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Theorem ndvdssub 13927
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 10803 . . . . . . . 8  |-  ( K  e.  NN  ->  K  e.  NN0 )
2 nnne0 10569 . . . . . . . 8  |-  ( K  e.  NN  ->  K  =/=  0 )
31, 2jca 532 . . . . . . 7  |-  ( K  e.  NN  ->  ( K  e.  NN0  /\  K  =/=  0 ) )
4 df-ne 2664 . . . . . . . . . . . 12  |-  ( K  =/=  0  <->  -.  K  =  0 )
54anbi2i 694 . . . . . . . . . . 11  |-  ( ( K  <  D  /\  K  =/=  0 )  <->  ( K  <  D  /\  -.  K  =  0 ) )
6 divalg2 13925 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E! r  e.  NN0  ( r  <  D  /\  D  ||  ( N  -  r ) ) )
7 breq1 4450 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( r  =  x  ->  (
r  <  D  <->  x  <  D ) )
8 oveq2 6293 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( r  =  x  ->  ( N  -  r )  =  ( N  -  x ) )
98breq2d 4459 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( r  =  x  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  x )
) )
107, 9anbi12d 710 . . . . . . . . . . . . . . . . . . . . 21  |-  ( r  =  x  ->  (
( r  <  D  /\  D  ||  ( N  -  r ) )  <-> 
( x  <  D  /\  D  ||  ( N  -  x ) ) ) )
1110reu4 3297 . . . . . . . . . . . . . . . . . . . 20  |-  ( 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 196 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 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 ) ) )
1312simprd 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  A. r  e.  NN0  A. x  e.  NN0  (
( ( r  < 
D  /\  D  ||  ( N  -  r )
)  /\  ( x  <  D  /\  D  ||  ( N  -  x
) ) )  -> 
r  =  x ) )
14 nngt0 10566 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( D  e.  NN  ->  0  <  D )
15143ad2ant2 1018 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  0  <  D )
16 zcn 10870 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( N  e.  ZZ  ->  N  e.  CC )
1716subid1d 9920 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  ZZ  ->  ( N  -  0 )  =  N )
1817breq2d 4459 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  e.  ZZ  ->  ( D  ||  ( N  - 
0 )  <->  D  ||  N
) )
1918biimpar 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( N  e.  ZZ  /\  D  ||  N )  ->  D  ||  ( N  - 
0 ) )
20193adant2 1015 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  D  ||  ( N  -  0 ) )
2115, 20jca 532 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  (
0  <  D  /\  D  ||  ( N  - 
0 ) ) )
22213expa 1196 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N
)  ->  ( 0  <  D  /\  D  ||  ( N  -  0 ) ) )
2322anim2i 569 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( r  <  D  /\  D  ||  ( N  -  r ) )  /\  ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N ) )  -> 
( ( r  < 
D  /\  D  ||  ( N  -  r )
)  /\  ( 0  <  D  /\  D  ||  ( N  -  0 ) ) ) )
2423ancoms 453 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N )  /\  (
r  <  D  /\  D  ||  ( N  -  r ) ) )  ->  ( ( r  <  D  /\  D  ||  ( N  -  r
) )  /\  (
0  <  D  /\  D  ||  ( N  - 
0 ) ) ) )
25 0nn0 10811 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  NN0
26 breq1 4450 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( x  =  0  ->  (
x  <  D  <->  0  <  D ) )
27 oveq2 6293 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( x  =  0  ->  ( N  -  x )  =  ( N  - 
0 ) )
2827breq2d 4459 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( x  =  0  ->  ( D  ||  ( N  -  x )  <->  D  ||  ( N  -  0 ) ) )
2926, 28anbi12d 710 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( x  =  0  ->  (
( x  <  D  /\  D  ||  ( N  -  x ) )  <-> 
( 0  <  D  /\  D  ||  ( N  -  0 ) ) ) )
3029anbi2d 703 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  0  ->  (
( ( r  < 
D  /\  D  ||  ( N  -  r )
)  /\  ( x  <  D  /\  D  ||  ( N  -  x
) ) )  <->  ( (
r  <  D  /\  D  ||  ( N  -  r ) )  /\  ( 0  <  D  /\  D  ||  ( N  -  0 ) ) ) ) )
31 eqeq2 2482 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  0  ->  (
