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Theorem gtndiv 10339
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 9999 . . . 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 10021 . . . 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 9083 . . . . . . . 8  |-  0  e.  RR
8 lttr 9144 . . . . . . . 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 9938 . 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 9082 . . . . . . 7  |-  1  e.  RR
17 ltdivmul2 9877 . . . . . . 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 9072 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
2120mulid2d 9098 . . . . . . 7  |-  ( A  e.  RR  ->  (
1  x.  A )  =  A )
2221breq2d 4216 . . . . . 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 10085 . . 3  |-  ( 0  +  1 )  =  1
2725, 26syl6breqr 4244 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  /  A )  < 
( 0  +  1 ) )
28 0z 10285 . . 3  |-  0  e.  ZZ
29 btwnnz 10338 . . 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 1725   class class class wbr 4204  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    / cdiv 9669   NNcn 9992   ZZcz 10274
This theorem is referenced by:  prime  10342
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-n0 10214  df-z 10275
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