Theorem mod2eq1n2dvds 15071

Description: An integer is 1 modulo 2 iff it is odd (i.e. not divisible by 2), see example 3 in [ApostolNT] p. 107. (Contributed by AV, 24-May-2020.) (Proof shortened by AV, 5-Jul-2020.)

Assertion

Ref	Expression
mod2eq1n2dvds	|- ( N e. ZZ -> ( ( N mod 2 ) = 1 <-> -. 2 || N ) )

Proof of Theorem mod2eq1n2dvds

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zeo 11463	. . . 4 |- ( N e. ZZ -> ( ( N / 2 ) e. ZZ \/ ( ( N + 1 ) / 2 ) e. ZZ ) )
2		zre 11381	. . . . . . . . 9 |- ( N e. ZZ -> N e. RR )
3		2rp 11837	. . . . . . . . 9 |- 2 e. RR+
4		mod0 12675	. . . . . . . . 9 |- ( ( N e. RR /\ 2 e. RR+ ) -> ( ( N mod 2 ) = 0 <-> ( N / 2 ) e. ZZ ) )
5	2, 3, 4	sylancl 694	. . . . . . . 8 |- ( N e. ZZ -> ( ( N mod 2 ) = 0 <-> ( N / 2 ) e. ZZ ) )
6	5	biimpar 502	. . . . . . 7 |- ( ( N e. ZZ /\ ( N / 2 ) e. ZZ ) -> ( N mod 2 ) = 0 )
7		eqeq1 2626	. . . . . . . 8 |- ( ( N mod 2 ) = 0 -> ( ( N mod 2 ) = 1 <-> 0 = 1 ) )
8		0ne1 11088	. . . . . . . . 9 |- 0 =/= 1
9		eqneqall 2805	. . . . . . . . 9 |- ( 0 = 1 -> ( 0 =/= 1 -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
10	8, 9	mpi 20	. . . . . . . 8 |- ( 0 = 1 -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
11	7, 10	syl6bi 243	. . . . . . 7 |- ( ( N mod 2 ) = 0 -> ( ( N mod 2 ) = 1 -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
12	6, 11	syl 17	. . . . . 6 |- ( ( N e. ZZ /\ ( N / 2 ) e. ZZ ) -> ( ( N mod 2 ) = 1 -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
13	12	expcom 451	. . . . 5 |- ( ( N / 2 ) e. ZZ -> ( N e. ZZ -> ( ( N mod 2 ) = 1 -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) ) )
14		peano2zm 11420	. . . . . . . . 9 |- ( ( ( N + 1 ) / 2 ) e. ZZ -> ( ( ( N + 1 ) / 2 ) - 1 ) e. ZZ )
15		zcn 11382	. . . . . . . . . . . 12 |- ( N e. ZZ -> N e. CC )
16		xp1d2m1eqxm1d2 11286	. . . . . . . . . . . 12 |- ( N e. CC -> ( ( ( N + 1 ) / 2 ) - 1 ) = ( ( N - 1 ) / 2 ) )
17	15, 16	syl 17	. . . . . . . . . . 11 |- ( N e. ZZ -> ( ( ( N + 1 ) / 2 ) - 1 ) = ( ( N - 1 ) / 2 ) )
18	17	eleq1d 2686	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( ( ( N + 1 ) / 2 ) - 1 ) e. ZZ <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
19	18	biimpd 219	. . . . . . . . 9 |- ( N e. ZZ -> ( ( ( ( N + 1 ) / 2 ) - 1 ) e. ZZ -> ( ( N - 1 ) / 2 ) e. ZZ ) )
20	14, 19	mpan9 486	. . . . . . . 8 |- ( ( ( ( N + 1 ) / 2 ) e. ZZ /\ N e. ZZ ) -> ( ( N - 1 ) / 2 ) e. ZZ )
21		oveq2 6658	. . . . . . . . . . 11 |- ( n = ( ( N - 1 ) / 2 ) -> ( 2 x. n ) = ( 2 x. ( ( N - 1 ) / 2 ) ) )
22	21	adantl 482	. . . . . . . . . 10 |- ( ( ( ( ( N + 1 ) / 2 ) e. ZZ /\ N e. ZZ ) /\ n = ( ( N - 1 ) / 2 ) ) -> ( 2 x. n ) = ( 2 x. ( ( N - 1 ) / 2 ) ) )
