Theorem nn0o1gt2 12453

Description: An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.)

Assertion

Ref	Expression
nn0o1gt2	|- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( N = 1 \/ 2 < N ) )

Proof of Theorem nn0o1gt2

Step	Hyp	Ref	Expression
1		elnn0 9392	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		elnnnn0c 9435	. . . . 5 |- ( N e. NN <-> ( N e. NN0 /\ 1 <_ N ) )
3		1z 9493	. . . . . . . 8 |- 1 e. ZZ
4		nn0z 9487	. . . . . . . 8 |- ( N e. NN0 -> N e. ZZ )
5		zleloe 9514	. . . . . . . 8 |- ( ( 1 e. ZZ /\ N e. ZZ ) -> ( 1 <_ N <-> ( 1 < N \/ 1 = N ) ) )
6	3, 4, 5	sylancr 414	. . . . . . 7 |- ( N e. NN0 -> ( 1 <_ N <-> ( 1 < N \/ 1 = N ) ) )
7		1zzd 9494	. . . . . . . . . . . . 13 |- ( N e. NN0 -> 1 e. ZZ )
8		zltp1le 9522	. . . . . . . . . . . . 13 |- ( ( 1 e. ZZ /\ N e. ZZ ) -> ( 1 < N <-> ( 1 + 1 ) <_ N ) )
9	7, 4, 8	syl2anc 411	. . . . . . . . . . . 12 |- ( N e. NN0 -> ( 1 < N <-> ( 1 + 1 ) <_ N ) )
10		1p1e2 9248	. . . . . . . . . . . . . 14 |- ( 1 + 1 ) = 2
11	10	breq1i 4091	. . . . . . . . . . . . 13 |- ( ( 1 + 1 ) <_ N <-> 2 <_ N )
12	11	a1i 9	. . . . . . . . . . . 12 |- ( N e. NN0 -> ( ( 1 + 1 ) <_ N <-> 2 <_ N ) )
13		2z 9495	. . . . . . . . . . . . 13 |- 2 e. ZZ
14		zleloe 9514	. . . . . . . . . . . . 13 |- ( ( 2 e. ZZ /\ N e. ZZ ) -> ( 2 <_ N <-> ( 2 < N \/ 2 = N ) ) )
15	13, 4, 14	sylancr 414	. . . . . . . . . . . 12 |- ( N e. NN0 -> ( 2 <_ N <-> ( 2 < N \/ 2 = N ) ) )
16	9, 12, 15	3bitrd 214	. . . . . . . . . . 11 |- ( N e. NN0 -> ( 1 < N <-> ( 2 < N \/ 2 = N ) ) )
17		olc 716	. . . . . . . . . . . . . 14 |- ( 2 < N -> ( N = 1 \/ 2 < N ) )
18	17	2a1d 23	. . . . . . . . . . . . 13 |- ( 2 < N -> ( N e. NN0 -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) ) )
19		oveq1 6018	. . . . . . . . . . . . . . . . . . . 20 |- ( N = 2 -> ( N + 1 ) = ( 2 + 1 ) )
20	19	oveq1d 6026	. . . . . . . . . . . . . . . . . . 19 |- ( N = 2 -> ( ( N + 1 ) / 2 ) = ( ( 2 + 1 ) / 2 ) )
21	20	eqcoms 2232	. . . . . . . . . . . . . . . . . 18 |- ( 2 = N -> ( ( N + 1 ) / 2 ) = ( ( 2 + 1 ) / 2 ) )
22	21	adantl 277	. . . . . . . . . . . . . . . . 17 |- ( ( N e. NN0 /\ 2 = N ) -> ( ( N + 1 ) / 2 ) = ( ( 2 + 1 ) / 2 ) )
23		2p1e3 9265	. . . . . . . . . . . . . . . . . 18 |- ( 2 + 1 ) = 3
24	23	oveq1i 6021	. . . . . . . . . . . . . . . . 17 |- ( ( 2 + 1 ) / 2 ) = ( 3 / 2 )
25	22, 24	eqtrdi 2278	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ 2 = N ) -> ( ( N + 1 ) / 2 ) = ( 3 / 2 ) )
26	25	eleq1d 2298	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ 2 = N ) -> ( ( ( N + 1 ) / 2 ) e. NN0 <-> ( 3 / 2 ) e. NN0 ) )
27		3halfnz 9565	. . . . . . . . . . . . . . . 16 |- -. ( 3 / 2 ) e. ZZ
28		nn0z 9487	. . . . . . . . . . . . . . . . 17 |- ( ( 3 / 2 ) e. NN0 -> ( 3 / 2 ) e. ZZ )
29	28	pm2.24d 625	. . . . . . . . . . . . . . . 16 |- ( ( 3 / 2 ) e. NN0 -> ( -. ( 3 / 2 ) e. ZZ -> ( N = 1 \/ 2 < N ) ) )
30	27, 29	mpi 15	. . . . . . . . . . . . . . 15 |- ( ( 3 / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) )
31	26, 30	biimtrdi 163	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ 2 = N ) -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) )
32	31	expcom 116	. . . . . . . . . . . . 13 |- ( 2 = N -> ( N e. NN0 -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) ) )
