Theorem ltdifltdiv 11929

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 11920	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. RR )
2		peano2re 9748	. . . . . 6 |- ( ( |_ ` ( A / B ) ) e. RR -> ( ( |_ ` ( A / B ) ) + 1 ) e. RR )
3	1, 2	syl 16	. . . . 5 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( |_ ` ( A / B ) ) + 1 ) e. RR )
4	3	3adant3 1016	. . . 4 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( |_ ` ( A / B ) ) + 1 ) e. RR )
5	4	adantr 465	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( |_ ` ( A / B ) ) + 1 ) e. RR )
6		rerpdivcl 11243	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR )
7		peano2re 9748	. . . . . 6 |- ( ( A / B ) e. RR -> ( ( A / B ) + 1 ) e. RR )
8	6, 7	syl 16	. . . . 5 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) + 1 ) e. RR )
9	8	3adant3 1016	. . . 4 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A / B ) + 1 ) e. RR )
10	9	adantr 465	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( A / B ) + 1 ) e. RR )
11		rerpdivcl 11243	. . . . . 6 |- ( ( C e. RR /\ B e. RR+ ) -> ( C / B ) e. RR )
12	11	ancoms 453	. . . . 5 |- ( ( B e. RR+ /\ C e. RR ) -> ( C / B ) e. RR )
13	12	3adant1 1014	. . . 4 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( C / B ) e. RR )
14	13	adantr 465	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( C / B ) e. RR )
15	1	3adant3 1016	. . . . 5 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( |_ ` ( A / B ) ) e. RR )
16	15	adantr 465	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( |_ ` ( A / B ) ) e. RR )
17	6	3adant3 1016	. . . . 5 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A / B ) e. RR )
18	17	adantr 465	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( A / B ) e. RR )
19		1red 9607	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> 1 e. RR )
20		3simpa 993	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A e. RR /\ B e. RR+ ) )
21	20	adantr 465	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( A e. RR /\ B e. RR+ ) )
22		fldivle 11926	. . . . 5 |- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) <_ ( A / B ) )
23	21, 22	syl 16	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( |_ ` ( A / B ) ) <_ ( A / B ) )
24	16, 18, 19, 23	leadd1dd 10162	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( |_ ` ( A / B ) ) + 1 ) <_ ( ( A / B ) + 1 ) )
25		rpre 11222	. . . . . . 7 |- ( B e. RR+ -> B e. RR )
26		ltaddsub 10022	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) < C <-> A < ( C - B ) ) )
27	25, 26	syl3an2 1262	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A + B ) < C <-> A < ( C - B ) ) )
28	27	biimpar 485	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( A + B ) < C )
29		recn 9578	. . . . . . . . . . 11 |- ( ( A / B ) e. RR -> ( A / B ) e. CC )
30	6, 29	syl 16	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. CC )
31	30	3adant3 1016	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A / B ) e. CC )
32		ax-1cn 9546	. . . . . . . . . 10 |- 1 e. CC
33	32	a1i 11	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> 1 e. CC )
34		rpcn 11224	. . . . . . . . . 10 |- ( B e. RR+ -> B e. CC )
35	34	3ad2ant2 1018	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B e. CC )
36	31, 33, 35	adddird 9617	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) + 1 ) x. B ) = ( ( ( A / B ) x. B ) + ( 1 x. B ) ) )
37		recn 9578	. . . . . . . . . . 11 |- ( A e. RR -> A e. CC )
38	37	3ad2ant1 1017	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> A e. CC )
39		rpne0 11231	. . . . . . . . . . 11 |- ( B e. RR+ -> B =/= 0 )
40	39	3ad2ant2 1018	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B =/= 0 )
41	38, 35, 40	divcan1d 10317	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A / B ) x. B ) = A )
42	34	mulid2d 9610	. . . . . . . . . 10 |- ( B e. RR+ -> ( 1 x. B ) = B )
43	42	3ad2ant2 1018	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( 1 x. B ) = B )
44	41, 43	oveq12d 6300	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) x. B ) + ( 1 x. B ) ) = ( A + B ) )
45	36, 44	eqtrd 2508	. . . . . . 7 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) + 1 ) x. B ) = ( A + B ) )
46		recn 9578	. . . . . . . . 9 |- ( C e. RR -> C e. CC )
47	46	3ad2ant3 1019	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> C e. CC )
48	47, 35, 40	divcan1d 10317	. . . . . . 7 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( C / B ) x. B ) = C )
49	45, 48	breq12d 4460	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( ( A / B ) + 1 ) x. B ) < ( ( C / B ) x. B ) <-> ( A + B ) < C ) )
50	49	adantr 465	. . . . 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 ) )
51	28, 50	mpbird 232	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( ( A / B ) + 1 ) x. B ) < ( ( C / B ) x. B ) )
52	17, 7	syl 16	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A / B ) + 1 ) e. RR )
53		simp2 997	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B e. RR+ )
54	52, 13, 53	ltmul1d 11289	. . . . 5 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) + 1 ) < ( C / B ) <-> ( ( ( A / B ) + 1 ) x. B ) < ( ( C / B ) x. B ) ) )
55	54	adantr 465	. . . 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 ) ) )
56	51, 55	mpbird 232	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( A / B ) + 1 ) < ( C / B ) )
57	5, 10, 14, 24, 56	lelttrd 9735	. 2 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( |_ ` ( A / B ) ) + 1 ) < ( C / B ) )
58	57	ex 434	1 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A < ( C - B ) -> ( ( |_ ` ( A / B ) ) + 1 ) < ( C / B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   + caddc 9491   x. cmul 9493   < clt 9624   <_ cle 9625   - cmin 9801   / cdiv 10202   RR+crp 11216   |_cfl 11891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fl 11893

This theorem is referenced by: (None)