Theorem dignn0flhalf 42412

Description: The digits of the rounded half of a nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 7-Jun-2010.)

Assertion

Ref	Expression
dignn0flhalf	|- ( ( A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( ( I + 1 ) (digit ` 2 ) A ) = ( I (digit ` 2 ) ( |_ ` ( A / 2 ) ) ) )

Proof of Theorem dignn0flhalf

Step	Hyp	Ref	Expression
1		eluzge2nn0 11727	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> A e. NN0 )
2		nn0eo 42322	. . . 4 |- ( A e. NN0 -> ( ( A / 2 ) e. NN0 \/ ( ( A + 1 ) / 2 ) e. NN0 ) )
3	1, 2	syl 17	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A / 2 ) e. NN0 \/ ( ( A + 1 ) / 2 ) e. NN0 ) )
4		dignn0ehalf 42411	. . . . . . 7 |- ( ( ( A / 2 ) e. NN0 /\ A e. NN0 /\ I e. NN0 ) -> ( ( I + 1 ) (digit ` 2 ) A ) = ( I (digit ` 2 ) ( A / 2 ) ) )
5	1, 4	syl3an2 1360	. . . . . 6 |- ( ( ( A / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( ( I + 1 ) (digit ` 2 ) A ) = ( I (digit ` 2 ) ( A / 2 ) ) )
6		eluzelz 11697	. . . . . . . . . 10 |- ( A e. ( ZZ>= ` 2 ) -> A e. ZZ )
7		nn0z 11400	. . . . . . . . . 10 |- ( ( A / 2 ) e. NN0 -> ( A / 2 ) e. ZZ )
8		zefldiv2 42324	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( A / 2 ) e. ZZ ) -> ( |_ ` ( A / 2 ) ) = ( A / 2 ) )
9	6, 7, 8	syl2anr 495	. . . . . . . . 9 |- ( ( ( A / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) ) -> ( |_ ` ( A / 2 ) ) = ( A / 2 ) )
10	9	eqcomd 2628	. . . . . . . 8 |- ( ( ( A / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) ) -> ( A / 2 ) = ( |_ ` ( A / 2 ) ) )
11	10	3adant3 1081	. . . . . . 7 |- ( ( ( A / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( A / 2 ) = ( |_ ` ( A / 2 ) ) )
12	11	oveq2d 6666	. . . . . 6 |- ( ( ( A / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( I (digit ` 2 ) ( A / 2 ) ) = ( I (digit ` 2 ) ( |_ ` ( A / 2 ) ) ) )
13	5, 12	eqtrd 2656	. . . . 5 |- ( ( ( A / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( ( I + 1 ) (digit ` 2 ) A ) = ( I (digit ` 2 ) ( |_ ` ( A / 2 ) ) ) )
14	13	3exp 1264	. . . 4 |- ( ( A / 2 ) e. NN0 -> ( A e. ( ZZ>= ` 2 ) -> ( I e. NN0 -> ( ( I + 1 ) (digit ` 2 ) A ) = ( I (digit ` 2 ) ( |_ ` ( A / 2 ) ) ) ) ) )
15	6	3ad2ant2 1083	. . . . . . . 8 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> A e. ZZ )
16		simp2 1062	. . . . . . . . 9 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> A e. ( ZZ>= ` 2 ) )
17		simp1 1061	. . . . . . . . 9 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( ( A + 1 ) / 2 ) e. NN0 )
18		nno 15098	. . . . . . . . 9 |- ( ( A e. ( ZZ>= ` 2 ) /\ ( ( A + 1 ) / 2 ) e. NN0 ) -> ( ( A - 1 ) / 2 ) e. NN )
19	16, 17, 18	syl2anc 693	. . . . . . . 8 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( ( A - 1 ) / 2 ) e. NN )
20		simp3 1063	. . . . . . . 8 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> I e. NN0 )
21		dignn0flhalflem2 42410	. . . . . . . 8 |- ( ( A e. ZZ /\ ( ( A - 1 ) / 2 ) e. NN /\ I e. NN0 ) -> ( |_ ` ( A / ( 2 ^ ( I + 1 ) ) ) ) = ( |_ ` ( ( |_ ` ( A / 2 ) ) / ( 2 ^ I ) ) ) )
22	15, 19, 20, 21	syl3anc 1326	. . . . . . 7 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( |_ ` ( A / ( 2 ^ ( I + 1 ) ) ) ) = ( |_ ` ( ( |_ ` ( A / 2 ) ) / ( 2 ^ I ) ) ) )
23	22	oveq1d 6665	. . . . . 6 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( ( |_ ` ( A / ( 2 ^ ( I + 1 ) ) ) ) mod 2 ) = ( ( |_ ` ( ( |_ ` ( A / 2 ) ) / ( 2 ^ I ) ) ) mod 2 ) )
24		2nn 11185	. . . . . . . 8 |- 2 e. NN
25	24	a1i 11	. . . . . . 7 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> 2 e. NN )
