Theorem flltdivnn0lt 10394

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 999	. . . . . . 7 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> K e. NN0 )
2	1	nn0zd 9446	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> K e. ZZ )
3		simp3 1001	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> L e. NN )
4		znq 9698	. . . . . . 7 |- ( ( K e. ZZ /\ L e. NN ) -> ( K / L ) e. QQ )
5	4	flqcld 10367	. . . . . 6 |- ( ( K e. ZZ /\ L e. NN ) -> ( |_ ` ( K / L ) ) e. ZZ )
6	2, 3, 5	syl2anc 411	. . . . 5 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> ( |_ ` ( K / L ) ) e. ZZ )
7	6	adantr 276	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( |_ ` ( K / L ) ) e. ZZ )
8	7	zred 9448	. . 3 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( |_ ` ( K / L ) ) e. RR )
9	2	adantr 276	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> K e. ZZ )
10	3	adantr 276	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> L e. NN )
11		qre 9699	. . . . 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 411	. . 3 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( K / L ) e. RR )
14		simp2 1000	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> N e. NN0 )
15	14	nn0zd 9446	. . . . 5 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> N e. ZZ )
16	15	adantr 276	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> N e. ZZ )
17		znq 9698	. . . . 5 |- ( ( N e. ZZ /\ L e. NN ) -> ( N / L ) e. QQ )
18		qre 9699	. . . . 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 411	. . 3 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( N / L ) e. RR )
21		fldivnn0le 10393	. . . . 5 |- ( ( K e. NN0 /\ L e. NN ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )
22	21	3adant2 1018	. . . 4 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )
23	22	adantr 276	. . 3 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )
24		simpr 110	. . . 4 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> K < N )
25		nn0re 9258	. . . . . . 7 |- ( K e. NN0 -> K e. RR )
26		nn0re 9258	. . . . . . 7 |- ( N e. NN0 -> N e. RR )
27		nnre 8997	. . . . . . . 8 |- ( L e. NN -> L e. RR )
28		nngt0 9015	. . . . . . . 8 |- ( L e. NN -> 0 < L )
29	27, 28	jca 306	. . . . . . 7 |- ( L e. NN -> ( L e. RR /\ 0 < L ) )
30	25, 26, 29	3anim123i 1186	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> ( K e. RR /\ N e. RR /\ ( L e. RR /\ 0 < L ) ) )
31	30	adantr 276	. . . . 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 8895	. . . . 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 147	. . 3 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( K / L ) < ( N / L ) )
35	8, 13, 20, 23, 34	lelttrd 8151	. 2 |- ( ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) /\ K < N ) -> ( |_ ` ( K / L ) ) < ( N / L ) )
36	35	ex 115	1 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> ( K < N -> ( |_ ` ( K / L ) ) < ( N / L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   RRcr 7878   0cc0 7879   < clt 8061   <_ cle 8062   / cdiv 8699   NNcn 8990   NN0cn0 9249   ZZcz 9326   QQcq 9693   |_cfl 10358

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 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998

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 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-n0 9250  df-z 9327  df-q 9694  df-rp 9729  df-fl 10360

This theorem is referenced by: (None)