dignn0ehalf - Mathbox for Alexander van der Vekens

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Theorem dignn0ehalf 47387

Description: The digits of the half of an even nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 3-Jun-2010.)

**Assertion**
Ref	Expression
dignn0ehalf	⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(𝐴 / 2)))

**Proof of Theorem dignn0ehalf**
Step	Hyp	Ref	Expression
1		nn0cn 12484	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ ℂ)
2	1	3ad2ant2 1134	. . . . . 6 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → 𝐴 ∈ ℂ)
3		2cnne0 12424	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
4	3	a1i 11	. . . . . 6 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (2 ∈ ℂ ∧ 2 ≠ 0))
5		2nn0 12491	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℕ₀ → 2 ∈ ℕ₀)
7		id 22	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℕ₀)
8	6, 7	nn0expcld 14211	. . . . . . . . 9 ⊢ (𝐼 ∈ ℕ₀ → (2↑𝐼) ∈ ℕ₀)
9	8	nn0cnd 12536	. . . . . . . 8 ⊢ (𝐼 ∈ ℕ₀ → (2↑𝐼) ∈ ℂ)
10		2cnd 12292	. . . . . . . . 9 ⊢ (𝐼 ∈ ℕ₀ → 2 ∈ ℂ)
11		2ne0 12318	. . . . . . . . . 10 ⊢ 2 ≠ 0
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝐼 ∈ ℕ₀ → 2 ≠ 0)
13		nn0z 12585	. . . . . . . . 9 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℤ)
14	10, 12, 13	expne0d 14119	. . . . . . . 8 ⊢ (𝐼 ∈ ℕ₀ → (2↑𝐼) ≠ 0)
15	9, 14	jca 512	. . . . . . 7 ⊢ (𝐼 ∈ ℕ₀ → ((2↑𝐼) ∈ ℂ ∧ (2↑𝐼) ≠ 0))
16	15	3ad2ant3 1135	. . . . . 6 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → ((2↑𝐼) ∈ ℂ ∧ (2↑𝐼) ≠ 0))
17		divdiv1 11927	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ ((2↑𝐼) ∈ ℂ ∧ (2↑𝐼) ≠ 0)) → ((𝐴 / 2) / (2↑𝐼)) = (𝐴 / (2 · (2↑𝐼))))
18	2, 4, 16, 17	syl3anc 1371	. . . . 5 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → ((𝐴 / 2) / (2↑𝐼)) = (𝐴 / (2 · (2↑𝐼))))
19	10, 9	mulcomd 11237	. . . . . . . 8 ⊢ (𝐼 ∈ ℕ₀ → (2 · (2↑𝐼)) = ((2↑𝐼) · 2))
20	19	3ad2ant3 1135	. . . . . . 7 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (2 · (2↑𝐼)) = ((2↑𝐼) · 2))
21		2cnd 12292	. . . . . . . 8 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → 2 ∈ ℂ)
22		simp3 1138	. . . . . . . 8 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → 𝐼 ∈ ℕ₀)
23	21, 22	expp1d 14114	. . . . . . 7 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (2↑(𝐼 + 1)) = ((2↑𝐼) · 2))
24	20, 23	eqtr4d 2775	. . . . . 6 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (2 · (2↑𝐼)) = (2↑(𝐼 + 1)))
25	24	oveq2d 7427	. . . . 5 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (𝐴 / (2 · (2↑𝐼))) = (𝐴 / (2↑(𝐼 + 1))))
26	18, 25	eqtr2d 2773	. . . 4 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (𝐴 / (2↑(𝐼 + 1))) = ((𝐴 / 2) / (2↑𝐼)))
27	26	fveq2d 6895	. . 3 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (⌊‘(𝐴 / (2↑(𝐼 + 1)))) = (⌊‘((𝐴 / 2) / (2↑𝐼))))
28	27	oveq1d 7426	. 2 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → ((⌊‘(𝐴 / (2↑(𝐼 + 1)))) mod 2) = ((⌊‘((𝐴 / 2) / (2↑𝐼))) mod 2))
29		2nn 12287	. . . 4 ⊢ 2 ∈ ℕ
30	29	a1i 11	. . 3 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → 2 ∈ ℕ)
31		peano2nn0 12514	. . . 4 ⊢ (𝐼 ∈ ℕ₀ → (𝐼 + 1) ∈ ℕ₀)
32	31	3ad2ant3 1135	. . 3 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (𝐼 + 1) ∈ ℕ₀)
33		nn0rp0 13434	. . . 4 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ (0[,)+∞))
34	33	3ad2ant2 1134	. . 3 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → 𝐴 ∈ (0[,)+∞))
35		nn0digval 47370	. . 3 ⊢ ((2 ∈ ℕ ∧ (𝐼 + 1) ∈ ℕ₀ ∧ 𝐴 ∈ (0[,)+∞)) → ((𝐼 + 1)(digit‘2)𝐴) = ((⌊‘(𝐴 / (2↑(𝐼 + 1)))) mod 2))
36	30, 32, 34, 35	syl3anc 1371	. 2 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → ((𝐼 + 1)(digit‘2)𝐴) = ((⌊‘(𝐴 / (2↑(𝐼 + 1)))) mod 2))
37		nn0rp0 13434	. . . 4 ⊢ ((𝐴 / 2) ∈ ℕ₀ → (𝐴 / 2) ∈ (0[,)+∞))
38	37	3ad2ant1 1133	. . 3 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (𝐴 / 2) ∈ (0[,)+∞))
39		nn0digval 47370	. . 3 ⊢ ((2 ∈ ℕ ∧ 𝐼 ∈ ℕ₀ ∧ (𝐴 / 2) ∈ (0[,)+∞)) → (𝐼(digit‘2)(𝐴 / 2)) = ((⌊‘((𝐴 / 2) / (2↑𝐼))) mod 2))
40	30, 22, 38, 39	syl3anc 1371	. 2 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (𝐼(digit‘2)(𝐴 / 2)) = ((⌊‘((𝐴 / 2) / (2↑𝐼))) mod 2))
41	28, 36, 40	3eqtr4d 2782	1 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(𝐴 / 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 +∞cpnf 11247 / cdiv 11873 ℕcn 12214 2c2 12269 ℕ₀cn0 12474 [,)cico 13328 ⌊cfl 13757 mod cmo 13836 ↑cexp 14029 digitcdig 47365

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-n0 12475 df-z 12561 df-uz 12825 df-ico 13332 df-seq 13969 df-exp 14030 df-dig 47366

This theorem is referenced by: dignn0flhalf 47388 nn0sumshdiglemA 47389

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