Theorem lefldiveq 39505

Description: A closed enough, smaller real C has the same floor of A when both are divided by B. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
lefldiveq.a	|- ( ph -> A e. RR )
lefldiveq.b	|- ( ph -> B e. RR+ )
lefldiveq.c	|- ( ph -> C e. ( ( A - ( A mod B ) ) [,] A ) )

Assertion

Ref	Expression
lefldiveq	|- ( ph -> ( |_ ` ( A / B ) ) = ( |_ ` ( C / B ) ) )

Proof of Theorem lefldiveq

Step	Hyp	Ref	Expression
1		lefldiveq.a	. . . . . . 7 |- ( ph -> A e. RR )
2		lefldiveq.b	. . . . . . 7 |- ( ph -> B e. RR+ )
3		moddiffl 12681	. . . . . . 7 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( A - ( A mod B ) ) / B ) = ( |_ ` ( A / B ) ) )
4	1, 2, 3	syl2anc 693	. . . . . 6 |- ( ph -> ( ( A - ( A mod B ) ) / B ) = ( |_ ` ( A / B ) ) )
5	1, 2	rerpdivcld 11903	. . . . . . 7 |- ( ph -> ( A / B ) e. RR )
6	5	flcld 12599	. . . . . 6 |- ( ph -> ( |_ ` ( A / B ) ) e. ZZ )
7	4, 6	eqeltrd 2701	. . . . 5 |- ( ph -> ( ( A - ( A mod B ) ) / B ) e. ZZ )
8		flid 12609	. . . . 5 |- ( ( ( A - ( A mod B ) ) / B ) e. ZZ -> ( |_ ` ( ( A - ( A mod B ) ) / B ) ) = ( ( A - ( A mod B ) ) / B ) )
9	7, 8	syl 17	. . . 4 |- ( ph -> ( |_ ` ( ( A - ( A mod B ) ) / B ) ) = ( ( A - ( A mod B ) ) / B ) )
10	9, 4	eqtr2d 2657	. . 3 |- ( ph -> ( |_ ` ( A / B ) ) = ( |_ ` ( ( A - ( A mod B ) ) / B ) ) )
11	1, 2	modcld 12674	. . . . . 6 |- ( ph -> ( A mod B ) e. RR )
12	1, 11	resubcld 10458	. . . . 5 |- ( ph -> ( A - ( A mod B ) ) e. RR )
13	12, 2	rerpdivcld 11903	. . . 4 |- ( ph -> ( ( A - ( A mod B ) ) / B ) e. RR )
14		iccssre 12255	. . . . . . 7 |- ( ( ( A - ( A mod B ) ) e. RR /\ A e. RR ) -> ( ( A - ( A mod B ) ) [,] A ) C_ RR )
15	12, 1, 14	syl2anc 693	. . . . . 6 |- ( ph -> ( ( A - ( A mod B ) ) [,] A ) C_ RR )
16		lefldiveq.c	. . . . . 6 |- ( ph -> C e. ( ( A - ( A mod B ) ) [,] A ) )
17	15, 16	sseldd 3604	. . . . 5 |- ( ph -> C e. RR )
18	17, 2	rerpdivcld 11903	. . . 4 |- ( ph -> ( C / B ) e. RR )
19	12	rexrd 10089	. . . . . 6 |- ( ph -> ( A - ( A mod B ) ) e. RR* )
20	1	rexrd 10089	. . . . . 6 |- ( ph -> A e. RR* )
21		iccgelb 12230	. . . . . 6 |- ( ( ( A - ( A mod B ) ) e. RR* /\ A e. RR* /\ C e. ( ( A - ( A mod B ) ) [,] A ) ) -> ( A - ( A mod B ) ) <_ C )
22	19, 20, 16, 21	syl3anc 1326	. . . . 5 |- ( ph -> ( A - ( A mod B ) ) <_ C )
23	12, 17, 2, 22	lediv1dd 11930	. . . 4 |- ( ph -> ( ( A - ( A mod B ) ) / B ) <_ ( C / B ) )
24		flwordi 12613	. . . 4 |- ( ( ( ( A - ( A mod B ) ) / B ) e. RR /\ ( C / B ) e. RR /\ ( ( A - ( A mod B ) ) / B ) <_ ( C / B ) ) -> ( |_ ` ( ( A - ( A mod B ) ) / B ) ) <_ ( |_ ` ( C / B ) ) )
25	13, 18, 23, 24	syl3anc 1326	. . 3 |- ( ph -> ( |_ ` ( ( A - ( A mod B ) ) / B ) ) <_ ( |_ ` ( C / B ) ) )
26	10, 25	eqbrtrd 4675	. 2 |- ( ph -> ( |_ ` ( A / B ) ) <_ ( |_ ` ( C / B ) ) )
27		iccleub 12229	. . . . 5 |- ( ( ( A - ( A mod B ) ) e. RR* /\ A e. RR* /\ C e. ( ( A - ( A mod B ) ) [,] A ) ) -> C <_ A )
28	19, 20, 16, 27	syl3anc 1326	. . . 4 |- ( ph -> C <_ A )
29	17, 1, 2, 28	lediv1dd 11930	. . 3 |- ( ph -> ( C / B ) <_ ( A / B ) )
30		flwordi 12613	. . 3 |- ( ( ( C / B ) e. RR /\ ( A / B ) e. RR /\ ( C / B ) <_ ( A / B ) ) -> ( |_ ` ( C / B ) ) <_ ( |_ ` ( A / B ) ) )
31	18, 5, 29, 30	syl3anc 1326	. 2 |- ( ph -> ( |_ ` ( C / B ) ) <_ ( |_ ` ( A / B ) ) )
32		reflcl 12597	. . . 4 |- ( ( A / B ) e. RR -> ( |_ ` ( A / B ) ) e. RR )
33	5, 32	syl 17	. . 3 |- ( ph -> ( |_ ` ( A / B ) ) e. RR )
34		reflcl 12597	. . . 4 |- ( ( C / B ) e. RR -> ( |_ ` ( C / B ) ) e. RR )
35	18, 34	syl 17	. . 3 |- ( ph -> ( |_ ` ( C / B ) ) e. RR )
36	33, 35	letri3d 10179	. 2 |- ( ph -> ( ( |_ ` ( A / B ) ) = ( |_ ` ( C / B ) ) <-> ( ( |_ ` ( A / B ) ) <_ ( |_ ` ( C / B ) ) /\ ( |_ ` ( C / B ) ) <_ ( |_ ` ( A / B ) ) ) ) )
37	26, 31, 36	mpbir2and 957	1 |- ( ph -> ( |_ ` ( A / B ) ) = ( |_ ` ( C / B ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   RR*cxr 10073   <_ cle 10075   - cmin 10266   / cdiv 10684   ZZcz 11377   RR+crp 11832   [,]cicc 12178   |_cfl 12591   mod cmo 12668

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-icc 12182  df-fl 12593  df-mod 12669

This theorem is referenced by:  ltmod  39870