Theorem flhalf 10534

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 9493	. . . . . . 7 |- ( N e. ZZ -> ( N + 1 ) e. ZZ )
2		2nn 9283	. . . . . . 7 |- 2 e. NN
3		znq 9831	. . . . . . 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 10511	. . . . . 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 9461	. . . . . . 7 |- ( N e. ZZ -> N e. RR )
8		peano2re 8293	. . . . . . 7 |- ( N e. RR -> ( N + 1 ) e. RR )
9	7, 8	syl 14	. . . . . 6 |- ( N e. ZZ -> ( N + 1 ) e. RR )
10	4	flqcld 10509	. . . . . . . 8 |- ( N e. ZZ -> ( |_ ` ( ( N + 1 ) / 2 ) ) e. ZZ )
11	10	zred 9580	. . . . . . 7 |- ( N e. ZZ -> ( |_ ` ( ( N + 1 ) / 2 ) ) e. RR )
12		1red 8172	. . . . . . 7 |- ( N e. ZZ -> 1 e. RR )
13	11, 12	readdcld 8187	. . . . . 6 |- ( N e. ZZ -> ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) e. RR )
14		2rp 9866	. . . . . . 7 |- 2 e. RR+
15	14	a1i 9	. . . . . 6 |- ( N e. ZZ -> 2 e. RR+ )
16	9, 13, 15	ltdivmuld 9956	. . . . 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 8186	. . . . . . 7 |- ( N e. ZZ -> 1 e. CC )
19	18	2timesd 9365	. . . . . 6 |- ( N e. ZZ -> ( 2 x. 1 ) = ( 1 + 1 ) )
20	19	oveq2d 6023	. . . . 5 |- ( N e. ZZ -> ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 2 x. 1 ) ) = ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 1 + 1 ) ) )
21		2cnd 9194	. . . . . 6 |- ( N e. ZZ -> 2 e. CC )
22	11	recnd 8186	. . . . . 6 |- ( N e. ZZ -> ( |_ ` ( ( N + 1 ) / 2 ) ) e. CC )
23	21, 22, 18	adddid 8182	. . . . 5 |- ( N e. ZZ -> ( 2 x. ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) ) = ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 2 x. 1 ) ) )
24		2re 9191	. . . . . . . . 9 |- 2 e. RR
25	24	a1i 9	. . . . . . . 8 |- ( N e. ZZ -> 2 e. RR )
26	25, 11	remulcld 8188	. . . . . . 7 |- ( N e. ZZ -> ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. RR )
27	26	recnd 8186	. . . . . 6 |- ( N e. ZZ -> ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. CC )
28	27, 18, 18	addassd 8180	. . . . 5 |- ( N e. ZZ -> ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) = ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 1 + 1 ) ) )
29	20, 23, 28	3eqtr4d 2272	. . . 4 |- ( N e. ZZ -> ( 2 x. ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) ) = ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) )
30	17, 29	breqtrd 4109	. . 3 |- ( N e. ZZ -> ( N + 1 ) < ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) )
31	26, 12	readdcld 8187	. . . 4 |- ( N e. ZZ -> ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) e. RR )
32	7, 31, 12	ltadd1d 8696	. . 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 9485	. . . . 5 |- 2 e. ZZ
35	34	a1i 9	. . . 4 |- ( N e. ZZ -> 2 e. ZZ )
36	35, 10	zmulcld 9586	. . 3 |- ( N e. ZZ -> ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. ZZ )
37		zleltp1 9513	. . 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 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   RRcr 8009   1c1 8011   + caddc 8013   x. cmul 8015   < clt 8192   <_ cle 8193   / cdiv 8830   NNcn 9121   2c2 9172   ZZcz 9457   QQcq 9826   RR+crp 9861   |_cfl 10500

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-n0 9381  df-z 9458  df-q 9827  df-rp 9862  df-fl 10502

This theorem is referenced by: (None)