26		peano2nn0 11333	. . . . . . . 8 |- ( I e. NN0 -> ( I + 1 ) e. NN0 )
27	26	3ad2ant3 1084	. . . . . . 7 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( I + 1 ) e. NN0 )
28		nn0rp0 12279	. . . . . . . . 9 |- ( A e. NN0 -> A e. ( 0 [,) +oo ) )
29	1, 28	syl 17	. . . . . . . 8 |- ( A e. ( ZZ>= ` 2 ) -> A e. ( 0 [,) +oo ) )
30	29	3ad2ant2 1083	. . . . . . 7 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> A e. ( 0 [,) +oo ) )
31		nn0digval 42394	. . . . . . 7 |- ( ( 2 e. NN /\ ( I + 1 ) e. NN0 /\ A e. ( 0 [,) +oo ) ) -> ( ( I + 1 ) (digit ` 2 ) A ) = ( ( |_ ` ( A / ( 2 ^ ( I + 1 ) ) ) ) mod 2 ) )
32	25, 27, 30, 31	syl3anc 1326	. . . . . 6 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( ( I + 1 ) (digit ` 2 ) A ) = ( ( |_ ` ( A / ( 2 ^ ( I + 1 ) ) ) ) mod 2 ) )
33		eluzelre 11698	. . . . . . . . . . 11 |- ( A e. ( ZZ>= ` 2 ) -> A e. RR )
34	33	rehalfcld 11279	. . . . . . . . . 10 |- ( A e. ( ZZ>= ` 2 ) -> ( A / 2 ) e. RR )
35	1	nn0ge0d 11354	. . . . . . . . . . 11 |- ( A e. ( ZZ>= ` 2 ) -> 0 <_ A )
36		2re 11090	. . . . . . . . . . . . 13 |- 2 e. RR
37		2pos 11112	. . . . . . . . . . . . 13 |- 0 < 2
38	36, 37	pm3.2i 471	. . . . . . . . . . . 12 |- ( 2 e. RR /\ 0 < 2 )
39	38	a1i 11	. . . . . . . . . . 11 |- ( A e. ( ZZ>= ` 2 ) -> ( 2 e. RR /\ 0 < 2 ) )
40		divge0 10892	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 <_ ( A / 2 ) )
41	33, 35, 39, 40	syl21anc 1325	. . . . . . . . . 10 |- ( A e. ( ZZ>= ` 2 ) -> 0 <_ ( A / 2 ) )
42		flge0nn0 12621	. . . . . . . . . 10 |- ( ( ( A / 2 ) e. RR /\ 0 <_ ( A / 2 ) ) -> ( |_ ` ( A / 2 ) ) e. NN0 )
43	34, 41, 42	syl2anc 693	. . . . . . . . 9 |- ( A e. ( ZZ>= ` 2 ) -> ( |_ ` ( A / 2 ) ) e. NN0 )
44	43	3ad2ant2 1083	. . . . . . . 8 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( |_ ` ( A / 2 ) ) e. NN0 )
45		nn0rp0 12279	. . . . . . . 8 |- ( ( |_ ` ( A / 2 ) ) e. NN0 -> ( |_ ` ( A / 2 ) ) e. ( 0 [,) +oo ) )
46	44, 45	syl 17	. . . . . . 7 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( |_ ` ( A / 2 ) ) e. ( 0 [,) +oo ) )
47		nn0digval 42394	. . . . . . 7 |- ( ( 2 e. NN /\ I e. NN0 /\ ( |_ ` ( A / 2 ) ) e. ( 0 [,) +oo ) ) -> ( I (digit ` 2 ) ( |_ ` ( A / 2 ) ) ) = ( ( |_ ` ( ( |_ ` ( A / 2 ) ) / ( 2 ^ I ) ) ) mod 2 ) )
48	25, 20, 46, 47	syl3anc 1326	. . . . . 6 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( I (digit ` 2 ) ( |_ ` ( A / 2 ) ) ) = ( ( |_ ` ( ( |_ ` ( A / 2 ) ) / ( 2 ^ I ) ) ) mod 2 ) )
49	23, 32, 48	3eqtr4d 2666	. . . . 5 |- ( ( ( ( A + 1 ) / 2 ) e. NN0 /\ A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( ( I + 1 ) (digit ` 2 ) A ) = ( I (digit ` 2 ) ( |_ ` ( A / 2 ) ) ) )
50	49	3exp 1264	. . . 4 |- ( ( ( A + 1 ) / 2 ) e. NN0 -> ( A e. ( ZZ>= ` 2 ) -> ( I e. NN0 -> ( ( I + 1 ) (digit ` 2 ) A ) = ( I (digit ` 2 ) ( |_ ` ( A / 2 ) ) ) ) ) )
51	14, 50	jaoi 394	. . 3 |- ( ( ( A / 2 ) e. NN0 \/ ( ( A + 1 ) / 2 ) e. NN0 ) -> ( A e. ( ZZ>= ` 2 ) -> ( I e. NN0 -> ( ( I + 1 ) (digit ` 2 ) A ) = ( I (digit ` 2 ) ( |_ ` ( A / 2 ) ) ) ) ) )
52	3, 51	mpcom 38	. 2 |- ( A e. ( ZZ>= ` 2 ) -> ( I e. NN0 -> ( ( I + 1 ) (digit ` 2 ) A ) = ( I (digit ` 2 ) ( |_ ` ( A / 2 ) ) ) ) )
53	52	imp 445	1 |- ( ( A e. ( ZZ>= ` 2 ) /\ I e. NN0 ) -> ( ( I + 1 ) (digit ` 2 ) A ) = ( I (digit ` 2 ) ( |_ ` ( A / 2 ) ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   [,)cico 12177   |_cfl 12591   mod cmo 12668   ^cexp 12860  digitcdig 42389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-dig 42390

This theorem is referenced by:  nn0sumshdiglemB  42414