dpexpp1 - Mathbox for Thierry Arnoux

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Theorem dpexpp1 32835

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 11183	. . . . . 6 ⊢ 0 ∈ ℝ
2		10pos 12673	. . . . . 6 ⊢ 0 < ;10
3	1, 2	gtneii 11293	. . . . 5 ⊢ ;10 ≠ 0
4		dpexpp1.a	. . . . . . . . . 10 ⊢ 𝐴 ∈ ℕ₀
5		dpexpp1.b	. . . . . . . . . 10 ⊢ 𝐵 ∈ ℝ⁺
6	4, 5	rpdp2cl 32809	. . . . . . . . 9 ⊢ _𝐴𝐵 ∈ ℝ⁺
7		rpre 12967	. . . . . . . . 9 ⊢ (_𝐴𝐵 ∈ ℝ⁺ → _𝐴𝐵 ∈ ℝ)
8	6, 7	ax-mp 5	. . . . . . . 8 ⊢ _𝐴𝐵 ∈ ℝ
9	8	recni 11195	. . . . . . 7 ⊢ _𝐴𝐵 ∈ ℂ
10		10re 12675	. . . . . . . . . . 11 ⊢ ;10 ∈ ℝ
11	10, 2	pm3.2i 470	. . . . . . . . . 10 ⊢ (;10 ∈ ℝ ∧ 0 < ;10)
12		elrp 12960	. . . . . . . . . 10 ⊢ (;10 ∈ ℝ⁺ ↔ (;10 ∈ ℝ ∧ 0 < ;10))
13	11, 12	mpbir 231	. . . . . . . . 9 ⊢ ;10 ∈ ℝ⁺
14		dpexpp1.p	. . . . . . . . 9 ⊢ 𝑃 ∈ ℤ
15		rpexpcl 14052	. . . . . . . . 9 ⊢ ((;10 ∈ ℝ⁺ ∧ 𝑃 ∈ ℤ) → (;10↑𝑃) ∈ ℝ⁺)
16	13, 14, 15	mp2an 692	. . . . . . . 8 ⊢ (;10↑𝑃) ∈ ℝ⁺
17		rpcn 12969	. . . . . . . 8 ⊢ ((;10↑𝑃) ∈ ℝ⁺ → (;10↑𝑃) ∈ ℂ)
18	16, 17	ax-mp 5	. . . . . . 7 ⊢ (;10↑𝑃) ∈ ℂ
19	9, 18	mulcli 11188	. . . . . 6 ⊢ (_𝐴𝐵 · (;10↑𝑃)) ∈ ℂ
20		10nn0 12674	. . . . . . 7 ⊢ ;10 ∈ ℕ₀
21	20	nn0cni 12461	. . . . . 6 ⊢ ;10 ∈ ℂ
22	19, 21	divcan1zi 11925	. . . . 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 11863	. . . . . 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 7400	. . . 4 ⊢ (((_𝐴𝐵 · (;10↑𝑃)) / ;10) · ;10) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10)
28	23, 27	eqtr3i 2755	. . 3 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10)
29	9, 21, 3	divcli 11931	. . . 4 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ
30	29, 18, 21	mulassi 11192	. . 3 ⊢ (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) = ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10))
31		expp1z 14083	. . . . . 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 7401	. . . . 5 ⊢ (;10↑(𝑃 + 1)) = (;10↑𝑄)
35	32, 34	eqtr3i 2755	. . . 4 ⊢ ((;10↑𝑃) · ;10) = (;10↑𝑄)
36	35	oveq2i 7401	. . 3 ⊢ ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄))
37	28, 30, 36	3eqtri 2757	. 2 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄))
38	4, 5	dpval3rp 32827	. . 3 ⊢ (𝐴.𝐵) = _𝐴𝐵
39	38	oveq1i 7400	. 2 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = (_𝐴𝐵 · (;10↑𝑃))
40		0nn0 12464	. . . . 5 ⊢ 0 ∈ ℕ₀
41	40, 6	dpval3rp 32827	. . . 4 ⊢ (0._𝐴𝐵) = _0_𝐴𝐵
42	6	dp20h 32806	. . . 4 ⊢ _0_𝐴𝐵 = (_𝐴𝐵 / ;10)
43	41, 42	eqtri 2753	. . 3 ⊢ (0._𝐴𝐵) = (_𝐴𝐵 / ;10)
44	43	oveq1i 7400	. 2 ⊢ ((0._𝐴𝐵) · (;10↑𝑄)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄))
45	37, 39, 44	3eqtr4i 2763	1 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 class class class wbr 5110 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 < clt 11215 / cdiv 11842 ℕ₀cn0 12449 ℤcz 12536 cdc 12656 ℝ⁺crp 12958 ↑cexp 14033 cdp2 32798 .cdp 32815

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-rp 12959 df-seq 13974 df-exp 14034 df-dp2 32799 df-dp 32816

This theorem is referenced by: 0dp2dp 32836 hgt750lemd 34646 hgt750lem 34649

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