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Theorem gtndiv 10279
Description: A larger number does not divide a smaller natural number. (Contributed by NM, 3-May-2005.)
Assertion
Ref Expression
gtndiv  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  -.  ( B  /  A
)  e.  ZZ )

Proof of Theorem gtndiv
StepHypRef Expression
1 nnre 9939 . . . 4  |-  ( B  e.  NN  ->  B  e.  RR )
213ad2ant2 979 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  B  e.  RR )
3 simp1 957 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  A  e.  RR )
4 nngt0 9961 . . . 4  |-  ( B  e.  NN  ->  0  <  B )
543ad2ant2 979 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  0  <  B )
64adantl 453 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN )  ->  0  <  B )
7 0re 9024 . . . . . . . 8  |-  0  e.  RR
8 lttr 9085 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  B  /\  B  <  A )  ->  0  <  A
) )
97, 8mp3an1 1266 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( ( 0  < 
B  /\  B  <  A )  ->  0  <  A ) )
101, 9sylan 458 . . . . . 6  |-  ( ( B  e.  NN  /\  A  e.  RR )  ->  ( ( 0  < 
B  /\  B  <  A )  ->  0  <  A ) )
1110ancoms 440 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN )  ->  ( ( 0  < 
B  /\  B  <  A )  ->  0  <  A ) )
126, 11mpand 657 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN )  ->  ( B  <  A  ->  0  <  A ) )
13123impia 1150 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  0  <  A )
142, 3, 5, 13divgt0d 9878 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  0  <  ( B  /  A
) )
15 simp3 959 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  B  <  A )
16 1re 9023 . . . . . . 7  |-  1  e.  RR
17 ltdivmul2 9817 . . . . . . 7  |-  ( ( B  e.  RR  /\  1  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( B  /  A )  <  1  <->  B  <  ( 1  x.  A ) ) )
1816, 17mp3an2 1267 . . . . . 6  |-  ( ( B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( ( B  /  A )  <  1  <->  B  <  ( 1  x.  A ) ) )
192, 3, 13, 18syl12anc 1182 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  (
( B  /  A
)  <  1  <->  B  <  ( 1  x.  A ) ) )
20 recn 9013 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
2120mulid2d 9039 . . . . . . 7  |-  ( A  e.  RR  ->  (
1  x.  A )  =  A )
2221breq2d 4165 . . . . . 6  |-  ( A  e.  RR  ->  ( B  <  ( 1  x.  A )  <->  B  <  A ) )
23223ad2ant1 978 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  <  ( 1  x.  A )  <->  B  <  A ) )
2419, 23bitrd 245 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  (
( B  /  A
)  <  1  <->  B  <  A ) )
2515, 24mpbird 224 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  /  A )  <  1 )
26 0p1e1 10025 . . 3  |-  ( 0  +  1 )  =  1
2725, 26syl6breqr 4193 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  /  A )  < 
( 0  +  1 ) )
28 0z 10225 . . 3  |-  0  e.  ZZ
29 btwnnz 10278 . . 3  |-  ( ( 0  e.  ZZ  /\  0  <  ( B  /  A )  /\  ( B  /  A )  < 
( 0  +  1 ) )  ->  -.  ( B  /  A
)  e.  ZZ )
3028, 29mp3an1 1266 . 2  |-  ( ( 0  <  ( B  /  A )  /\  ( B  /  A
)  <  ( 0  +  1 ) )  ->  -.  ( B  /  A )  e.  ZZ )
3114, 27, 30syl2anc 643 1  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  -.  ( B  /  A
)  e.  ZZ )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1717   class class class wbr 4153  (class class class)co 6020   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    < clt 9053    / cdiv 9609   NNcn 9932   ZZcz 10214
This theorem is referenced by:  prime  10282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-n0 10154  df-z 10215
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