Theorem ltdifltdiv 12635

Description: If the dividend of a division is less than the difference between a real number and the divisor, the floor function of the division plus 1 is less than the division of the real number by the divisor. (Contributed by Alexander van der Vekens, 14-Apr-2018.)

Assertion

Ref	Expression
ltdifltdiv	|- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A < ( C - B ) -> ( ( |_ ` ( A / B ) ) + 1 ) < ( C / B ) ) )

Proof of Theorem ltdifltdiv

Step	Hyp	Ref	Expression
1		refldivcl 12624	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. RR )
2		peano2re 10209	. . . . . 6 |- ( ( |_ ` ( A / B ) ) e. RR -> ( ( |_ ` ( A / B ) ) + 1 ) e. RR )
3	1, 2	syl 17	. . . . 5 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( |_ ` ( A / B ) ) + 1 ) e. RR )
4	3	3adant3 1081	. . . 4 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( |_ ` ( A / B ) ) + 1 ) e. RR )
5	4	adantr 481	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( |_ ` ( A / B ) ) + 1 ) e. RR )
6		rerpdivcl 11861	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR )
7		peano2re 10209	. . . . . 6 |- ( ( A / B ) e. RR -> ( ( A / B ) + 1 ) e. RR )
8	6, 7	syl 17	. . . . 5 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) + 1 ) e. RR )
9	8	3adant3 1081	. . . 4 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A / B ) + 1 ) e. RR )
10	9	adantr 481	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( A / B ) + 1 ) e. RR )
11		rerpdivcl 11861	. . . . . 6 |- ( ( C e. RR /\ B e. RR+ ) -> ( C / B ) e. RR )
12	11	ancoms 469	. . . . 5 |- ( ( B e. RR+ /\ C e. RR ) -> ( C / B ) e. RR )
13	12	3adant1 1079	. . . 4 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( C / B ) e. RR )
14	13	adantr 481	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( C / B ) e. RR )
15	1	3adant3 1081	. . . . 5 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( |_ ` ( A / B ) ) e. RR )
16	15	adantr 481	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( |_ ` ( A / B ) ) e. RR )
17	6	3adant3 1081	. . . . 5 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A / B ) e. RR )
18	17	adantr 481	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( A / B ) e. RR )
19		1red 10055	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> 1 e. RR )
20		3simpa 1058	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A e. RR /\ B e. RR+ ) )
21	20	adantr 481	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( A e. RR /\ B e. RR+ ) )
22		fldivle 12632	. . . . 5 |- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) <_ ( A / B ) )
23	21, 22	syl 17	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( |_ ` ( A / B ) ) <_ ( A / B ) )
24	16, 18, 19, 23	leadd1dd 10641	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( |_ ` ( A / B ) ) + 1 ) <_ ( ( A / B ) + 1 ) )
25		rpre 11839	. . . . . . 7 |- ( B e. RR+ -> B e. RR )
26		ltaddsub 10502	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) < C <-> A < ( C - B ) ) )
27	25, 26	syl3an2 1360	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A + B ) < C <-> A < ( C - B ) ) )
28	27	biimpar 502	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( A + B ) < C )
29		recn 10026	. . . . . . . . . . 11 |- ( ( A / B ) e. RR -> ( A / B ) e. CC )
30	6, 29	syl 17	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. CC )
31	30	3adant3 1081	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A / B ) e. CC )
32		1cnd 10056	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> 1 e. CC )
33		rpcn 11841	. . . . . . . . . 10 |- ( B e. RR+ -> B e. CC )
34	33	3ad2ant2 1083	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B e. CC )
35	31, 32, 34	adddird 10065	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) + 1 ) x. B ) = ( ( ( A / B ) x. B ) + ( 1 x. B ) ) )
36		recn 10026	. . . . . . . . . . 11 |- ( A e. RR -> A e. CC )
37	36	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> A e. CC )
38		rpne0 11848	. . . . . . . . . . 11 |- ( B e. RR+ -> B =/= 0 )
39	38	3ad2ant2 1083	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B =/= 0 )
40	37, 34, 39	divcan1d 10802	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A / B ) x. B ) = A )
41	33	mulid2d 10058	. . . . . . . . . 10 |- ( B e. RR+ -> ( 1 x. B ) = B )
42	41	3ad2ant2 1083	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( 1 x. B ) = B )
43	40, 42	oveq12d 6668	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) x. B ) + ( 1 x. B ) ) = ( A + B ) )
44	35, 43	eqtrd 2656	. . . . . . 7 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) + 1 ) x. B ) = ( A + B ) )
45		recn 10026	. . . . . . . . 9 |- ( C e. RR -> C e. CC )
46	45	3ad2ant3 1084	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> C e. CC )
47	46, 34, 39	divcan1d 10802	. . . . . . 7 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( C / B ) x. B ) = C )
48	44, 47	breq12d 4666	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( ( A / B ) + 1 ) x. B ) < ( ( C / B ) x. B ) <-> ( A + B ) < C ) )
49	48	adantr 481	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( ( ( A / B ) + 1 ) x. B ) < ( ( C / B ) x. B ) <-> ( A + B ) < C ) )
50	28, 49	mpbird 247	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( ( A / B ) + 1 ) x. B ) < ( ( C / B ) x. B ) )
51	17, 7	syl 17	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A / B ) + 1 ) e. RR )
52		simp2 1062	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B e. RR+ )
53	51, 13, 52	ltmul1d 11913	. . . . 5 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) + 1 ) < ( C / B ) <-> ( ( ( A / B ) + 1 ) x. B ) < ( ( C / B ) x. B ) ) )
54	53	adantr 481	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( ( A / B ) + 1 ) < ( C / B ) <-> ( ( ( A / B ) + 1 ) x. B ) < ( ( C / B ) x. B ) ) )
55	50, 54	mpbird 247	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( A / B ) + 1 ) < ( C / B ) )
56	5, 10, 14, 24, 55	lelttrd 10195	. 2 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( |_ ` ( A / B ) ) + 1 ) < ( C / B ) )
57	56	ex 450	1 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A < ( C - B ) -> ( ( |_ ` ( A / B ) ) + 1 ) < ( C / B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   RR+crp 11832   |_cfl 12591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593

This theorem is referenced by: (None)