Theorem flltdivnn0lt 10239

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 987	. . . . . . 7 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> K e. NN0 )
2	1	nn0zd 9311	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> K e. ZZ )
3		simp3 989	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> L e. NN )
4		znq 9562	. . . . . . 7 |- ( ( K e. ZZ /\ L e. NN ) -> ( K / L ) e. QQ )
5	4	flqcld 10212	. . . . . 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 9313	. . 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 9563	. . . . 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 988	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> N e. NN0 )
15	14	nn0zd 9311	. . . . 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 9562	. . . . 5 |- ( ( N e. ZZ /\ L e. NN ) -> ( N / L ) e. QQ )
18		qre 9563	. . . . 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 10238	. . . . 5 |- ( ( K e. NN0 /\ L e. NN ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )
22	21	3adant2 1006	. . . 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 9123	. . . . . . 7 |- ( K e. NN0 -> K e. RR )
26		nn0re 9123	. . . . . . 7 |- ( N e. NN0 -> N e. RR )
27		nnre 8864	. . . . . . . 8 |- ( L e. NN -> L e. RR )
28		nngt0 8882	. . . . . . . 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 1174	. . . . . 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 8763	. . . . 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 8023	. 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 968   e. wcel 2136   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   RRcr 7752   0cc0 7753   < clt 7933   <_ cle 7934   / cdiv 8568   NNcn 8857   NN0cn0 9114   ZZcz 9191   QQcq 9557   |_cfl 10203

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-n0 9115  df-z 9192  df-q 9558  df-rp 9590  df-fl 10205

This theorem is referenced by: (None)