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Theorem mod0 11257
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 11254 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
21eqeq1d 2446 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  0 ) )
3 recn 9082 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
43adantr 453 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  e.  CC )
5 rpre 10620 . . . . . . . 8  |-  ( B  e.  RR+  ->  B  e.  RR )
65adantl 454 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR )
7 rerpdivcl 10641 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
8 reflcl 11207 . . . . . . . 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 9118 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( |_ `  ( A  /  B ) ) )  e.  RR )
1110recnd 9116 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( |_ `  ( A  /  B ) ) )  e.  CC )
124, 11subeq0ad 9423 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  0  <->  A  =  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
132, 12bitrd 246 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  A  =  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
14 eqcom 2440 . . . 4  |-  ( ( A  /  B )  =  ( |_ `  ( A  /  B
) )  <->  ( |_ `  ( A  /  B
) )  =  ( A  /  B ) )
159recnd 9116 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  CC )
16 rpcnne0 10631 . . . . . 6  |-  ( B  e.  RR+  ->  ( B  e.  CC  /\  B  =/=  0 ) )
1716adantl 454 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
18 divmul2 9684 . . . . 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 1185 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  =  ( |_ `  ( A  /  B ) )  <-> 
A  =  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
2014, 19syl5rbbr 253 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  =  ( B  x.  ( |_
`  ( A  /  B ) ) )  <-> 
( |_ `  ( A  /  B ) )  =  ( A  /  B ) ) )
2113, 20bitrd 246 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( |_ `  ( A  /  B ) )  =  ( A  /  B
) ) )
22 flidz 11220 . . 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 246 1  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( A  /  B )  e.  ZZ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992    x. cmul 8997    - cmin 9293    / cdiv 9679   ZZcz 10284   RR+crp 10614   |_cfl 11203    mod cmo 11252
This theorem is referenced by:  negmod0  11258  dvdsval3  12858  elqaalem2  20239  elqaalem3  20240  sineq0  20431  pellexlem6  26899  modid0  28170  modidmul0  28171  2txmodxeq0  28173  sineq0ALT  29111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fl 11204  df-mod 11253
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