MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  modid Structured version   Visualization version   Unicode version
Theorem modid 12695
Description: Identity law for modulo. (Contributed by NM, 29-Dec-2008.)
Assertion
Ref Expression
modid |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A )
Proof of Theorem modid
StepHypRef Expression
1 modval 12670 . . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
21adantr 481 . 2 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
3 rerpdivcl 11861 . . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR )
43adantr 481 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A / B ) e. RR )
54recnd 10068 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A / B ) e. CC )
6 addid2 10219 . . . . . . . . 9 |- ( ( A / B ) e. CC -> ( 0 + ( A / B ) ) = ( A / B ) )
76fveq2d 6195 . . . . . . . 8 |- ( ( A / B ) e. CC -> ( |_ ` ( 0 + ( A / B ) ) ) = ( |_ ` ( A / B ) ) )
85, 7syl 17 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( |_ ` ( 0 + ( A / B ) ) ) = ( |_ ` ( A / B ) ) )
9 rpregt0 11846 . . . . . . . . . . 11 |- ( B e. RR+ -> ( B e. RR /\ 0 < B ) )
10 divge0 10892 . . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 <_ ( A / B ) )
119, 10sylan2 491 . . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> 0 <_ ( A / B ) )
1211an32s 846 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR+ ) /\ 0 <_ A ) -> 0 <_ ( A / B ) )
1312adantrr 753 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> 0 <_ ( A / B ) )
14 simpr 477 . . . . . . . . . . 11 |- ( ( B e. RR+ /\ A < B ) -> A < B )
15 rpcn 11841 . . . . . . . . . . . . 13 |- ( B e. RR+ -> B e. CC )
1615mulid1d 10057 . . . . . . . . . . . 12 |- ( B e. RR+ -> ( B x. 1 ) = B )
1716adantr 481 . . . . . . . . . . 11 |- ( ( B e. RR+ /\ A < B ) -> ( B x. 1 ) = B )
1814, 17breqtrrd 4681 . . . . . . . . . 10 |- ( ( B e. RR+ /\ A < B ) -> A < ( B x. 1 ) )
1918ad2ant2l 782 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> A < ( B x. 1 ) )
20 simpll 790 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> A e. RR )
219ad2antlr 763 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( B e. RR /\ 0 < B ) )
22 1re 10039 . . . . . . . . . . 11 |- 1 e. RR
23 ltdivmul 10898 . . . . . . . . . . 11 |- ( ( A e. RR /\ 1 e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) < 1 <-> A < ( B x. 1 ) ) )
2422, 23mp3an2 1412 . . . . . . . . . 10 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) < 1 <-> A < ( B x. 1 ) ) )
2520, 21, 24syl2anc 693 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( ( A / B ) < 1 <-> A < ( B x. 1 ) ) )
2619, 25mpbird 247 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A / B ) < 1 )
27 0z 11388 . . . . . . . . 9 |- 0 e. ZZ
28 flbi2 12618 . . . . . . . . 9 |- ( ( 0 e. ZZ /\ ( A / B ) e. RR ) -> ( ( |_ ` ( 0 + ( A / B ) ) ) = 0 <-> ( 0 <_ ( A / B ) /\ ( A / B ) < 1 ) ) )
2927, 4, 28sylancr 695 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( ( |_ ` ( 0 + ( A / B ) ) ) = 0 <-> ( 0 <_ ( A / B ) /\ ( A / B ) < 1 ) ) )
3013, 26, 29mpbir2and 957 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( |_ ` ( 0 + ( A / B ) ) ) = 0 )
318, 30eqtr3d 2658 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( |_ ` ( A / B ) ) = 0 )
3231oveq2d 6666 . . . . 5 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( B x. ( |_ ` ( A / B ) ) ) = ( B x. 0 ) )
3315mul01d 10235 . . . . . 6 |- ( B e. RR+ -> ( B x. 0 ) = 0 )
3433ad2antlr 763 . . . . 5 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( B x. 0 ) = 0 )
3532, 34eqtrd 2656 . . . 4 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( B x. ( |_ ` ( A / B ) ) ) = 0 )
3635oveq2d 6666 . . 3 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A - ( B x. ( |_ ` ( A / B ) ) ) ) = ( A - 0 ) )
37 recn 10026 . . . . 5 |- ( A e. RR -> A e. CC )
3837subid1d 10381 . . . 4 |- ( A e. RR -> ( A - 0 ) = A )
3938ad2antrr 762 . . 3 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A - 0 ) = A )
4036, 39eqtrd 2656 . 2 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A - ( B x. ( |_ ` ( A / B ) ) ) ) = A )
412, 40eqtrd 2656 1 |- ( ( ( A e. RR /\ B e. RR+ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   ZZcz 11377   RR+crp 11832   |_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-fl 12593  df-mod 12669
This theorem is referenced by:  modid2  12697  0mod  12701  1mod  12702  modabs  12703  muladdmodid  12710  m1modnnsub1  12716  modltm1p1mod  12722  2submod  12731  modifeq2int  12732  modaddmodlo  12734  modsubdir  12739  modsumfzodifsn  12743  digit1  12998  cshwidxm1  13553  bitsinv1  15164  sadaddlem  15188  sadasslem  15192  sadeq  15194  crth  15483  eulerthlem2  15487  prmdiveq  15491  modprm0  15510  4sqlem12  15660  dfod2  17981  znf1o  19900  wilthlem1  24794  ppiub  24929  lgslem1  25022  lgsdir2lem1  25050  lgsdirprm  25056  lgsqrlem2  25072  lgseisenlem1  25100  lgseisenlem2  25101  lgseisen  25104  m1lgs  25113  2lgslem1a1  25114  2lgslem4  25131  2sqlem11  25154  sqwvfoura  40445  sqwvfourb  40446  fourierswlem  40447  fouriersw  40448  m1modmmod  42316  nnpw2pmod  42377
  Copyright terms: Public domain W3C validator