Theorem flhalf 10316

Description: Ordering relation for the floor of half of an integer. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)

Assertion

Ref	Expression
flhalf	|- ( N e. ZZ -> N <_ ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) )

Proof of Theorem flhalf

Step	Hyp	Ref	Expression
1		peano2z 9303	. . . . . . 7 |- ( N e. ZZ -> ( N + 1 ) e. ZZ )
2		2nn 9094	. . . . . . 7 |- 2 e. NN
3		znq 9638	. . . . . . 7 |- ( ( ( N + 1 ) e. ZZ /\ 2 e. NN ) -> ( ( N + 1 ) / 2 ) e. QQ )
4	1, 2, 3	sylancl 413	. . . . . 6 |- ( N e. ZZ -> ( ( N + 1 ) / 2 ) e. QQ )
5		flqltp1 10293	. . . . . 6 |- ( ( ( N + 1 ) / 2 ) e. QQ -> ( ( N + 1 ) / 2 ) < ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) )
6	4, 5	syl 14	. . . . 5 |- ( N e. ZZ -> ( ( N + 1 ) / 2 ) < ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) )
7		zre 9271	. . . . . . 7 |- ( N e. ZZ -> N e. RR )
8		peano2re 8107	. . . . . . 7 |- ( N e. RR -> ( N + 1 ) e. RR )
9	7, 8	syl 14	. . . . . 6 |- ( N e. ZZ -> ( N + 1 ) e. RR )
10	4	flqcld 10291	. . . . . . . 8 |- ( N e. ZZ -> ( |_ ` ( ( N + 1 ) / 2 ) ) e. ZZ )
11	10	zred 9389	. . . . . . 7 |- ( N e. ZZ -> ( |_ ` ( ( N + 1 ) / 2 ) ) e. RR )
12		1red 7986	. . . . . . 7 |- ( N e. ZZ -> 1 e. RR )
13	11, 12	readdcld 8001	. . . . . 6 |- ( N e. ZZ -> ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) e. RR )
14		2rp 9672	. . . . . . 7 |- 2 e. RR+
15	14	a1i 9	. . . . . 6 |- ( N e. ZZ -> 2 e. RR+ )
16	9, 13, 15	ltdivmuld 9762	. . . . 5 |- ( N e. ZZ -> ( ( ( N + 1 ) / 2 ) < ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) <-> ( N + 1 ) < ( 2 x. ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) ) ) )
17	6, 16	mpbid 147	. . . 4 |- ( N e. ZZ -> ( N + 1 ) < ( 2 x. ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) ) )
18	12	recnd 8000	. . . . . . 7 |- ( N e. ZZ -> 1 e. CC )
19	18	2timesd 9175	. . . . . 6 |- ( N e. ZZ -> ( 2 x. 1 ) = ( 1 + 1 ) )
20	19	oveq2d 5904	. . . . 5 |- ( N e. ZZ -> ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 2 x. 1 ) ) = ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 1 + 1 ) ) )
21		2cnd 9006	. . . . . 6 |- ( N e. ZZ -> 2 e. CC )
22	11	recnd 8000	. . . . . 6 |- ( N e. ZZ -> ( |_ ` ( ( N + 1 ) / 2 ) ) e. CC )
23	21, 22, 18	adddid 7996	. . . . 5 |- ( N e. ZZ -> ( 2 x. ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) ) = ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 2 x. 1 ) ) )
24		2re 9003	. . . . . . . . 9 |- 2 e. RR
25	24	a1i 9	. . . . . . . 8 |- ( N e. ZZ -> 2 e. RR )
26	25, 11	remulcld 8002	. . . . . . 7 |- ( N e. ZZ -> ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. RR )
27	26	recnd 8000	. . . . . 6 |- ( N e. ZZ -> ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. CC )
28	27, 18, 18	addassd 7994	. . . . 5 |- ( N e. ZZ -> ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) = ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 1 + 1 ) ) )
29	20, 23, 28	3eqtr4d 2230	. . . 4 |- ( N e. ZZ -> ( 2 x. ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) ) = ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) )
30	17, 29	breqtrd 4041	. . 3 |- ( N e. ZZ -> ( N + 1 ) < ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) )
31	26, 12	readdcld 8001	. . . 4 |- ( N e. ZZ -> ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) e. RR )
32	7, 31, 12	ltadd1d 8509	. . 3 |- ( N e. ZZ -> ( N < ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) <-> ( N + 1 ) < ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) ) )
33	30, 32	mpbird 167	. 2 |- ( N e. ZZ -> N < ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) )
34		2z 9295	. . . . 5 |- 2 e. ZZ
35	34	a1i 9	. . . 4 |- ( N e. ZZ -> 2 e. ZZ )
36	35, 10	zmulcld 9395	. . 3 |- ( N e. ZZ -> ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. ZZ )
37		zleltp1 9322	. . 3 |- ( ( N e. ZZ /\ ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. ZZ ) -> ( N <_ ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) <-> N < ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) ) )
38	36, 37	mpdan 421	. 2 |- ( N e. ZZ -> ( N <_ ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) <-> N < ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) ) )
39	33, 38	mpbird 167	1 |- ( N e. ZZ -> N <_ ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2158   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   RRcr 7824   1c1 7826   + caddc 7828   x. cmul 7830   < clt 8006   <_ cle 8007   / cdiv 8643   NNcn 8933   2c2 8984   ZZcz 9267   QQcq 9633   RR+crp 9667   |_cfl 10282

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-n0 9191  df-z 9268  df-q 9634  df-rp 9668  df-fl 10284

This theorem is referenced by: (None)