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Theorem ndvdssub 11023
Description: Corollary of the division algorithm. If an integer  D greater than  1 divides  N, then it does not divide any of  N  -  1,  N  -  2...  N  -  ( D  -  1 ). (Contributed by Paul Chapman, 31-Mar-2011.)
Assertion
Ref Expression
ndvdssub  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  -> 
( D  ||  N  ->  -.  D  ||  ( N  -  K )
) )

Proof of Theorem ndvdssub
Dummy variables  r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnnn0 8650 . . . . . . . 8  |-  ( K  e.  NN  ->  K  e.  NN0 )
2 nnne0 8422 . . . . . . . 8  |-  ( K  e.  NN  ->  K  =/=  0 )
31, 2jca 300 . . . . . . 7  |-  ( K  e.  NN  ->  ( K  e.  NN0  /\  K  =/=  0 ) )
4 df-ne 2256 . . . . . . . . . . . 12  |-  ( K  =/=  0  <->  -.  K  =  0 )
54anbi2i 445 . . . . . . . . . . 11  |-  ( ( K  <  D  /\  K  =/=  0 )  <->  ( K  <  D  /\  -.  K  =  0 ) )
6 divalg2 11019 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E! r  e.  NN0  ( r  <  D  /\  D  ||  ( N  -  r ) ) )
7 breq1 3840 . . . . . . . . . . . . . . . . . . . . 21  |-  ( r  =  x  ->  (
r  <  D  <->  x  <  D ) )
8 oveq2 5642 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( r  =  x  ->  ( N  -  r )  =  ( N  -  x ) )
98breq2d 3849 . . . . . . . . . . . . . . . . . . . . 21  |-  ( r  =  x  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  x )
) )
107, 9anbi12d 457 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  =  x  ->  (
( r  <  D  /\  D  ||  ( N  -  r ) )  <-> 
( x  <  D  /\  D  ||  ( N  -  x ) ) ) )
1110reu4 2807 . . . . . . . . . . . . . . . . . . 19  |-  ( E! r  e.  NN0  (
r  <  D  /\  D  ||  ( N  -  r ) )  <->  ( E. r  e.  NN0  ( r  <  D  /\  D  ||  ( N  -  r
) )  /\  A. r  e.  NN0  A. x  e.  NN0  ( ( ( r  <  D  /\  D  ||  ( N  -  r ) )  /\  ( x  <  D  /\  D  ||  ( N  -  x ) ) )  ->  r  =  x ) ) )
126, 11sylib 120 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( E. r  e. 
NN0  ( r  < 
D  /\  D  ||  ( N  -  r )
)  /\  A. r  e.  NN0  A. x  e. 
NN0  ( ( ( r  <  D  /\  D  ||  ( N  -  r ) )  /\  ( x  <  D  /\  D  ||  ( N  -  x ) ) )  ->  r  =  x ) ) )
13 nngt0 8419 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( D  e.  NN  ->  0  <  D )
14133ad2ant2 965 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  0  <  D )
15 zcn 8725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( N  e.  ZZ  ->  N  e.  CC )
1615subid1d 7761 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  ZZ  ->  ( N  -  0 )  =  N )
1716breq2d 3849 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  e.  ZZ  ->  ( D  ||  ( N  - 
0 )  <->  D  ||  N
) )
1817biimpar 291 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( N  e.  ZZ  /\  D  ||  N )  ->  D  ||  ( N  - 
0 ) )
19183adant2 962 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  D  ||  ( N  -  0 ) )
2014, 19jca 300 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  (
0  <  D  /\  D  ||  ( N  - 
0 ) ) )
21203expa 1143 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N
)  ->  ( 0  <  D  /\  D  ||  ( N  -  0 ) ) )
2221anim2i 334 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( r  <  D  /\  D  ||  ( N  -  r ) )  /\  ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N ) )  -> 
( ( r  < 
D  /\  D  ||  ( N  -  r )
)  /\  ( 0  <  D  /\  D  ||  ( N  -  0 ) ) ) )
2322ancoms 264 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N )  /\  (
r  <  D  /\  D  ||  ( N  -  r ) ) )  ->  ( ( r  <  D  /\  D  ||  ( N  -  r
) )  /\  (
0  <  D  /\  D  ||  ( N  - 
0 ) ) ) )