33	18, 32	jaoi 721	. . . . . . . . . . . 12 |- ( ( 2 < N \/ 2 = N ) -> ( N e. NN0 -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) ) )
34	33	com12 30	. . . . . . . . . . 11 |- ( N e. NN0 -> ( ( 2 < N \/ 2 = N ) -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) ) )
35	16, 34	sylbid 150	. . . . . . . . . 10 |- ( N e. NN0 -> ( 1 < N -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) ) )
36	35	com12 30	. . . . . . . . 9 |- ( 1 < N -> ( N e. NN0 -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) ) )
37		orc 717	. . . . . . . . . . 11 |- ( N = 1 -> ( N = 1 \/ 2 < N ) )
38	37	eqcoms 2232	. . . . . . . . . 10 |- ( 1 = N -> ( N = 1 \/ 2 < N ) )
39	38	2a1d 23	. . . . . . . . 9 |- ( 1 = N -> ( N e. NN0 -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) ) )
40	36, 39	jaoi 721	. . . . . . . 8 |- ( ( 1 < N \/ 1 = N ) -> ( N e. NN0 -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) ) )
41	40	com12 30	. . . . . . 7 |- ( N e. NN0 -> ( ( 1 < N \/ 1 = N ) -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) ) )
42	6, 41	sylbid 150	. . . . . 6 |- ( N e. NN0 -> ( 1 <_ N -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) ) )
43	42	imp 124	. . . . 5 |- ( ( N e. NN0 /\ 1 <_ N ) -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) )
44	2, 43	sylbi 121	. . . 4 |- ( N e. NN -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) )
45		oveq1 6018	. . . . . . . 8 |- ( N = 0 -> ( N + 1 ) = ( 0 + 1 ) )
46		0p1e1 9245	. . . . . . . 8 |- ( 0 + 1 ) = 1
47	45, 46	eqtrdi 2278	. . . . . . 7 |- ( N = 0 -> ( N + 1 ) = 1 )
48	47	oveq1d 6026	. . . . . 6 |- ( N = 0 -> ( ( N + 1 ) / 2 ) = ( 1 / 2 ) )
49	48	eleq1d 2298	. . . . 5 |- ( N = 0 -> ( ( ( N + 1 ) / 2 ) e. NN0 <-> ( 1 / 2 ) e. NN0 ) )
50		halfnz 9564	. . . . . 6 |- -. ( 1 / 2 ) e. ZZ
51		nn0z 9487	. . . . . . 7 |- ( ( 1 / 2 ) e. NN0 -> ( 1 / 2 ) e. ZZ )
52	51	pm2.24d 625	. . . . . 6 |- ( ( 1 / 2 ) e. NN0 -> ( -. ( 1 / 2 ) e. ZZ -> ( N = 1 \/ 2 < N ) ) )
53	50, 52	mpi 15	. . . . 5 |- ( ( 1 / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) )
54	49, 53	biimtrdi 163	. . . 4 |- ( N = 0 -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) )
55	44, 54	jaoi 721	. . 3 |- ( ( N e. NN \/ N = 0 ) -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) )
56	1, 55	sylbi 121	. 2 |- ( N e. NN0 -> ( ( ( N + 1 ) / 2 ) e. NN0 -> ( N = 1 \/ 2 < N ) ) )
57	56	imp 124	1 |- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( N = 1 \/ 2 < N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   class class class wbr 4084  (class class class)co 6011   0cc0 8020   1c1 8021   + caddc 8023   < clt 8202   <_ cle 8203   / cdiv 8840   NNcn 9131   2c2 9182   3c3 9183   NN0cn0 9390   ZZcz 9467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4203  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-mulrcl 8119  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-precex 8130  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136  ax-pre-mulgt0 8137  ax-pre-mulext 8138

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-br 4085  df-opab 4147  df-id 4386  df-po 4389  df-iso 4390  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-iota 5282  df-fun 5324  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-reap 8743  df-ap 8750  df-div 8841  df-inn 9132  df-2 9190  df-3 9191  df-4 9192  df-n0 9391  df-z 9468

This theorem is referenced by:  nno  12454  nn0o  12455