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Theorem gtndiv 10105
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 9769 . . . 4  |-  ( B  e.  NN  ->  B  e.  RR )
213ad2ant2 977 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  B  e.  RR )
3 simp1 955 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  A  e.  RR )
4 nngt0 9791 . . . 4  |-  ( B  e.  NN  ->  0  <  B )
543ad2ant2 977 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  0  <  B )
64adantl 452 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN )  ->  0  <  B )
7 0re 8854 . . . . . . . 8  |-  0  e.  RR
8 lttr 8915 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  B  /\  B  <  A )  ->  0  <  A
) )
97, 8mp3an1 1264 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( ( 0  < 
B  /\  B  <  A )  ->  0  <  A ) )
101, 9sylan 457 . . . . . 6  |-  ( ( B  e.  NN  /\  A  e.  RR )  ->  ( ( 0  < 
B  /\  B  <  A )  ->  0  <  A ) )
1110ancoms 439 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN )  ->  ( ( 0  < 
B  /\  B  <  A )  ->  0  <  A ) )
126, 11mpand 656 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN )  ->  ( B  <  A  ->  0  <  A ) )
13123impia 1148 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  0  <  A )
142, 3, 5, 13divgt0d 9708 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  0  <  ( B  /  A
) )
15 simp3 957 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  B  <  A )
16 1re 8853 . . . . . . 7  |-  1  e.  RR
17 ltdivmul2 9647 . . . . . . 7  |-  ( ( B  e.  RR  /\  1  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( B  /  A )  <  1  <->  B  <  ( 1  x.  A ) ) )
1816, 17mp3an2 1265 . . . . . 6  |-  ( ( B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( ( B  /  A )  <  1  <->  B  <  ( 1  x.  A ) ) )
192, 3, 13, 18syl12anc 1180 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  (
( B  /  A
)  <  1  <->  B  <  ( 1  x.  A ) ) )
20 recn 8843 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
2120mulid2d 8869 . . . . . . 7  |-  ( A  e.  RR  ->  (
1  x.  A )  =  A )
2221breq2d 4051 . . . . . 6  |-  ( A  e.  RR  ->  ( B  <  ( 1  x.  A )  <->  B  <  A ) )
23223ad2ant1 976 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  <  ( 1  x.  A )  <->  B  <  A ) )
2419, 23bitrd 244 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  (
( B  /  A
)  <  1  <->  B  <  A ) )
2515, 24mpbird 223 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  /  A )  <  1 )
26 0p1e1 9855 . . 3  |-  ( 0  +  1 )  =  1
2725, 26syl6breqr 4079 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  /  A )  < 
( 0  +  1 ) )
28 0z 10051 . . 3  |-  0  e.  ZZ
29 btwnnz 10104 . . 3  |-  ( ( 0  e.  ZZ  /\  0  <  ( B  /  A )  /\  ( B  /  A )  < 
( 0  +  1 ) )  ->  -.  ( B  /  A
)  e.  ZZ )
3028, 29mp3an1 1264 . 2  |-  ( ( 0  <  ( B  /  A )  /\  ( B  /  A
)  <  ( 0  +  1 ) )  ->  -.  ( B  /  A )  e.  ZZ )
3114, 27, 30syl2anc 642 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 176    /\ wa 358    /\ w3a 934    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    / cdiv 9439   NNcn 9762   ZZcz 10040
This theorem is referenced by:  prime  10108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041
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