dpexpp1 - Mathbox for Thierry Arnoux

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Theorem dpexpp1 32625

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 11240	. . . . . 6 ⊢ 0 ∈ ℝ
2		10pos 12718	. . . . . 6 ⊢ 0 < ;10
3	1, 2	gtneii 11350	. . . . 5 ⊢ ;10 ≠ 0
4		dpexpp1.a	. . . . . . . . . 10 ⊢ 𝐴 ∈ ℕ₀
5		dpexpp1.b	. . . . . . . . . 10 ⊢ 𝐵 ∈ ℝ⁺
6	4, 5	rpdp2cl 32599	. . . . . . . . 9 ⊢ _𝐴𝐵 ∈ ℝ⁺
7		rpre 13008	. . . . . . . . 9 ⊢ (_𝐴𝐵 ∈ ℝ⁺ → _𝐴𝐵 ∈ ℝ)
8	6, 7	ax-mp 5	. . . . . . . 8 ⊢ _𝐴𝐵 ∈ ℝ
9	8	recni 11252	. . . . . . 7 ⊢ _𝐴𝐵 ∈ ℂ
10		10re 12720	. . . . . . . . . . 11 ⊢ ;10 ∈ ℝ
11	10, 2	pm3.2i 470	. . . . . . . . . 10 ⊢ (;10 ∈ ℝ ∧ 0 < ;10)
12		elrp 13002	. . . . . . . . . 10 ⊢ (;10 ∈ ℝ⁺ ↔ (;10 ∈ ℝ ∧ 0 < ;10))
13	11, 12	mpbir 230	. . . . . . . . 9 ⊢ ;10 ∈ ℝ⁺
14		dpexpp1.p	. . . . . . . . 9 ⊢ 𝑃 ∈ ℤ
15		rpexpcl 14071	. . . . . . . . 9 ⊢ ((;10 ∈ ℝ⁺ ∧ 𝑃 ∈ ℤ) → (;10↑𝑃) ∈ ℝ⁺)
16	13, 14, 15	mp2an 691	. . . . . . . 8 ⊢ (;10↑𝑃) ∈ ℝ⁺
17		rpcn 13010	. . . . . . . 8 ⊢ ((;10↑𝑃) ∈ ℝ⁺ → (;10↑𝑃) ∈ ℂ)
18	16, 17	ax-mp 5	. . . . . . 7 ⊢ (;10↑𝑃) ∈ ℂ
19	9, 18	mulcli 11245	. . . . . 6 ⊢ (_𝐴𝐵 · (;10↑𝑃)) ∈ ℂ
20		10nn0 12719	. . . . . . 7 ⊢ ;10 ∈ ℕ₀
21	20	nn0cni 12508	. . . . . 6 ⊢ ;10 ∈ ℂ
22	19, 21	divcan1zi 11974	. . . . 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 11915	. . . . . 6 ⊢ ((_𝐴𝐵 ∈ ℂ ∧ (;10↑𝑃) ∈ ℂ ∧ (;10 ∈ ℂ ∧ ;10 ≠ 0)) → ((_𝐴𝐵 · (;10↑𝑃)) / ;10) = ((_𝐴𝐵 / ;10) · (;10↑𝑃)))
26	9, 18, 24, 25	mp3an 1458	. . . . 5 ⊢ ((_𝐴𝐵 · (;10↑𝑃)) / ;10) = ((_𝐴𝐵 / ;10) · (;10↑𝑃))
27	26	oveq1i 7424	. . . 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 11980	. . . 4 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ
30	29, 18, 21	mulassi 11249	. . 3 ⊢ (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) = ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10))
31		expp1z 14102	. . . . . 6 ⊢ ((;10 ∈ ℂ ∧ ;10 ≠ 0 ∧ 𝑃 ∈ ℤ) → (;10↑(𝑃 + 1)) = ((;10↑𝑃) · ;10))
32	21, 3, 14, 31	mp3an 1458	. . . . 5 ⊢ (;10↑(𝑃 + 1)) = ((;10↑𝑃) · ;10)
33		dpexpp1.1	. . . . . 6 ⊢ (𝑃 + 1) = 𝑄
34	33	oveq2i 7425	. . . . 5 ⊢ (;10↑(𝑃 + 1)) = (;10↑𝑄)
35	32, 34	eqtr3i 2758	. . . 4 ⊢ ((;10↑𝑃) · ;10) = (;10↑𝑄)
36	35	oveq2i 7425	. . 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 32617	. . 3 ⊢ (𝐴.𝐵) = _𝐴𝐵
39	38	oveq1i 7424	. 2 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = (_𝐴𝐵 · (;10↑𝑃))
40		0nn0 12511	. . . . 5 ⊢ 0 ∈ ℕ₀
41	40, 6	dpval3rp 32617	. . . 4 ⊢ (0._𝐴𝐵) = _0_𝐴𝐵
42	6	dp20h 32596	. . . 4 ⊢ _0_𝐴𝐵 = (_𝐴𝐵 / ;10)
43	41, 42	eqtri 2756	. . 3 ⊢ (0._𝐴𝐵) = (_𝐴𝐵 / ;10)
44	43	oveq1i 7424	. 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 1534 ∈ wcel 2099 ≠ wne 2936 class class class wbr 5142 (class class class)co 7414 ℂcc 11130 ℝcr 11131 0cc0 11132 1c1 11133 + caddc 11135 · cmul 11137 < clt 11272 / cdiv 11895 ℕ₀cn0 12496 ℤcz 12582 cdc 12701 ℝ⁺crp 13000 ↑cexp 14052 cdp2 32588 .cdp 32605

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-rp 13001 df-seq 13993 df-exp 14053 df-dp2 32589 df-dp 32606

This theorem is referenced by: 0dp2dp 32626 hgt750lemd 34274 hgt750lem 34277

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