24 0nn0 8658 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  NN0
25 breq1 3840 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( x  =  0  ->  (
x  <  D  <->  0  <  D ) )
26 oveq2 5642 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( x  =  0  ->  ( N  -  x )  =  ( N  - 
0 ) )
2726breq2d 3849 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( x  =  0  ->  ( D  ||  ( N  -  x )  <->  D  ||  ( N  -  0 ) ) )
2825, 27anbi12d 457 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( x  =  0  ->  (
( x  <  D  /\  D  ||  ( N  -  x ) )  <-> 
( 0  <  D  /\  D  ||  ( N  -  0 ) ) ) )
2928anbi2d 452 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  0  ->  (
( ( r  < 
D  /\  D  ||  ( N  -  r )
)  /\  ( x  <  D  /\  D  ||  ( N  -  x
) ) )  <->  ( (
r  <  D  /\  D  ||  ( N  -  r ) )  /\  ( 0  <  D  /\  D  ||  ( N  -  0 ) ) ) ) )
30 eqeq2 2097 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  0  ->  (
r  =  x  <->  r  = 
0 ) )
3129, 30imbi12d 232 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  =  0  ->  (
( ( ( r  <  D  /\  D  ||  ( N  -  r
) )  /\  (
x  <  D  /\  D  ||  ( N  -  x ) ) )  ->  r  =  x )  <->  ( ( ( r  <  D  /\  D  ||  ( N  -  r ) )  /\  ( 0  <  D  /\  D  ||  ( N  -  0 ) ) )  ->  r  = 
0 ) ) )
3231rspcv 2718 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  e.  NN0  ->  ( A. x  e.  NN0  ( ( ( r  <  D  /\  D  ||  ( N  -  r ) )  /\  ( x  < 
D  /\  D  ||  ( N  -  x )
) )  ->  r  =  x )  ->  (
( ( r  < 
D  /\  D  ||  ( N  -  r )
)  /\  ( 0  <  D  /\  D  ||  ( N  -  0 ) ) )  -> 
r  =  0 ) ) )
3324, 32ax-mp 7 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A. x  e.  NN0  ( ( ( r  <  D  /\  D  ||  ( N  -  r ) )  /\  ( x  < 
D  /\  D  ||  ( N  -  x )
) )  ->  r  =  x )  ->  (
( ( r  < 
D  /\  D  ||  ( N  -  r )
)  /\  ( 0  <  D  /\  D  ||  ( N  -  0 ) ) )  -> 
r  =  0 ) )
3423, 33syl5 32 . . . . . . . . . . . . . . . . . . . 20  |-  ( A. x  e.  NN0  ( ( ( r  <  D  /\  D  ||  ( N  -  r ) )  /\  ( x  < 
D  /\  D  ||  ( N  -  x )
) )  ->  r  =  x )  ->  (
( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N )  /\  (
r  <  D  /\  D  ||  ( N  -  r ) ) )  ->  r  =  0 ) )
3534expd 254 . . . . . . . . . . . . . . . . . . 19  |-  ( A. x  e.  NN0  ( ( ( r  <  D  /\  D  ||  ( N  -  r ) )  /\  ( x  < 
D  /\  D  ||  ( N  -  x )
) )  ->  r  =  x )  ->  (
( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N )  ->  (
( r  <  D  /\  D  ||  ( N  -  r ) )  ->  r  =  0 ) ) )
3635ralimi 2438 . . . . . . . . . . . . . . . . . 18  |-  ( A. r  e.  NN0  A. x  e.  NN0  ( ( ( r  <  D  /\  D  ||  ( N  -  r ) )  /\  ( x  <  D  /\  D  ||  ( N  -  x ) ) )  ->  r  =  x )  ->  A. r  e.  NN0  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N )  ->  ( ( r  <  D  /\  D  ||  ( N  -  r
) )  ->  r  =  0 ) ) )
3712, 36simpl2im 378 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  A. r  e.  NN0  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N )  ->  (
( r  <  D  /\  D  ||  ( N  -  r ) )  ->  r  =  0 ) ) )
38 r19.21v 2450 . . . . . . . . . . . . . . . . 17  |-  ( A. r  e.  NN0  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N
)  ->  ( (
r  <  D  /\  D  ||  ( N  -  r ) )  -> 
r  =  0 ) )  <->  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N )  ->  A. r  e.  NN0  ( ( r  < 
D  /\  D  ||  ( N  -  r )
)  ->  r  = 
0 ) ) )
3937, 38sylib 120 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N )  ->  A. r  e.  NN0  ( ( r  <  D  /\  D  ||  ( N  -  r
) )  ->  r  =  0 ) ) )
4039expd 254 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  ||  N  ->  A. r  e.  NN0  ( ( r  <  D  /\  D  ||  ( N  -  r
