 5	. . . . . . . 8 ⊢ _𝐴𝐵 ∈ ℝ
9	8	recni 11133	. . . . . . 7 ⊢ _𝐴𝐵 ∈ ℂ
10		10re 12613	. . . . . . . . . . 11 ⊢ ;10 ∈ ℝ
11	10, 2	pm3.2i 470	. . . . . . . . . 10 ⊢ (;10 ∈ ℝ ∧ 0 < ;10)
12		elrp 12894	. . . . . . . . . 10 ⊢ (;10 ∈ ℝ⁺ ↔ (;10 ∈ ℝ ∧ 0 < ;10))
13	11, 12	mpbir 231	. . . . . . . . 9 ⊢ ;10 ∈ ℝ⁺
14		dpexpp1.p	. . . . . . . . 9 ⊢ 𝑃 ∈ ℤ
15		rpexpcl 13989	. . . . . . . . 9 ⊢ ((;10 ∈ ℝ⁺ ∧ 𝑃 ∈ ℤ) → (;10↑𝑃) ∈ ℝ⁺)
16	13, 14, 15	mp2an 692	. . . . . . . 8 ⊢ (;10↑𝑃) ∈ ℝ⁺
17		rpcn 12903	. . . . . . . 8 ⊢ ((;10↑𝑃) ∈ ℝ⁺ → (;10↑𝑃) ∈ ℂ)
18	16, 17	ax-mp 5	. . . . . . 7 ⊢ (;10↑𝑃) ∈ ℂ
19	9, 18	mulcli 11126	. . . . . 6 ⊢ (_𝐴𝐵 · (;10↑𝑃)) ∈ ℂ
20		10nn0 12612	. . . . . . 7 ⊢ ;10 ∈ ℕ₀
21	20	nn0cni 12400	. . . . . 6 ⊢ ;10 ∈ ℂ
22	19, 21	divcan1zi 11864	. . . . 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 11802	. . . . . 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 7362	. . . 4 ⊢ (((_𝐴𝐵 · (;10↑𝑃)) / ;10) · ;10) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10)
28	23, 27	eqtr3i 2758	. . 3 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10)
29	9, 21, 3	divcli 11870	. . . 4 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ
30	29, 18, 21	mulassi 11130	. . 3 ⊢ (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) = ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10))
31		expp1z 14020	. . . . . 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 7363	. . . . 5 ⊢ (;10↑(𝑃 + 1)) = (;10↑𝑄)
35	32, 34	eqtr3i 2758	. . . 4 ⊢ ((;10↑𝑃) · ;10) = (;10↑𝑄)
36	35	oveq2i 7363	. . 3 ⊢ ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄))
37	28, 30, 36	3eqtri 2760	. 2 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄))
38	4, 5	dpval3rp 32887	. . 3 ⊢ (𝐴.𝐵) = _𝐴𝐵
39	38	oveq1i 7362	. 2 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = (_𝐴𝐵 · (;10↑𝑃))
40		0nn0 12403	. . . . 5 ⊢ 0 ∈ ℕ₀
41	40, 6	dpval3rp 32887	. . . 4 ⊢ (0._𝐴𝐵) = _0_𝐴𝐵
42	6	dp20h 32866	. . . 4 ⊢ _0_𝐴𝐵 = (_𝐴𝐵 / ;10)
43	41, 42	eqtri 2756	. . 3 ⊢ (0._𝐴𝐵) = (_𝐴𝐵 / ;10)
44	43	oveq1i 7362	. 2 ⊢ ((0._𝐴𝐵) · (;10↑𝑄)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄))
45	37, 39, 44	3eqtr4i 2766	1 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 class class class wbr 5093 (class class class)co 7352 ℂcc 11011 ℝcr 11012 0cc0 11013 1c1 11014 + caddc 11016 · cmul 11018 < clt 11153 / cdiv 11781 ℕ₀cn0 12388 ℤcz 12475 cdc 12594 ℝ⁺crp 12892 ↑cexp 13970 cdp2 32858 .cdp 32875

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-rp 12893 df-seq 13911 df-exp 13971 df-dp2 32859 df-dp 32876

This theorem is referenced by: 0dp2dp 32896 hgt750lemd 34682 hgt750lem 34685

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