Theorem flltdivnn0lt 10524

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 1021	. . . . . . 7 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> K e. NN0 )
2	1	nn0zd 9567	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> K e. ZZ )
3		simp3 1023	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> L e. NN )
4		znq 9819	. . . . . . 7 |- ( ( K e. ZZ /\ L e. NN ) -> ( K / L ) e. QQ )
5	4	flqcld 10497	. . . . . 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 9569	. . 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 9820	. . . . 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 1022	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> N e. NN0 )
15	14	nn0zd 9567	. . . . 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 9819	. . . . 5 |- ( ( N e. ZZ /\ L e. NN ) -> ( N / L ) e. QQ )
18		qre 9820	. . . . 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 10523	. . . . 5 |- ( ( K e. NN0 /\ L e. NN ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )
22	21	3adant2 1040	. . . 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 9378	. . . . . . 7 |- ( K e. NN0 -> K e. RR )
26		nn0re 9378	. . . . . . 7 |- ( N e. NN0 -> N e. RR )
27		nnre 9117	. . . . . . . 8 |- ( L e. NN -> L e. RR )
28		nngt0 9135	. . . . . . . 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 1208	. . . . . 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 9015	. . . . 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 8271	. 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 1002   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   RRcr 7998   0cc0 7999   < clt 8181   <_ cle 8182   / cdiv 8819   NNcn 9110   NN0cn0 9369   ZZcz 9446   QQcq 9814   |_cfl 10488

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-n0 9370  df-z 9447  df-q 9815  df-rp 9850  df-fl 10490

This theorem is referenced by: (None)