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Theorem mod0 11959
Description:  A  mod  B is zero iff  A is evenly divisible by 
B. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)
Assertion
Ref Expression
mod0  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( A  /  B )  e.  ZZ ) )

Proof of Theorem mod0
StepHypRef Expression
1 modval 11954 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
21eqeq1d 2462 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  0 ) )
3 recn 9571 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
43adantr 465 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  e.  CC )
5 rpre 11215 . . . . . . . 8  |-  ( B  e.  RR+  ->  B  e.  RR )
65adantl 466 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR )
7 rerpdivcl 11236 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
8 reflcl 11890 . . . . . . . 8  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  e.  RR )
97, 8syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  RR )
106, 9remulcld 9613 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( |_ `  ( A  /  B ) ) )  e.  RR )
1110recnd 9611 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( |_ `  ( A  /  B ) ) )  e.  CC )
124, 11subeq0ad 9929 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  0  <->  A  =  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
132, 12bitrd 253 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  A  =  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
14 eqcom 2469 . . . 4  |-  ( ( A  /  B )  =  ( |_ `  ( A  /  B
) )  <->  ( |_ `  ( A  /  B
) )  =  ( A  /  B ) )
159recnd 9611 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  CC )
16 rpcnne0 11226 . . . . . 6  |-  ( B  e.  RR+  ->  ( B  e.  CC  /\  B  =/=  0 ) )
1716adantl 466 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
18 divmul2 10200 . . . . 5  |-  ( ( A  e.  CC  /\  ( |_ `  ( A  /  B ) )  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( A  /  B )  =  ( |_ `  ( A  /  B ) )  <-> 
A  =  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
194, 15, 17, 18syl3anc 1223 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  =  ( |_ `  ( A  /  B ) )  <-> 
A  =  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
2014, 19syl5rbbr 260 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  =  ( B  x.  ( |_
`  ( A  /  B ) ) )  <-> 
( |_ `  ( A  /  B ) )  =  ( A  /  B ) ) )
2113, 20bitrd 253 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( |_ `  ( A  /  B ) )  =  ( A  /  B
) ) )
22 flidz 11903 . . 3  |-  ( ( A  /  B )  e.  RR  ->  (
( |_ `  ( A  /  B ) )  =  ( A  /  B )  <->  ( A  /  B )  e.  ZZ ) )
237, 22syl 16 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( |_ `  ( A  /  B
) )  =  ( A  /  B )  <-> 
( A  /  B
)  e.  ZZ ) )
2421, 23bitrd 253 1  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( A  /  B )  e.  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486    - cmin 9794    / cdiv 10195   ZZcz 10853   RR+crp 11209   |_cfl 11884    mod cmo 11952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fl 11886  df-mod 11953
This theorem is referenced by:  negmod0  11960  modid0  11977  modidmul0  11978  2txmodxeq0  12003  dvdsval3  13840  elqaalem2  22443  elqaalem3  22444  sineq0  22640  pellexlem6  30361  oddfl  30991  dirker2re  31347  dirkerdenne0  31348  dirkerval2  31349  dirkertrigeqlem3  31355  dirkertrigeq  31356  dirkercncflem1  31358  dirkercncflem2  31359  dirkercncflem4  31361  fourierdlem24  31386  fourierswlem  31486  sineq0ALT  32692
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