23	22	oveq1d 6665	. . . . . . . . 9 |- ( ( ( ( ( N + 1 ) / 2 ) e. ZZ /\ N e. ZZ ) /\ n = ( ( N - 1 ) / 2 ) ) -> ( ( 2 x. n ) + 1 ) = ( ( 2 x. ( ( N - 1 ) / 2 ) ) + 1 ) )
24		peano2zm 11420	. . . . . . . . . . . . . 14 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
25	24	zcnd 11483	. . . . . . . . . . . . 13 |- ( N e. ZZ -> ( N - 1 ) e. CC )
26		2cnd 11093	. . . . . . . . . . . . 13 |- ( N e. ZZ -> 2 e. CC )
27		2ne0 11113	. . . . . . . . . . . . . 14 |- 2 =/= 0
28	27	a1i 11	. . . . . . . . . . . . 13 |- ( N e. ZZ -> 2 =/= 0 )
29	25, 26, 28	divcan2d 10803	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( 2 x. ( ( N - 1 ) / 2 ) ) = ( N - 1 ) )
30	29	oveq1d 6665	. . . . . . . . . . 11 |- ( N e. ZZ -> ( ( 2 x. ( ( N - 1 ) / 2 ) ) + 1 ) = ( ( N - 1 ) + 1 ) )
31		npcan1 10455	. . . . . . . . . . . 12 |- ( N e. CC -> ( ( N - 1 ) + 1 ) = N )
32	15, 31	syl 17	. . . . . . . . . . 11 |- ( N e. ZZ -> ( ( N - 1 ) + 1 ) = N )
33	30, 32	eqtrd 2656	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 x. ( ( N - 1 ) / 2 ) ) + 1 ) = N )
34	33	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ( ( N + 1 ) / 2 ) e. ZZ /\ N e. ZZ ) /\ n = ( ( N - 1 ) / 2 ) ) -> ( ( 2 x. ( ( N - 1 ) / 2 ) ) + 1 ) = N )
35	23, 34	eqtrd 2656	. . . . . . . 8 |- ( ( ( ( ( N + 1 ) / 2 ) e. ZZ /\ N e. ZZ ) /\ n = ( ( N - 1 ) / 2 ) ) -> ( ( 2 x. n ) + 1 ) = N )
36	20, 35	rspcedeq1vd 3318	. . . . . . 7 |- ( ( ( ( N + 1 ) / 2 ) e. ZZ /\ N e. ZZ ) -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
37	36	a1d 25	. . . . . 6 |- ( ( ( ( N + 1 ) / 2 ) e. ZZ /\ N e. ZZ ) -> ( ( N mod 2 ) = 1 -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
38	37	ex 450	. . . . 5 |- ( ( ( N + 1 ) / 2 ) e. ZZ -> ( N e. ZZ -> ( ( N mod 2 ) = 1 -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) ) )
39	13, 38	jaoi 394	. . . 4 |- ( ( ( N / 2 ) e. ZZ \/ ( ( N + 1 ) / 2 ) e. ZZ ) -> ( N e. ZZ -> ( ( N mod 2 ) = 1 -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) ) )
40	1, 39	mpcom 38	. . 3 |- ( N e. ZZ -> ( ( N mod 2 ) = 1 -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
41		oveq1 6657	. . . . . . 7 |- ( N = ( ( 2 x. n ) + 1 ) -> ( N mod 2 ) = ( ( ( 2 x. n ) + 1 ) mod 2 ) )
42	41	eqcoms 2630	. . . . . 6 |- ( ( ( 2 x. n ) + 1 ) = N -> ( N mod 2 ) = ( ( ( 2 x. n ) + 1 ) mod 2 ) )
43		2cnd 11093	. . . . . . . . . . . . . 14 |- ( n e. ZZ -> 2 e. CC )
44		zcn 11382	. . . . . . . . . . . . . 14 |- ( n e. ZZ -> n e. CC )
45	43, 44	mulcomd 10061	. . . . . . . . . . . . 13 |- ( n e. ZZ -> ( 2 x. n ) = ( n x. 2 ) )
46	45	oveq1d 6665	. . . . . . . . . . . 12 |- ( n e. ZZ -> ( ( 2 x. n ) mod 2 ) = ( ( n x. 2 ) mod 2 ) )
47		mulmod0 12676	. . . . . . . . . . . . 13 |- ( ( n e. ZZ /\ 2 e. RR+ ) -> ( ( n x. 2 ) mod 2 ) = 0 )
48	3, 47	mpan2 707	. . . . . . . . . . . 12 |- ( n e. ZZ -> ( ( n x. 2 ) mod 2 ) = 0 )
