dpexpp1 - Mathbox for Thierry Arnoux

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Theorem dpexpp1 32989

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 11134	. . . . . 6 ⊢ 0 ∈ ℝ
2		10pos 12624	. . . . . 6 ⊢ 0 < ;10
3	1, 2	gtneii 11245	. . . . 5 ⊢ ;10 ≠ 0
4		dpexpp1.a	. . . . . . . . . 10 ⊢ 𝐴 ∈ ℕ₀
5		dpexpp1.b	. . . . . . . . . 10 ⊢ 𝐵 ∈ ℝ⁺
6	4, 5	rpdp2cl 32963	. . . . . . . . 9 ⊢ _𝐴𝐵 ∈ ℝ⁺
7		rpre 12914	. . . . . . . . 9 ⊢ (_𝐴𝐵 ∈ ℝ⁺ → _𝐴𝐵 ∈ ℝ)
8	6, 7	ax-mp 5	. . . . . . . 8 ⊢ _𝐴𝐵 ∈ ℝ
9	8	recni 11146	. . . . . . 7 ⊢ _𝐴𝐵 ∈ ℂ
10		10re 12626	. . . . . . . . . . 11 ⊢ ;10 ∈ ℝ
11	10, 2	pm3.2i 470	. . . . . . . . . 10 ⊢ (;10 ∈ ℝ ∧ 0 < ;10)
12		elrp 12907	. . . . . . . . . 10 ⊢ (;10 ∈ ℝ⁺ ↔ (;10 ∈ ℝ ∧ 0 < ;10))
13	11, 12	mpbir 231	. . . . . . . . 9 ⊢ ;10 ∈ ℝ⁺
14		dpexpp1.p	. . . . . . . . 9 ⊢ 𝑃 ∈ ℤ
15		rpexpcl 14003	. . . . . . . . 9 ⊢ ((;10 ∈ ℝ⁺ ∧ 𝑃 ∈ ℤ) → (;10↑𝑃) ∈ ℝ⁺)
16	13, 14, 15	mp2an 692	. . . . . . . 8 ⊢ (;10↑𝑃) ∈ ℝ⁺
17		rpcn 12916	. . . . . . . 8 ⊢ ((;10↑𝑃) ∈ ℝ⁺ → (;10↑𝑃) ∈ ℂ)
18	16, 17	ax-mp 5	. . . . . . 7 ⊢ (;10↑𝑃) ∈ ℂ
19	9, 18	mulcli 11139	. . . . . 6 ⊢ (_𝐴𝐵 · (;10↑𝑃)) ∈ ℂ
20		10nn0 12625	. . . . . . 7 ⊢ ;10 ∈ ℕ₀
21	20	nn0cni 12413	. . . . . 6 ⊢ ;10 ∈ ℂ
22	19, 21	divcan1zi 11877	. . . . 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 11815	. . . . . 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 7368	. . . 4 ⊢ (((_𝐴𝐵 · (;10↑𝑃)) / ;10) · ;10) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10)
28	23, 27	eqtr3i 2761	. . 3 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10)
29	9, 21, 3	divcli 11883	. . . 4 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ
30	29, 18, 21	mulassi 11143	. . 3 ⊢ (((_𝐴𝐵 / ;10) · (;10↑𝑃)) · ;10) = ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10))
31		expp1z 14034	. . . . . 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 7369	. . . . 5 ⊢ (;10↑(𝑃 + 1)) = (;10↑𝑄)
35	32, 34	eqtr3i 2761	. . . 4 ⊢ ((;10↑𝑃) · ;10) = (;10↑𝑄)
36	35	oveq2i 7369	. . 3 ⊢ ((_𝐴𝐵 / ;10) · ((;10↑𝑃) · ;10)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄))
37	28, 30, 36	3eqtri 2763	. 2 ⊢ (_𝐴𝐵 · (;10↑𝑃)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄))
38	4, 5	dpval3rp 32981	. . 3 ⊢ (𝐴.𝐵) = _𝐴𝐵
39	38	oveq1i 7368	. 2 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = (_𝐴𝐵 · (;10↑𝑃))
40		0nn0 12416	. . . . 5 ⊢ 0 ∈ ℕ₀
41	40, 6	dpval3rp 32981	. . . 4 ⊢ (0._𝐴𝐵) = _0_𝐴𝐵
42	6	dp20h 32960	. . . 4 ⊢ _0_𝐴𝐵 = (_𝐴𝐵 / ;10)
43	41, 42	eqtri 2759	. . 3 ⊢ (0._𝐴𝐵) = (_𝐴𝐵 / ;10)
44	43	oveq1i 7368	. 2 ⊢ ((0._𝐴𝐵) · (;10↑𝑄)) = ((_𝐴𝐵 / ;10) · (;10↑𝑄))
45	37, 39, 44	3eqtr4i 2769	1 ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 class class class wbr 5098 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 < clt 11166 / cdiv 11794 ℕ₀cn0 12401 ℤcz 12488 cdc 12607 ℝ⁺crp 12905 ↑cexp 13984 cdp2 32952 .cdp 32969

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 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-rp 12906 df-seq 13925 df-exp 13985 df-dp2 32953 df-dp 32970

This theorem is referenced by: 0dp2dp 32990 hgt750lemd 34805 hgt750lem 34808

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