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Theorem modirr 11765
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 3335 . . 3  |-  ( ( A  /  B )  e.  ( RR  \  QQ )  <->  ( ( A  /  B )  e.  RR  /\  -.  ( A  /  B )  e.  QQ ) )
2 modval 11706 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
32eqeq1d 2449 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  0 ) )
4 recn 9368 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
54adantr 462 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  e.  CC )
6 rpre 10993 . . . . . . . . . . 11  |-  ( B  e.  RR+  ->  B  e.  RR )
76adantl 463 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR )
8 rerpdivcl 11014 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
9 reflcl 11642 . . . . . . . . . . 11  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  e.  RR )
108, 9syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  RR )
117, 10remulcld 9410 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( |_ `  ( A  /  B ) ) )  e.  RR )
1211recnd 9408 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( |_ `  ( A  /  B ) ) )  e.  CC )
135, 12subeq0ad 9725 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  =  0  <->  A  =  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
14 eqcom 2443 . . . . . . . 8  |-  ( ( A  /  B )  =  ( |_ `  ( A  /  B
) )  <->  ( |_ `  ( A  /  B
) )  =  ( A  /  B ) )
159recnd 9408 . . . . . . . . . 10  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  e.  CC )
168, 15syl 16 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  CC )
17 rpcnne0 11004 . . . . . . . . . 10  |-  ( B  e.  RR+  ->  ( B  e.  CC  /\  B  =/=  0 ) )
1817adantl 463 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
19 divmul2 9994 . . . . . . . . 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 1213 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  =  ( |_ `  ( A  /  B ) )  <-> 
A  =  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
2114, 20syl5rbbr 260 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  =  ( B  x.  ( |_
`  ( A  /  B ) ) )  <-> 
( |_ `  ( A  /  B ) )  =  ( A  /  B ) ) )
223, 13, 213bitrd 279 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( |_ `  ( A  /  B ) )  =  ( A  /  B
) ) )
23 flidz 11655 . . . . . . . 8  |-  ( ( A  /  B )  e.  RR  ->  (
( |_ `  ( A  /  B ) )  =  ( A  /  B )  <->  ( A  /  B )  e.  ZZ ) )
24 zq 10955 . . . . . . . 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 ) )
268, 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 2643 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( -.  ( A  /  B )  e.  QQ  ->  ( A  mod  B )  =/=  0
) )
2928adantld 464 . . 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 1179 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604    \ cdif 3322   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278    x. cmul 9283    - cmin 9591    / cdiv 9989   ZZcz 10642   QQcq 10949   RR+crp 10987   |_cfl 11636    mod cmo 11704
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-fl 11638  df-mod 11705
This theorem is referenced by: (None)
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