49	46, 48	eqtrd 2656	. . . . . . . . . . 11 |- ( n e. ZZ -> ( ( 2 x. n ) mod 2 ) = 0 )
50	49	oveq1d 6665	. . . . . . . . . 10 |- ( n e. ZZ -> ( ( ( 2 x. n ) mod 2 ) + 1 ) = ( 0 + 1 ) )
51		0p1e1 11132	. . . . . . . . . 10 |- ( 0 + 1 ) = 1
52	50, 51	syl6eq 2672	. . . . . . . . 9 |- ( n e. ZZ -> ( ( ( 2 x. n ) mod 2 ) + 1 ) = 1 )
53	52	oveq1d 6665	. . . . . . . 8 |- ( n e. ZZ -> ( ( ( ( 2 x. n ) mod 2 ) + 1 ) mod 2 ) = ( 1 mod 2 ) )
54		2z 11409	. . . . . . . . . . . 12 |- 2 e. ZZ
55	54	a1i 11	. . . . . . . . . . 11 |- ( n e. ZZ -> 2 e. ZZ )
56		id 22	. . . . . . . . . . 11 |- ( n e. ZZ -> n e. ZZ )
57	55, 56	zmulcld 11488	. . . . . . . . . 10 |- ( n e. ZZ -> ( 2 x. n ) e. ZZ )
58	57	zred 11482	. . . . . . . . 9 |- ( n e. ZZ -> ( 2 x. n ) e. RR )
59		1red 10055	. . . . . . . . 9 |- ( n e. ZZ -> 1 e. RR )
60	3	a1i 11	. . . . . . . . 9 |- ( n e. ZZ -> 2 e. RR+ )
61		modaddmod 12709	. . . . . . . . 9 |- ( ( ( 2 x. n ) e. RR /\ 1 e. RR /\ 2 e. RR+ ) -> ( ( ( ( 2 x. n ) mod 2 ) + 1 ) mod 2 ) = ( ( ( 2 x. n ) + 1 ) mod 2 ) )
62	58, 59, 60, 61	syl3anc 1326	. . . . . . . 8 |- ( n e. ZZ -> ( ( ( ( 2 x. n ) mod 2 ) + 1 ) mod 2 ) = ( ( ( 2 x. n ) + 1 ) mod 2 ) )
63		2re 11090	. . . . . . . . . 10 |- 2 e. RR
64		1lt2 11194	. . . . . . . . . 10 |- 1 < 2
65	63, 64	pm3.2i 471	. . . . . . . . 9 |- ( 2 e. RR /\ 1 < 2 )
66		1mod 12702	. . . . . . . . 9 |- ( ( 2 e. RR /\ 1 < 2 ) -> ( 1 mod 2 ) = 1 )
67	65, 66	mp1i 13	. . . . . . . 8 |- ( n e. ZZ -> ( 1 mod 2 ) = 1 )
68	53, 62, 67	3eqtr3d 2664	. . . . . . 7 |- ( n e. ZZ -> ( ( ( 2 x. n ) + 1 ) mod 2 ) = 1 )
69	68	adantl 482	. . . . . 6 |- ( ( N e. ZZ /\ n e. ZZ ) -> ( ( ( 2 x. n ) + 1 ) mod 2 ) = 1 )
70	42, 69	sylan9eqr 2678	. . . . 5 |- ( ( ( N e. ZZ /\ n e. ZZ ) /\ ( ( 2 x. n ) + 1 ) = N ) -> ( N mod 2 ) = 1 )
71	70	ex 450	. . . 4 |- ( ( N e. ZZ /\ n e. ZZ ) -> ( ( ( 2 x. n ) + 1 ) = N -> ( N mod 2 ) = 1 ) )
72	71	rexlimdva 3031	. . 3 |- ( N e. ZZ -> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N -> ( N mod 2 ) = 1 ) )
73	40, 72	impbid 202	. 2 |- ( N e. ZZ -> ( ( N mod 2 ) = 1 <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
74		odd2np1 15065	. 2 |- ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
75	73, 74	bitr4d 271	1 |- ( N e. ZZ -> ( ( N mod 2 ) = 1 <-> -. 2 || N ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   2c2 11070   ZZcz 11377   RR+crp 11832   mod cmo 12668   || cdvds 14983

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-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-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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-dvds 14984

This theorem is referenced by:  2lgslem3b1  25126  2lgslem3c1  25127  ex-mod  27306  dig2nn1st  42399  0dig2nn0o  42407  dig2bits  42408