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Theorem gtndiv 10879
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 10481 . . . 4  |-  ( B  e.  NN  ->  B  e.  RR )
213ad2ant2 1016 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  B  e.  RR )
3 simp1 994 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  A  e.  RR )
4 nngt0 10503 . . . 4  |-  ( B  e.  NN  ->  0  <  B )
543ad2ant2 1016 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  0  <  B )
64adantl 464 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN )  ->  0  <  B )
7 0re 9529 . . . . . . . 8  |-  0  e.  RR
8 lttr 9594 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  B  /\  B  <  A )  ->  0  <  A
) )
97, 8mp3an1 1309 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( ( 0  < 
B  /\  B  <  A )  ->  0  <  A ) )
101, 9sylan 469 . . . . . 6  |-  ( ( B  e.  NN  /\  A  e.  RR )  ->  ( ( 0  < 
B  /\  B  <  A )  ->  0  <  A ) )
1110ancoms 451 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN )  ->  ( ( 0  < 
B  /\  B  <  A )  ->  0  <  A ) )
126, 11mpand 673 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN )  ->  ( B  <  A  ->  0  <  A ) )
13123impia 1191 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  0  <  A )
142, 3, 5, 13divgt0d 10419 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  0  <  ( B  /  A
) )
15 simp3 996 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  B  <  A )
16 1re 9528 . . . . . . 7  |-  1  e.  RR
17 ltdivmul2 10358 . . . . . . 7  |-  ( ( B  e.  RR  /\  1  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( B  /  A )  <  1  <->  B  <  ( 1  x.  A ) ) )
1816, 17mp3an2 1310 . . . . . 6  |-  ( ( B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( ( B  /  A )  <  1  <->  B  <  ( 1  x.  A ) ) )
192, 3, 13, 18syl12anc 1224 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  (
( B  /  A
)  <  1  <->  B  <  ( 1  x.  A ) ) )
20 recn 9515 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
2120mulid2d 9547 . . . . . . 7  |-  ( A  e.  RR  ->  (
1  x.  A )  =  A )
2221breq2d 4396 . . . . . 6  |-  ( A  e.  RR  ->  ( B  <  ( 1  x.  A )  <->  B  <  A ) )
23223ad2ant1 1015 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  <  ( 1  x.  A )  <->  B  <  A ) )
2419, 23bitrd 253 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  (
( B  /  A
)  <  1  <->  B  <  A ) )
2515, 24mpbird 232 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  /  A )  <  1 )
26 0p1e1 10586 . . 3  |-  ( 0  +  1 )  =  1
2725, 26syl6breqr 4424 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  ( B  /  A )  < 
( 0  +  1 ) )
28 0z 10814 . . 3  |-  0  e.  ZZ
29 btwnnz 10878 . . 3  |-  ( ( 0  e.  ZZ  /\  0  <  ( B  /  A )  /\  ( B  /  A )  < 
( 0  +  1 ) )  ->  -.  ( B  /  A
)  e.  ZZ )
3028, 29mp3an1 1309 . 2  |-  ( ( 0  <  ( B  /  A )  /\  ( B  /  A
)  <  ( 0  +  1 ) )  ->  -.  ( B  /  A )  e.  ZZ )
3114, 27, 30syl2anc 659 1  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A )  ->  -.  ( B  /  A
)  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    e. wcel 1836   class class class wbr 4384  (class class class)co 6218   RRcr 9424   0cc0 9425   1c1 9426    + caddc 9428    x. cmul 9430    < clt 9561    / cdiv 10145   NNcn 10474   ZZcz 10803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-recs 6982  df-rdg 7016  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-div 10146  df-nn 10475  df-n0 10735  df-z 10804
This theorem is referenced by:  prime  10882
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