dignn0ehalf - Mathbox for Alexander van der Vekens

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Theorem dignn0ehalf 47303

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 12482	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ ℂ)
2	1	3ad2ant2 1135	. . . . . 6 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → 𝐴 ∈ ℂ)
3		2cnne0 12422	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
4	3	a1i 11	. . . . . 6 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (2 ∈ ℂ ∧ 2 ≠ 0))
5		2nn0 12489	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℕ₀ → 2 ∈ ℕ₀)
7		id 22	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℕ₀)
8	6, 7	nn0expcld 14209	. . . . . . . . 9 ⊢ (𝐼 ∈ ℕ₀ → (2↑𝐼) ∈ ℕ₀)
9	8	nn0cnd 12534	. . . . . . . 8 ⊢ (𝐼 ∈ ℕ₀ → (2↑𝐼) ∈ ℂ)
10		2cnd 12290	. . . . . . . . 9 ⊢ (𝐼 ∈ ℕ₀ → 2 ∈ ℂ)
11		2ne0 12316	. . . . . . . . . 10 ⊢ 2 ≠ 0
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝐼 ∈ ℕ₀ → 2 ≠ 0)
13		nn0z 12583	. . . . . . . . 9 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℤ)
14	10, 12, 13	expne0d 14117	. . . . . . . 8 ⊢ (𝐼 ∈ ℕ₀ → (2↑𝐼) ≠ 0)
15	9, 14	jca 513	. . . . . . 7 ⊢ (𝐼 ∈ ℕ₀ → ((2↑𝐼) ∈ ℂ ∧ (2↑𝐼) ≠ 0))
16	15	3ad2ant3 1136	. . . . . 6 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → ((2↑𝐼) ∈ ℂ ∧ (2↑𝐼) ≠ 0))
17		divdiv1 11925	. . . . . 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 1372	. . . . 5 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → ((𝐴 / 2) / (2↑𝐼)) = (𝐴 / (2 · (2↑𝐼))))
19	10, 9	mulcomd 11235	. . . . . . . 8 ⊢ (𝐼 ∈ ℕ₀ → (2 · (2↑𝐼)) = ((2↑𝐼) · 2))
20	19	3ad2ant3 1136	. . . . . . 7 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (2 · (2↑𝐼)) = ((2↑𝐼) · 2))
21		2cnd 12290	. . . . . . . 8 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → 2 ∈ ℂ)
22		simp3 1139	. . . . . . . 8 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → 𝐼 ∈ ℕ₀)
23	21, 22	expp1d 14112	. . . . . . 7 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (2↑(𝐼 + 1)) = ((2↑𝐼) · 2))
24	20, 23	eqtr4d 2776	. . . . . 6 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (2 · (2↑𝐼)) = (2↑(𝐼 + 1)))
25	24	oveq2d 7425	. . . . 5 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (𝐴 / (2 · (2↑𝐼))) = (𝐴 / (2↑(𝐼 + 1))))
26	18, 25	eqtr2d 2774	. . . 4 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (𝐴 / (2↑(𝐼 + 1))) = ((𝐴 / 2) / (2↑𝐼)))
27	26	fveq2d 6896	. . 3 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (⌊‘(𝐴 / (2↑(𝐼 + 1)))) = (⌊‘((𝐴 / 2) / (2↑𝐼))))
28	27	oveq1d 7424	. 2 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → ((⌊‘(𝐴 / (2↑(𝐼 + 1)))) mod 2) = ((⌊‘((𝐴 / 2) / (2↑𝐼))) mod 2))
29		2nn 12285	. . . 4 ⊢ 2 ∈ ℕ
30	29	a1i 11	. . 3 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → 2 ∈ ℕ)
31		peano2nn0 12512	. . . 4 ⊢ (𝐼 ∈ ℕ₀ → (𝐼 + 1) ∈ ℕ₀)
32	31	3ad2ant3 1136	. . 3 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (𝐼 + 1) ∈ ℕ₀)
33		nn0rp0 13432	. . . 4 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ (0[,)+∞))
34	33	3ad2ant2 1135	. . 3 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → 𝐴 ∈ (0[,)+∞))
35		nn0digval 47286	. . 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 1372	. 2 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → ((𝐼 + 1)(digit‘2)𝐴) = ((⌊‘(𝐴 / (2↑(𝐼 + 1)))) mod 2))
37		nn0rp0 13432	. . . 4 ⊢ ((𝐴 / 2) ∈ ℕ₀ → (𝐴 / 2) ∈ (0[,)+∞))
38	37	3ad2ant1 1134	. . 3 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (𝐴 / 2) ∈ (0[,)+∞))
39		nn0digval 47286	. . 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 1372	. 2 ⊢ (((𝐴 / 2) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ 𝐼 ∈ ℕ₀) → (𝐼(digit‘2)(𝐴 / 2)) = ((⌊‘((𝐴 / 2) / (2↑𝐼))) mod 2))
41	28, 36, 40	3eqtr4d 2783	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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 +∞cpnf 11245 / cdiv 11871 ℕcn 12212 2c2 12267 ℕ₀cn0 12472 [,)cico 13326 ⌊cfl 13755 mod cmo 13834 ↑cexp 14027 digitcdig 47281

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-ico 13330 df-seq 13967 df-exp 14028 df-dig 47282

This theorem is referenced by: dignn0flhalf 47304 nn0sumshdiglemA 47305

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