Theorem flltdivnn0lt 10298

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 997	. . . . . . 7 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> K e. NN0 )
2	1	nn0zd 9368	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> K e. ZZ )
3		simp3 999	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> L e. NN )
4		znq 9619	. . . . . . 7 |- ( ( K e. ZZ /\ L e. NN ) -> ( K / L ) e. QQ )
5	4	flqcld 10271	. . . . . 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 9370	. . 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 9620	. . . . 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 998	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> N e. NN0 )
15	14	nn0zd 9368	. . . . 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 9619	. . . . 5 |- ( ( N e. ZZ /\ L e. NN ) -> ( N / L ) e. QQ )
18		qre 9620	. . . . 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 10297	. . . . 5 |- ( ( K e. NN0 /\ L e. NN ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )
22	21	3adant2 1016	. . . 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 9180	. . . . . . 7 |- ( K e. NN0 -> K e. RR )
26		nn0re 9180	. . . . . . 7 |- ( N e. NN0 -> N e. RR )
27		nnre 8921	. . . . . . . 8 |- ( L e. NN -> L e. RR )
28		nngt0 8939	. . . . . . . 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 1184	. . . . . 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 8820	. . . . 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 8077	. 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 978   e. wcel 2148   class class class wbr 4002   ` cfv 5214  (class class class)co 5871   RRcr 7806   0cc0 7807   < clt 7987   <_ cle 7988   / cdiv 8624   NNcn 8914   NN0cn0 9171   ZZcz 9248   QQcq 9614   |_cfl 10262

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7898  ax-resscn 7899  ax-1cn 7900  ax-1re 7901  ax-icn 7902  ax-addcl 7903  ax-addrcl 7904  ax-mulcl 7905  ax-mulrcl 7906  ax-addcom 7907  ax-mulcom 7908  ax-addass 7909  ax-mulass 7910  ax-distr 7911  ax-i2m1 7912  ax-0lt1 7913  ax-1rid 7914  ax-0id 7915  ax-rnegex 7916  ax-precex 7917  ax-cnre 7918  ax-pre-ltirr 7919  ax-pre-ltwlin 7920  ax-pre-lttrn 7921  ax-pre-apti 7922  ax-pre-ltadd 7923  ax-pre-mulgt0 7924  ax-pre-mulext 7925  ax-arch 7926

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-po 4295  df-iso 4296  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-fv 5222  df-riota 5827  df-ov 5874  df-oprab 5875  df-mpo 5876  df-1st 6137  df-2nd 6138  df-pnf 7989  df-mnf 7990  df-xr 7991  df-ltxr 7992  df-le 7993  df-sub 8125  df-neg 8126  df-reap 8527  df-ap 8534  df-div 8625  df-inn 8915  df-n0 9172  df-z 9249  df-q 9615  df-rp 9649  df-fl 10264

This theorem is referenced by: (None)