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Theorem gtndiv 9100
Description: A larger number does not divide a smaller positive integer. (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 8687 . . . 4  |-  ( B  e.  NN  ->  B  e.  RR )
213ad2ant2 986 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  B  e.  RR )
3 simp1 964 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  A  e.  RR )
4 nngt0 8705 . . . 4  |-  ( B  e.  NN  ->  0  <  B )
543ad2ant2 986 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  0  <  B )
64adantl 273 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN )  ->  0  <  B )
7 0re 7730 . . . . . . . 8  |-  0  e.  RR
8 lttr 7802 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  B  /\  B  <  A )  ->  0  <  A
) )
97, 8mp3an1 1285 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( ( 0  < 
B  /\  B  <  A )  ->  0  <  A ) )
101, 9sylan 279 . . . . . 6  |-  ( ( B  e.  NN  /\  A  e.  RR )  ->  ( ( 0  < 
B  /\  B  <  A )  ->  0  <  A ) )
1110ancoms 266 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN )  ->  ( ( 0  < 
B  /\  B  <  A )  ->  0  <  A ) )
126, 11mpand 423 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN )  ->  ( B  <  A  ->  0  <  A ) )
13123impia 1161 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  0  <  A )
142, 3, 5, 13divgt0d 8653 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  0  <  ( B  /  A
) )
15 simp3 966 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  B  <  A )
16 1re 7729 . . . . . . 7  |-  1  e.  RR
17 ltdivmul2 8596 . . . . . . 7  |-  ( ( B  e.  RR  /\  1  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( B  /  A )  <  1  <->  B  <  ( 1  x.  A ) ) )
1816, 17mp3an2 1286 . . . . . 6  |-  ( ( B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( ( B  /  A )  <  1  <->  B  <  ( 1  x.  A ) ) )
192, 3, 13, 18syl12anc 1197 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  (
( B  /  A
)  <  1  <->  B  <  ( 1  x.  A ) ) )
20 recn 7717 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
2120mulid2d 7748 . . . . . . 7  |-  ( A  e.  RR  ->  (
1  x.  A )  =  A )
2221breq2d 3909 . . . . . 6  |-  ( A  e.  RR  ->  ( B  <  ( 1  x.  A )  <->  B  <  A ) )
23223ad2ant1 985 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  <  ( 1  x.  A )  <->  B  <  A ) )
2419, 23bitrd 187 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  (
( B  /  A
)  <  1  <->  B  <  A ) )
2515, 24mpbird 166 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  /  A )  <  1 )
26 0p1e1 8794 . . 3  |-  ( 0  +  1 )  =  1
2725, 26breqtrrdi 3938 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  /  A )  < 
( 0  +  1 ) )
28 0z 9019 . . 3  |-  0  e.  ZZ
29 btwnnz 9099 . . 3  |-  ( ( 0  e.  ZZ  /\  0  <  ( B  /  A )  /\  ( B  /  A )  < 
( 0  +  1 ) )  ->  -.  ( B  /  A
)  e.  ZZ )
3028, 29mp3an1 1285 . 2  |-  ( ( 0  <  ( B  /  A )  /\  ( B  /  A
)  <  ( 0  +  1 ) )  ->  -.  ( B  /  A )  e.  ZZ )
3114, 27, 30syl2anc 406 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    /\ wa 103    <-> wb 104    /\ w3a 945    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   RRcr 7583   0cc0 7584   1c1 7585    + caddc 7587    x. cmul 7589    < clt 7764    / cdiv 8395   NNcn 8680   ZZcz 9008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-id 4183  df-po 4186  df-iso 4187  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-n0 8932  df-z 9009
This theorem is referenced by:  prime  9104
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