Theorem ltdifltdiv 11765

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 11756	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. RR )
2		peano2re 9629	. . . . . 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 1008	. . . 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 11105	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR )
7		peano2re 9629	. . . . . 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 1008	. . . 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 11105	. . . . . 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 1006	. . . 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 1008	. . . . 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 1008	. . . . 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 9488	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> 1 e. RR )
20		3simpa 985	. . . . . 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 11762	. . . . 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 10040	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( |_ ` ( A / B ) ) + 1 ) <_ ( ( A / B ) + 1 ) )
25		rpre 11084	. . . . . . 7 |- ( B e. RR+ -> B e. RR )
26		ltaddsub 9900	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) < C <-> A < ( C - B ) ) )
27	25, 26	syl3an2 1253	. . . . . 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 9459	. . . . . . . . . . 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 1008	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A / B ) e. CC )
32		ax-1cn 9427	. . . . . . . . . 10 |- 1 e. CC
33	32	a1i 11	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> 1 e. CC )
34		rpcn 11086	. . . . . . . . . 10 |- ( B e. RR+ -> B e. CC )
35	34	3ad2ant2 1010	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B e. CC )
36	31, 33, 35	adddird 9498	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) + 1 ) x. B ) = ( ( ( A / B ) x. B ) + ( 1 x. B ) ) )
37		recn 9459	. . . . . . . . . . 11 |- ( A e. RR -> A e. CC )
38	37	3ad2ant1 1009	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> A e. CC )
39		rpne0 11093	. . . . . . . . . . 11 |- ( B e. RR+ -> B =/= 0 )
40	39	3ad2ant2 1010	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B =/= 0 )
41	38, 35, 40	divcan1d 10195	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A / B ) x. B ) = A )
42	34	mulid2d 9491	. . . . . . . . . 10 |- ( B e. RR+ -> ( 1 x. B ) = B )
43	42	3ad2ant2 1010	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( 1 x. B ) = B )
44	41, 43	oveq12d 6194	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) x. B ) + ( 1 x. B ) ) = ( A + B ) )
45	36, 44	eqtrd 2490	. . . . . . 7 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) + 1 ) x. B ) = ( A + B ) )
46		recn 9459	. . . . . . . . 9 |- ( C e. RR -> C e. CC )
47	46	3ad2ant3 1011	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> C e. CC )
48	47, 35, 40	divcan1d 10195	. . . . . . 7 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( C / B ) x. B ) = C )
49	45, 48	breq12d 4389	. . . . . 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 989	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B e. RR+ )
54	52, 13, 53	ltmul1d 11151	. . . . 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 9616	. 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 965   = wceq 1370   e. wcel 1757   =/= wne 2641   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   CCcc 9367   RRcr 9368   0cc0 9369   1c1 9370   + caddc 9372   x. cmul 9374   < clt 9505   <_ cle 9506   - cmin 9682   / cdiv 10080   RR+crp 11078   |_cfl 11727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446  ax-pre-sup 9447

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-recs 6918  df-rdg 6952  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-sup 7778  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-div 10081  df-nn 10410  df-n0 10667  df-z 10734  df-uz 10949  df-rp 11079  df-fl 11729

This theorem is referenced by: (None)