Theorem flhalf 10409

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 9379	. . . . . . 7 |- ( N e. ZZ -> ( N + 1 ) e. ZZ )
2		2nn 9169	. . . . . . 7 |- 2 e. NN
3		znq 9715	. . . . . . 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 10386	. . . . . 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 9347	. . . . . . 7 |- ( N e. ZZ -> N e. RR )
8		peano2re 8179	. . . . . . 7 |- ( N e. RR -> ( N + 1 ) e. RR )
9	7, 8	syl 14	. . . . . 6 |- ( N e. ZZ -> ( N + 1 ) e. RR )
10	4	flqcld 10384	. . . . . . . 8 |- ( N e. ZZ -> ( |_ ` ( ( N + 1 ) / 2 ) ) e. ZZ )
11	10	zred 9465	. . . . . . 7 |- ( N e. ZZ -> ( |_ ` ( ( N + 1 ) / 2 ) ) e. RR )
12		1red 8058	. . . . . . 7 |- ( N e. ZZ -> 1 e. RR )
13	11, 12	readdcld 8073	. . . . . 6 |- ( N e. ZZ -> ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) e. RR )
14		2rp 9750	. . . . . . 7 |- 2 e. RR+
15	14	a1i 9	. . . . . 6 |- ( N e. ZZ -> 2 e. RR+ )
16	9, 13, 15	ltdivmuld 9840	. . . . 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 8072	. . . . . . 7 |- ( N e. ZZ -> 1 e. CC )
19	18	2timesd 9251	. . . . . 6 |- ( N e. ZZ -> ( 2 x. 1 ) = ( 1 + 1 ) )
20	19	oveq2d 5941	. . . . 5 |- ( N e. ZZ -> ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 2 x. 1 ) ) = ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 1 + 1 ) ) )
21		2cnd 9080	. . . . . 6 |- ( N e. ZZ -> 2 e. CC )
22	11	recnd 8072	. . . . . 6 |- ( N e. ZZ -> ( |_ ` ( ( N + 1 ) / 2 ) ) e. CC )
23	21, 22, 18	adddid 8068	. . . . 5 |- ( N e. ZZ -> ( 2 x. ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) ) = ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 2 x. 1 ) ) )
24		2re 9077	. . . . . . . . 9 |- 2 e. RR
25	24	a1i 9	. . . . . . . 8 |- ( N e. ZZ -> 2 e. RR )
26	25, 11	remulcld 8074	. . . . . . 7 |- ( N e. ZZ -> ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. RR )
27	26	recnd 8072	. . . . . 6 |- ( N e. ZZ -> ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. CC )
28	27, 18, 18	addassd 8066	. . . . 5 |- ( N e. ZZ -> ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) = ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + ( 1 + 1 ) ) )
29	20, 23, 28	3eqtr4d 2239	. . . 4 |- ( N e. ZZ -> ( 2 x. ( ( |_ ` ( ( N + 1 ) / 2 ) ) + 1 ) ) = ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) )
30	17, 29	breqtrd 4060	. . 3 |- ( N e. ZZ -> ( N + 1 ) < ( ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) + 1 ) )
31	26, 12	readdcld 8073	. . . 4 |- ( N e. ZZ -> ( ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) + 1 ) e. RR )
32	7, 31, 12	ltadd1d 8582	. . 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 9371	. . . . 5 |- 2 e. ZZ
35	34	a1i 9	. . . 4 |- ( N e. ZZ -> 2 e. ZZ )
36	35, 10	zmulcld 9471	. . 3 |- ( N e. ZZ -> ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) e. ZZ )
37		zleltp1 9398	. . 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 2167   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   RRcr 7895   1c1 7897   + caddc 7899   x. cmul 7901   < clt 8078   <_ cle 8079   / cdiv 8716   NNcn 9007   2c2 9058   ZZcz 9343   QQcq 9710   RR+crp 9745   |_cfl 10375

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-q 9711  df-rp 9746  df-fl 10377

This theorem is referenced by: (None)