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Theorem modirr 12037
Description: A number modulo an irrational multiple of it is nonzero. (Contributed by NM, 11-Nov-2008.)
Assertion
Ref Expression
modirr  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( A  /  B )  e.  ( RR  \  QQ ) )  ->  ( A  mod  B )  =/=  0 )

Proof of Theorem modirr
StepHypRef Expression
1 eldif 3491 . . 3  |-  ( ( A  /  B )  e.  ( RR  \  QQ )  <->  ( ( A  /  B )  e.  RR  /\  -.  ( A  /  B )  e.  QQ ) )
2 modval 11978 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
32eqeq1d 2469 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  0 ) )
4 recn 9594 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
54adantr 465 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  e.  CC )
6 rpre 11238 . . . . . . . . . . 11  |-  ( B  e.  RR+  ->  B  e.  RR )
76adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR )
8 refldivcl 11937 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  RR )
97, 8remulcld 9636 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( |_ `  ( A  /  B ) ) )  e.  RR )
109recnd 9634 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( |_ `  ( A  /  B ) ) )  e.  CC )
115, 10subeq0ad 9952 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  0  <->  A  =  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
12 eqcom 2476 . . . . . . . 8  |-  ( ( A  /  B )  =  ( |_ `  ( A  /  B
) )  <->  ( |_ `  ( A  /  B
) )  =  ( A  /  B ) )
13 rerpdivcl 11259 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
14 reflcl 11913 . . . . . . . . . . 11  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  e.  RR )
1514recnd 9634 . . . . . . . . . 10  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  e.  CC )
1613, 15syl 16 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  CC )
17 rpcnne0 11249 . . . . . . . . . 10  |-  ( B  e.  RR+  ->  ( B  e.  CC  /\  B  =/=  0 ) )
1817adantl 466 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
19 divmul2 10223 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( |_ `  ( A  /  B ) )  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( A  /  B )  =  ( |_ `  ( A  /  B ) )  <-> 
A  =  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
205, 16, 18, 19syl3anc 1228 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  =  ( |_ `  ( A  /  B ) )  <-> 
A  =  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
2112, 20syl5rbbr 260 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  =  ( B  x.  ( |_
`  ( A  /  B ) ) )  <-> 
( |_ `  ( A  /  B ) )  =  ( A  /  B ) ) )
223, 11, 213bitrd 279 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( |_ `  ( A  /  B ) )  =  ( A  /  B
) ) )
23 flidz 11927 . . . . . . . 8  |-  ( ( A  /  B )  e.  RR  ->  (
( |_ `  ( A  /  B ) )  =  ( A  /  B )  <->  ( A  /  B )  e.  ZZ ) )
24 zq 11200 . . . . . . . 8  |-  ( ( A  /  B )  e.  ZZ  ->  ( A  /  B )  e.  QQ )
2523, 24syl6bi 228 . . . . . . 7  |-  ( ( A  /  B )  e.  RR  ->  (
( |_ `  ( A  /  B ) )  =  ( A  /  B )  ->  ( A  /  B )  e.  QQ ) )
2613, 25syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( |_ `  ( A  /  B
) )  =  ( A  /  B )  ->  ( A  /  B )  e.  QQ ) )
2722, 26sylbid 215 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  -> 
( A  /  B
)  e.  QQ ) )
2827necon3bd 2679 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( -.  ( A  /  B )  e.  QQ  ->  ( A  mod  B )  =/=  0
) )
2928adantld 467 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( ( A  /  B )  e.  RR  /\  -.  ( A  /  B )  e.  QQ )  ->  ( A  mod  B )  =/=  0 ) )
301, 29syl5bi 217 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  e.  ( RR  \  QQ )  ->  ( A  mod  B )  =/=  0 ) )
31303impia 1193 1  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( A  /  B )  e.  ( RR  \  QQ ) )  ->  ( A  mod  B )  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3478   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504    x. cmul 9509    - cmin 9817    / cdiv 10218   ZZcz 10876   QQcq 11194   RR+crp 11232   |_cfl 11907    mod cmo 11976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-fl 11909  df-mod 11977
This theorem is referenced by: (None)
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