Theorem flltdivnn0lt 10260

Description: The floor function of a division of a nonnegative integer by a positive integer is less than the division of a greater dividend by the same positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)

Assertion

Ref	Expression
flltdivnn0lt	|- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> ( K < N -> ( |_ ` ( K / L ) ) < ( N / L ) ) )

Proof of Theorem flltdivnn0lt

Step	Hyp	Ref	Expression
1		simp1 992	. . . . . . 7 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> K e. NN0 )
2	1	nn0zd 9332	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> K e. ZZ )
3		simp3 994	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> L e. NN )
4		znq 9583	. . . . . . 7 |- ( ( K e. ZZ /\ L e. NN ) -> ( K / L ) e. QQ )
5	4	flqcld 10233	. . . . . 6 |- ( ( K e. ZZ /\ L e. NN ) -> ( |_ ` ( K / L ) ) e. ZZ )
6	2, 3, 5	syl2anc 409	. . . . 5 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> ( |_ ` ( K / L ) ) e. ZZ )
7	6	adantr 274	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( |_ ` ( K / L ) ) e. ZZ )
8	7	zred 9334	. . 3 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( |_ ` ( K / L ) ) e. RR )
9	2	adantr 274	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> K e. ZZ )
10	3	adantr 274	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> L e. NN )
11		qre 9584	. . . . 5 |- ( ( K / L ) e. QQ -> ( K / L ) e. RR )
12	4, 11	syl 14	. . . 4 |- ( ( K e. ZZ /\ L e. NN ) -> ( K / L ) e. RR )
13	9, 10, 12	syl2anc 409	. . 3 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( K / L ) e. RR )
14		simp2 993	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> N e. NN0 )
15	14	nn0zd 9332	. . . . 5 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> N e. ZZ )
16	15	adantr 274	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> N e. ZZ )
17		znq 9583	. . . . 5 |- ( ( N e. ZZ /\ L e. NN ) -> ( N / L ) e. QQ )
18		qre 9584	. . . . 5 |- ( ( N / L ) e. QQ -> ( N / L ) e. RR )
19	17, 18	syl 14	. . . 4 |- ( ( N e. ZZ /\ L e. NN ) -> ( N / L ) e. RR )
20	16, 10, 19	syl2anc 409	. . 3 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( N / L ) e. RR )
21		fldivnn0le 10259	. . . . 5 |- ( ( K e. NN0 /\ L e. NN ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )
22	21	3adant2 1011	. . . 4 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )
23	22	adantr 274	. . 3 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )
24		simpr 109	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> K < N )
25		nn0re 9144	. . . . . . 7 |- ( K e. NN0 -> K e. RR )
26		nn0re 9144	. . . . . . 7 |- ( N e. NN0 -> N e. RR )
27		nnre 8885	. . . . . . . 8 |- ( L e. NN -> L e. RR )
28		nngt0 8903	. . . . . . . 8 |- ( L e. NN -> 0 < L )
29	27, 28	jca 304	. . . . . . 7 |- ( L e. NN -> ( L e. RR /\ 0 < L ) )
30	25, 26, 29	3anim123i 1179	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> ( K e. RR /\ N e. RR /\ ( L e. RR /\ 0 < L ) ) )
31	30	adantr 274	. . . . 5 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( K e. RR /\ N e. RR /\ ( L e. RR /\ 0 < L ) ) )
32		ltdiv1 8784	. . . . 5 |- ( ( K e. RR /\ N e. RR /\ ( L e. RR /\ 0 < L ) ) -> ( K < N <-> ( K / L ) < ( N / L ) ) )
33	31, 32	syl 14	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( K < N <-> ( K / L ) < ( N / L ) ) )
34	24, 33	mpbid 146	. . 3 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( K / L ) < ( N / L ) )
35	8, 13, 20, 23, 34	lelttrd 8044	. 2 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( |_ ` ( K / L ) ) < ( N / L ) )
36	35	ex 114	1 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> ( K < N -> ( |_ ` ( K / L ) ) < ( N / L ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   RRcr 7773   0cc0 7774   < clt 7954   <_ cle 7955   / cdiv 8589   NNcn 8878   NN0cn0 9135   ZZcz 9212   QQcq 9578   |_cfl 10224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-q 9579  df-rp 9611  df-fl 10226

This theorem is referenced by: (None)