) )  ->  r  =  0 ) ) ) )
4140pm2.43i 48 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  ||  N  ->  A. r  e.  NN0  ( ( r  < 
D  /\  D  ||  ( N  -  r )
)  ->  r  = 
0 ) ) )
42413impia 1140 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  A. r  e.  NN0  ( ( r  <  D  /\  D  ||  ( N  -  r
) )  ->  r  =  0 ) )
43 breq1 3840 . . . . . . . . . . . . . . . 16  |-  ( r  =  K  ->  (
r  <  D  <->  K  <  D ) )
44 oveq2 5642 . . . . . . . . . . . . . . . . 17  |-  ( r  =  K  ->  ( N  -  r )  =  ( N  -  K ) )
4544breq2d 3849 . . . . . . . . . . . . . . . 16  |-  ( r  =  K  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  K )
) )
4643, 45anbi12d 457 . . . . . . . . . . . . . . 15  |-  ( r  =  K  ->  (
( r  <  D  /\  D  ||  ( N  -  r ) )  <-> 
( K  <  D  /\  D  ||  ( N  -  K ) ) ) )
47 eqeq1 2094 . . . . . . . . . . . . . . 15  |-  ( r  =  K  ->  (
r  =  0  <->  K  =  0 ) )
4846, 47imbi12d 232 . . . . . . . . . . . . . 14  |-  ( r  =  K  ->  (
( ( r  < 
D  /\  D  ||  ( N  -  r )
)  ->  r  = 
0 )  <->  ( ( K  <  D  /\  D  ||  ( N  -  K
) )  ->  K  =  0 ) ) )
4948rspcv 2718 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  ( A. r  e.  NN0  ( ( r  <  D  /\  D  ||  ( N  -  r ) )  -> 
r  =  0 )  ->  ( ( K  <  D  /\  D  ||  ( N  -  K
) )  ->  K  =  0 ) ) )
5042, 49syl5com 29 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN0  ->  (
( K  <  D  /\  D  ||  ( N  -  K ) )  ->  K  =  0 ) ) )
51 pm3.37 826 . . . . . . . . . . . 12  |-  ( ( ( K  <  D  /\  D  ||  ( N  -  K ) )  ->  K  =  0 )  ->  ( ( K  <  D  /\  -.  K  =  0 )  ->  -.  D  ||  ( N  -  K )
) )
5250, 51syl6 33 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN0  ->  (
( K  <  D  /\  -.  K  =  0 )  ->  -.  D  ||  ( N  -  K
) ) ) )
535, 52syl7bi 163 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN0  ->  (
( K  <  D  /\  K  =/=  0
)  ->  -.  D  ||  ( N  -  K
) ) ) )
5453exp4a 358 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN0  ->  ( K  <  D  ->  ( K  =/=  0  ->  -.  D  ||  ( N  -  K ) ) ) ) )
5554com23 77 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  <  D  ->  ( K  e.  NN0  ->  ( K  =/=  0  ->  -.  D  ||  ( N  -  K ) ) ) ) )
5655imp4a 341 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  <  D  ->  (
( K  e.  NN0  /\  K  =/=  0 )  ->  -.  D  ||  ( N  -  K )
) ) )
573, 56syl7 68 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  <  D  ->  ( K  e.  NN  ->  -.  D  ||  ( N  -  K ) ) ) )
5857com23 77 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN  ->  ( K  <  D  ->  -.  D  ||  ( N  -  K ) ) ) )
5958impd 251 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  (
( K  e.  NN  /\  K  <  D )  ->  -.  D  ||  ( N  -  K )
) )
60593expia 1145 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  ||  N  ->  ( ( K  e.  NN  /\  K  < 
D )  ->  -.  D  ||  ( N  -  K ) ) ) )
6160com23 77 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( K  e.  NN  /\  K  < 
D )  ->  ( D  ||  N  ->  -.  D  ||  ( N  -  K ) ) ) )
62613impia 1140 1  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  -> 
( D  ||  N  ->  -.  D  ||  ( N  -  K )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438    =/= wne 2255   A.wral 2359   E.wrex 2360   E!wreu 2361   class class class wbr 3837  (class class class)co 5634   0cc0 7329    < clt 7501    - cmin 7632   NNcn 8394   NN0cn0 8643   ZZcz 8720    || cdvds 10889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10890
This theorem is referenced by:  ndvdsadd  11024
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