Theorem flhalf 10737

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 9680	. . . . . . 7 |- ( N e. ZZ -> ( N + 1 ) e. ZZ )
2		2nn 9466	. . . . . . 7 |- 2 e. NN
3		znq 10024	. . . . . . 7 |- ( ( ( N + 1 ) e. ZZ /\ 2 e. NN ) -> ( ( N + 1 ) / 2 ) e. QQ )
4	1, 2, 3	sylancl 417	. . . . . 6 |- ( N e. ZZ -> ( ( N + 1 ) / 2 ) e. QQ )
5		flqltp1 10714	. . . . . 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 9648	. . . . . . 7 |- ( N e. ZZ -> N e. RR )
8		peano2re 8462	. . . . . . 7 |- ( N e. RR -> ( N + 1 ) e. RR )
9	7, 8	syl 14	. . . . . 6 |- ( N e. ZZ -> ( N + 1 ) e. RR )
10	4	flqcld 10712	. . . . . . . 8 |- ( N e. ZZ -> ( |_ ` ( ( N + 1 ) / 2 ) ) e. ZZ )
11	10	zred 9768	. . . . . . 7 |- ( N e. ZZ -> ( |_ ` ( ( N + 1 ) / 2 ) ) e. RR )
12		1red 8341	. . . . . . 7 |- ( N e. ZZ -> 1 e. RR )
13	11, 12	readdcld 8355	. . . . . 6 |- ( N e. ZZ -> ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) e. RR )
14		2rp 10059	. . . . . . 7 |- 2 e. RR+
15	14	a1i 9	. . . . . 6 |- ( N e. ZZ -> 2 e. RR+ )
16	9, 13, 15	ltdivmuld 10149	. . . . 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 8354	. . . . . . 7 |- ( N e. ZZ -> 1 e. CC )
19	18	2timesd 9548	. . . . . 6 |- ( N e. ZZ -> ( 2 x. 1 ) = ( 1 + 1 ) )
20	19	oveq2d 6101	. . . . 5 |- ( N e. ZZ -> ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 2 x. 1 ) ) = ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 1 + 1 ) ) )
21		2cnd 9377	. . . . . 6 |- ( N e. ZZ -> 2 e. CC )
22	11	recnd 8354	. . . . . 6 |- ( N e. ZZ -> ( |_ ` ( ( N + 1 ) / 2 ) ) e. CC )
23	21, 22, 18	adddid 8350	. . . . 5 |- ( N e. ZZ -> ( 2 x. ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) ) = ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 2 x. 1 ) ) )
24		2re 9374	. . . . . . . . 9 |- 2 e. RR
25	24	a1i 9	. . . . . . . 8 |- ( N e. ZZ -> 2 e. RR )
26	25, 11	remulcld 8356	. . . . . . 7 |- ( N e. ZZ -> ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. RR )
27	26	recnd 8354	. . . . . 6 |- ( N e. ZZ -> ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. CC )
28	27, 18, 18	addassd 8348	. . . . 5 |- ( N e. ZZ -> ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) = ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 1 + 1 ) ) )
29	20, 23, 28	3eqtr4d 2281	. . . 4 |- ( N e. ZZ -> ( 2 x. ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) ) = ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) )
30	17, 29	breqtrd 4156	. . 3 |- ( N e. ZZ -> ( N + 1 ) < ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) )
31	26, 12	readdcld 8355	. . . 4 |- ( N e. ZZ -> ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) e. RR )
32	7, 31, 12	ltadd1d 8866	. . 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 9672	. . . . 5 |- 2 e. ZZ
35	34	a1i 9	. . . 4 |- ( N e. ZZ -> 2 e. ZZ )
36	35, 10	zmulcld 9774	. . 3 |- ( N e. ZZ -> ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. ZZ )
37		zleltp1 9700	. . 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 425	. 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 ) ) ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   RRcr 8178   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   <_ cle 8361   / cdiv 9002   NNcn 9304   2c2 9355   ZZcz 9644   QQcq 10019   RR+crp 10054   |_cfl 10703

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-q 10020  df-rp 10055  df-fl 10705

This theorem is used by: (None)