dpexpp1 - Mathbox for Thierry Arnoux

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Theorem dpexpp1 33520

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 11313	. . . . . 6 ⊢ 0 ∈ ℝ
2		10pos 12838	. . . . . 6 ⊢ 0 < ;10
3	1, 2	gtneii 11425	. . . . 5 ⊢ ;10 ≠ 0
4		dpexpp1.a	. . . . . . . . . 10 ⊢ 𝐴 ∈ ℕ₀
5		dpexpp1.b	. . . . . . . . . 10 ⊢ 𝐵 ∈ ℝ⁺
6	4, 5	rpdp2cl 33494	. . . . . . . . 9 ⊢ _𝐴𝐵 ∈ ℝ⁺
7		rpre 13132	. . . . . . . . 9 ⊢ (_𝐴𝐵 ∈ ℝ⁺ → _𝐴𝐵 ∈ ℝ)
8	6, 7	ax-mp 5	. . . . . . . 8 ⊢ _𝐴𝐵 ∈ ℝ
9	8	recni 11326	. . . . . . 7 ⊢ _𝐴𝐵 ∈ ℂ
10		10re 12840	. . . . . . . . . . 11 ⊢ ;10 ∈ ℝ
11	10, 2	pm3.2i 476	. . . . . . . . . 10 ⊢ (;10 ∈ ℝ ∧ 0 < ;10)
12		elrp 13125	. . . . . . . . . 10 ⊢ (;10 ∈ ℝ⁺ ↔ (;10 ∈ ℝ ∧ 0 < ;10))
13	11, 12	mpbir 234	. . . . . . . . 9 ⊢ ;10 ∈ ℝ⁺
14		dpexpp1.p	. . . . . . . . 9 ⊢ 𝑃 ∈ ℤ
15		rpexpcl 14226	. . . . . . . . 9 ⊢ ((;10 ∈ ℝ⁺ ∧ 𝑃 ∈ ℤ) → (;10↑𝑃) ∈ ℝ⁺)
16	13, 14, 15	mp2an 705	. . . . . . . 8 ⊢ (;10↑𝑃) ∈ ℝ⁺
17		rpcn 13134	. . . . . . . 8 ⊢ ((;10↑𝑃) ∈ ℝ⁺ → (;10↑𝑃) ∈ ℂ)
18	16, 17	ax-mp 5	. . . . . . 7 ⊢ (;10↑𝑃) ∈ ℂ
19	9, 18	mulcli 11319	. . . . . 6 ⊢ (_𝐴𝐵 · (;10↑𝑃)) ∈ ℂ
20		10nn0 12839	. . . . . . 7 ⊢ ;10 ∈ ℕ₀
21	20	nn0cni 12621	. . . . . 6 ⊢ ;10 ∈ ℂ
22	19, 21	divcan1zi 12056	. . . . 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 476	. . . . . 6 ⊢ (;10 ∈ ℂ ∧ ;10 ≠ 0)
25		div23 11996	. . . . . 6 ⊢ ((_𝐴𝐵 ∈ ℂ ∧ (;10↑𝑃) ∈ ℂ ∧ (;10 ∈ ℂ ∧ ;10 ≠ 0)) → ((_𝐴𝐵 · (;10↑𝑃)) / ;10) = ((_𝐴𝐵 / ;10) · (;10↑𝑃)))
26	9, 18, 24, 25	mp3an 1490	. . . . 5 ⊢ ((_𝐴𝐵 · (;10↑𝑃)) / ;10) = ((_𝐴𝐵 / ;10) · (;10↑𝑃))
27	26	oveq1i 7430	. . . 4 ⊢ (((_𝐴𝐵 · (;10↑𝑃)) / ;10) · ;10) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10)
28	23, 27	eqtr3i 2785	. . 3 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10)
29	9, 21, 3	divcli 12062	. . . 4 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ
30	29, 18, 21	mulassi 11323	. . 3 ⊢ (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) = ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10))
31		expp1z 14257	. . . . . 6 ⊢ ((;10 ∈ ℂ ∧ ;10 ≠ 0 ∧ 𝑃 ∈ ℤ) → (;10↑(𝑃 + 1)) = ((;10↑𝑃) · ;10))
32	21, 3, 14, 31	mp3an 1490	. . . . 5 ⊢ (;10↑(𝑃 + 1)) = ((;10↑𝑃) · ;10)
33		dpexpp1.1	. . . . . 6 ⊢ (𝑃 + 1) = 𝑄
34	33	oveq2i 7431	. . . . 5 ⊢ (;10↑(𝑃 + 1)) = (;10↑𝑄)
35	32, 34	eqtr3i 2785	. . . 4 ⊢ ((;10↑𝑃) · ;10) = (;10↑𝑄)
36	35	oveq2i 7431	. . 3 ⊢ ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄))
37	28, 30, 36	3eqtri 2787	. 2 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄))
38	4, 5	dpval3rp 33512	. . 3 ⊢ (𝐴.𝐵) = _𝐴𝐵
39	38	oveq1i 7430	. 2 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = (_𝐴𝐵 · (;10↑𝑃))
40		0nn0 12624	. . . . 5 ⊢ 0 ∈ ℕ₀
41	40, 6	dpval3rp 33512	. . . 4 ⊢ (0._𝐴𝐵) = _0_𝐴𝐵
42	6	dp20h 33491	. . . 4 ⊢ _0_𝐴𝐵 = (_𝐴𝐵 / ;10)
43	41, 42	eqtri 2783	. . 3 ⊢ (0._𝐴𝐵) = (_𝐴𝐵 / ;10)
44	43	oveq1i 7430	. 2 ⊢ ((0._𝐴𝐵) · (;10↑𝑄)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄))
45	37, 39, 44	3eqtr4i 2793	1 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 (class class class)co 7420 ℂcc 11201 ℝcr 11202 0cc0 11203 1c1 11204 + caddc 11206 · cmul 11208 < clt 11346 / cdiv 11976 ℕ₀cn0 12609 ℤcz 12696 cdc 12817 ℝ⁺crp 13123 ↑cexp 14207 cdp2 33483 .cdp 33500

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-er 8720 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-rp 13124 df-seq 14148 df-exp 14208 df-dp2 33484 df-dp 33501

This theorem is used by: 0dp2dp 33521 hgt750lemd 35323 hgt750lem 35326

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