Theorem ltdifltdiv 11969

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 11960	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. RR )
2		peano2re 9770	. . . . . 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 11272	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR )
7		peano2re 9770	. . . . . 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 11272	. . . . . 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 9628	. . . 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 11966	. . . . 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 10187	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( |_ ` ( A / B ) ) + 1 ) <_ ( ( A / B ) + 1 ) )
25		rpre 11251	. . . . . . 7 |- ( B e. RR+ -> B e. RR )
26		ltaddsub 10047	. . . . . . 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 9599	. . . . . . . . . . 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		1cnd 9629	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> 1 e. CC )
33		rpcn 11253	. . . . . . . . . 10 |- ( B e. RR+ -> B e. CC )
34	33	3ad2ant2 1018	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B e. CC )
35	31, 32, 34	adddird 9638	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) + 1 ) x. B ) = ( ( ( A / B ) x. B ) + ( 1 x. B ) ) )
36		recn 9599	. . . . . . . . . . 11 |- ( A e. RR -> A e. CC )
37	36	3ad2ant1 1017	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> A e. CC )
38		rpne0 11260	. . . . . . . . . . 11 |- ( B e. RR+ -> B =/= 0 )
39	38	3ad2ant2 1018	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B =/= 0 )
40	37, 34, 39	divcan1d 10342	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A / B ) x. B ) = A )
41	33	mulid2d 9631	. . . . . . . . . 10 |- ( B e. RR+ -> ( 1 x. B ) = B )
42	41	3ad2ant2 1018	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( 1 x. B ) = B )
43	40, 42	oveq12d 6314	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) x. B ) + ( 1 x. B ) ) = ( A + B ) )
44	35, 43	eqtrd 2498	. . . . . . 7 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( ( A / B ) + 1 ) x. B ) = ( A + B ) )
45		recn 9599	. . . . . . . . 9 |- ( C e. RR -> C e. CC )
46	45	3ad2ant3 1019	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> C e. CC )
47	46, 34, 39	divcan1d 10342	. . . . . . 7 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( C / B ) x. B ) = C )
48	44, 47	breq12d 4469	. . . . . 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 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 ) )
50	28, 49	mpbird 232	. . . 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 16	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> ( ( A / B ) + 1 ) e. RR )
52		simp2 997	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ C e. RR ) -> B e. RR+ )
53	51, 13, 52	ltmul1d 11318	. . . . 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 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 ) ) )
55	50, 54	mpbird 232	. . 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 9757	. 2 |- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR ) /\ A < ( C - B ) ) -> ( ( |_ ` ( A / B ) ) + 1 ) < ( C / B ) )
57	56	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 1395   e. wcel 1819   =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510   + caddc 9512   x. cmul 9514   < clt 9645   <_ cle 9646   - cmin 9824   / cdiv 10227   RR+crp 11245   |_cfl 11930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fl 11932

This theorem is referenced by: (None)