Theorem ltdifltdiv 12016

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 12007	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. RR )
2		peano2re 9757	. . . . . 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 1025	. . . 4 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( |_ ` ( A / B ) ) + 1 ) e. RR )
5	4	adantr 466	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( |_ ` ( A / B ) ) + 1 ) e. RR )
6		rerpdivcl 11281	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR )
7		peano2re 9757	. . . . . 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 1025	. . . 4 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A / B ) + 1 ) e. RR )
10	9	adantr 466	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( A / B ) + 1 ) e. RR )
11		rerpdivcl 11281	. . . . . 6 |- ( ( C e. RR /\ B e. RR+ ) -> ( C / B ) e. RR )
12	11	ancoms 454	. . . . 5 |- ( ( B e. RR+ /\ C e. RR ) -> ( C / B ) e. RR )
13	12	3adant1 1023	. . . 4 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( C / B ) e. RR )
14	13	adantr 466	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( C / B ) e. RR )
15	1	3adant3 1025	. . . . 5 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( |_ ` ( A / B ) ) e. RR )
16	15	adantr 466	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( |_ ` ( A / B ) ) e. RR )
17	6	3adant3 1025	. . . . 5 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A / B ) e. RR )
18	17	adantr 466	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( A / B ) e. RR )
19		1red 9609	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> 1 e. RR )
20		3simpa 1002	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A e. RR /\ B e. RR+ ) )
21	20	adantr 466	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( A e. RR /\ B e. RR+ ) )
22		fldivle 12013	. . . . 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 10178	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( |_ ` ( A / B ) ) + 1 ) <_ ( ( A / B ) + 1 ) )
25		rpre 11259	. . . . . . 7 |- ( B e. RR+ -> B e. RR )
26		ltaddsub 10039	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) < C <-> A < ( C - B ) ) )
27	25, 26	syl3an2 1298	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A + B ) < C <-> A < ( C - B ) ) )
28	27	biimpar 487	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( A + B ) < C )
29		recn 9580	. . . . . . . . . . 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 1025	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A / B ) e. CC )
32		1cnd 9610	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> 1 e. CC )
33		rpcn 11261	. . . . . . . . . 10 |- ( B e. RR+ -> B e. CC )
34	33	3ad2ant2 1027	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B e. CC )
35	31, 32, 34	adddird 9619	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) + 1 ) x. B ) = ( ( ( A / B ) x. B ) + ( 1 x. B ) ) )
36		recn 9580	. . . . . . . . . . 11 |- ( A e. RR -> A e. CC )
37	36	3ad2ant1 1026	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> A e. CC )
38		rpne0 11268	. . . . . . . . . . 11 |- ( B e. RR+ -> B =/= 0 )
39	38	3ad2ant2 1027	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B =/= 0 )
40	37, 34, 39	divcan1d 10335	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A / B ) x. B ) = A )
41	33	mulid2d 9612	. . . . . . . . . 10 |- ( B e. RR+ -> ( 1 x. B ) = B )
42	41	3ad2ant2 1027	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( 1 x. B ) = B )
43	40, 42	oveq12d 6267	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) x. B ) + ( 1 x. B ) ) = ( A + B ) )
44	35, 43	eqtrd 2462	. . . . . . 7 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) + 1 ) x. B ) = ( A + B ) )
45		recn 9580	. . . . . . . . 9 |- ( C e. RR -> C e. CC )
46	45	3ad2ant3 1028	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> C e. CC )
47	46, 34, 39	divcan1d 10335	. . . . . . 7 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( C / B ) x. B ) = C )
48	44, 47	breq12d 4379	. . . . . 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 466	. . . . 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 235	. . . 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 1006	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B e. RR+ )
53	51, 13, 52	ltmul1d 11330	. . . . 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 466	. . . 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 235	. . 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 9744	. 2 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( |_ ` ( A / B ) ) + 1 ) < ( C / B ) )
57	56	ex 435	1 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( A < ( C - B ) -> ( ( |_ ` ( A / B ) ) + 1 ) < ( C / B ) ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1872   =/= wne 2599   class class class wbr 4366   ` cfv 5544  (class class class)co 6249   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491   + caddc 9493   x. cmul 9495   < clt 9626   <_ cle 9627   - cmin 9811   / cdiv 10220   RR+crp 11253   |_cfl 11976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-sup 7909  df-inf 7910  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-n0 10821  df-z 10889  df-uz 11111  df-rp 11254  df-fl 11978

This theorem is referenced by: (None)