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Theorem modid 12153
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 12130 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
21adantr 471 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( A  mod  B )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
3 rerpdivcl 11359 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
43adantr 471 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( A  /  B )  e.  RR )
54recnd 9695 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( A  /  B )  e.  CC )
6 addid2 9842 . . . . . . . . 9  |-  ( ( A  /  B )  e.  CC  ->  (
0  +  ( A  /  B ) )  =  ( A  /  B ) )
76fveq2d 5892 . . . . . . . 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 11344 . . . . . . . . . . 11  |-  ( B  e.  RR+  ->  ( B  e.  RR  /\  0  <  B ) )
10 divge0 10502 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  0  <_  ( A  /  B ) )
119, 10sylan2 481 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR+ )  ->  0  <_  ( A  /  B ) )
1211an32s 818 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  0  <_  A )  ->  0  <_  ( A  /  B ) )
1312adantrr 728 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  0  <_  ( A  /  B ) )
14 simpr 467 . . . . . . . . . . 11  |-  ( ( B  e.  RR+  /\  A  <  B )  ->  A  <  B )
15 rpcn 11339 . . . . . . . . . . . . 13  |-  ( B  e.  RR+  ->  B  e.  CC )
1615mulid1d 9686 . . . . . . . . . . . 12  |-  ( B  e.  RR+  ->  ( B  x.  1 )  =  B )
1716adantr 471 . . . . . . . . . . 11  |-  ( ( B  e.  RR+  /\  A  <  B )  ->  ( B  x.  1 )  =  B )
1814, 17breqtrrd 4443 . . . . . . . . . 10  |-  ( ( B  e.  RR+  /\  A  <  B )  ->  A  <  ( B  x.  1 ) )
1918ad2ant2l 757 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  A  <  ( B  x.  1 ) )
20 simpll 765 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  A  e.  RR )
219ad2antlr 738 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( B  e.  RR  /\  0  < 
B ) )
22 1re 9668 . . . . . . . . . . 11  |-  1  e.  RR
23 ltdivmul 10508 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( A  /  B )  <  1  <->  A  <  ( B  x.  1 ) ) )
2422, 23mp3an2 1361 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  <  1  <->  A  <  ( B  x.  1 ) ) )
2520, 21, 24syl2anc 671 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( ( A  /  B )  <  1  <->  A  <  ( B  x.  1 ) ) )
2619, 25mpbird 240 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( A  /  B )  <  1
)
27 0z 10977 . . . . . . . . 9  |-  0  e.  ZZ
28 flbi2 12084 . . . . . . . . 9  |-  ( ( 0  e.  ZZ  /\  ( A  /  B
)  e.  RR )  ->  ( ( |_
`  ( 0  +  ( A  /  B
) ) )  =  0  <->  ( 0  <_ 
( A  /  B
)  /\  ( A  /  B )  <  1
) ) )
2927, 4, 28sylancr 674 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( ( |_ `  ( 0  +  ( A  /  B
) ) )  =  0  <->  ( 0  <_ 
( A  /  B
)  /\  ( A  /  B )  <  1
) ) )
3013, 26, 29mpbir2and 938 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( |_ `  ( 0  +  ( A  /  B ) ) )  =  0 )
318, 30eqtr3d 2498 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( |_ `  ( A  /  B
) )  =  0 )
3231oveq2d 6331 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( B  x.  ( |_ `  ( A  /  B ) ) )  =  ( B  x.  0 ) )
3315mul01d 9858 . . . . . 6  |-  ( B  e.  RR+  ->  ( B  x.  0 )  =  0 )
3433ad2antlr 738 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( B  x.  0 )  =  0 )
3532, 34eqtrd 2496 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( B  x.  ( |_ `  ( A  /  B ) ) )  =  0 )
3635oveq2d 6331 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  ( A  -  0 ) )
37 recn 9655 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
3837subid1d 10001 . . . 4  |-  ( A  e.  RR  ->  ( A  -  0 )  =  A )
3938ad2antrr 737 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( A  -  0 )  =  A )
4036, 39eqtrd 2496 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  A )
412, 40eqtrd 2496 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 189    /\ wa 375    = wceq 1455    e. wcel 1898   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   CCcc 9563   RRcr 9564   0cc0 9565   1c1 9566    + caddc 9568    x. cmul 9570    < clt 9701    <_ cle 9702    - cmin 9886    / cdiv 10297   ZZcz 10966   RR+crp 11331   |_cfl 12058    mod cmo 12128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-sup 7982  df-inf 7983  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-n0 10899  df-z 10967  df-uz 11189  df-rp 11332  df-fl 12060  df-mod 12129
This theorem is referenced by:  modid2  12156  0mod  12160  1mod  12161  modabs  12162  muladdmodid  12169  modltm1p1mod  12174  2submod  12183  modifeq2int  12184  modaddmodlo  12186  modsubdir  12190  digit1  12438  cshwidxm1  12945  bitsinv1  14465  sadaddlem  14489  sadasslem  14493  sadeq  14495  crt  14775  eulerthlem2  14779  prmdiveq  14783  modprm0  14805  4sqlem12  14949  dfod2  17264  znf1o  19171  wilthlem1  24042  ppiub  24181  lgslem1  24273  lgsdir2lem1  24300  lgsdirprm  24306  lgsqrlem2  24319  lgseisenlem1  24326  lgseisenlem2  24327  lgseisen  24330  m1lgs  24339  2sqlem11  24352  sqwvfoura  38130  sqwvfourb  38131  fourierswlem  38132  fouriersw  38133  m1modmmod  40597  nnpw2pmod  40667
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