r  =  x  <->  r  = 
0 ) )
3230, 31imbi12d 320 . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
3332rspcv 3210 . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
3425, 33ax-mp 5 . . . . . . . . . . . . . . . . . . . . 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 ) )
3524, 34syl5 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 ) )
3635expd 436 . . . . . . . . . . . . . . . . . . 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 ) ) )
3736ralimi 2857 . . . . . . . . . . . . . . . . . 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 ) ) )
3813, 37syl 16 . . . . . . . . . . . . . . . . 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 ) ) )
39 r19.21v 2869 . . . . . . . . . . . . . . . . 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 ) ) )
4038, 39sylib 196 . . . . . . . . . . . . . . . 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 ) ) )
4140expd 436 . . . . . . . . . . . . . . 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 ) ) ) )
4241pm2.43i 47 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  ||  N  ->  A. r  e.  NN0  ( ( r  < 
D  /\  D  ||  ( N  -  r )
)  ->  r  = 
0 ) ) )
43423impia 1193 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  A. r  e.  NN0  ( ( r  <  D  /\  D  ||  ( N  -  r
) )  ->  r  =  0 ) )
44 breq1 4450 . . . . . . . . . . . . . . . 16  |-  ( r  =  K  ->  (
r  <  D  <->  K  <  D ) )
45 oveq2 6293 . . . . . . . . . . . . . . . . 17  |-  ( r  =  K  ->  ( N  -  r )  =  ( N  -  K ) )
4645breq2d 4459 . . . . . . . . . . . . . . . 16  |-  ( r  =  K  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  K )
) )
4744, 46anbi12d 710 . . . . . . . . . . . . . . 15  |-  ( r  =  K  ->  (
( r  <  D  /\  D  ||  ( N  -  r ) )  <-> 
( K  <  D  /\  D  ||  ( N  -  K ) ) ) )
48 eqeq1 2471 . . . . . . . . . . . . . . 15  |-  ( r  =  K  ->  (
r  =  0  <->  K  =  0 ) )
4947, 48imbi12d 320 . . . . . . . . . . . . . 14  |-  ( r  =  K  ->  (
( ( r  < 
D  /\  D  ||  ( N  -  r )
)  ->  r  = 
0 )  <->  ( ( K  <  D  /\  D  ||  ( N  -  K
) )  ->  K  =  0 ) ) )
5049rspcv 3210 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  ( A. r  e.  NN0  ( ( r  <  D  /\  D  ||  ( N  -  r ) )  -> 
r  =  0 )  ->  ( ( K  <  D  /\  D  ||  ( N  -  K
) )  ->  K  =  0 ) ) )
5143, 50syl5com 30 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN0  ->  (
( K  <  D  /\  D  ||  ( N  -  K ) )  ->  K  =  0 ) ) )
52 pm4.14 578 . . . . . . . . . . . 12  |-  ( ( ( K  <  D  /\  D  ||  ( N  -  K ) )  ->  K  =  0 )  <->  ( ( K  <  D  /\  -.  K  =  0 )  ->  -.  D  ||  ( N  -  K )
) )
5351, 52syl6ib 226 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN0  ->  (
( K  <  D  /\  -.  K  =  0 )  ->  -.  D  ||  ( N  -  K
) ) ) )
545, 53syl7bi 230 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN0  ->  (
( K  <  D  /\  K  =/=  0
)  ->  -.  D  ||  ( N  -  K
) ) ) )
5554exp4a 606 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN0  ->  ( K  <  D  ->  ( K  =/=  0  ->  -.  D  ||  ( N  -  K ) ) ) ) )
5655com23 78 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  <  D  ->  ( K  e.  NN0  ->  ( K  =/=  0  ->  -.  D  ||  ( N  -  K ) ) ) ) )
5756imp4a 589 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  <  D  ->  (
( K  e.  NN0  /\  K  =/=  0 )  ->  -.  D  ||  ( N  -  K )
) ) )
583, 57syl7 68 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  <  D  ->  ( K  e.  NN  ->  -.  D  ||  ( N  -  K ) ) ) )
5958com23 78 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN  ->  ( K  <  D  ->  -.  D  ||  ( N  -  K ) ) ) )
6059impd 431 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  (
( K  e.  NN  /\  K  <  D )  ->  -.  D  ||  ( N  -  K )
) )
61603expia 1198 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  ||  N  ->  ( ( K  e.  NN  /\  K  < 
D )  ->  -.  D  ||  ( N  -  K ) ) ) )
6261com23 78 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( K  e.  NN  /\  K  < 
D )  ->  ( D  ||  N  ->  -.  D  ||  ( N  -  K ) ) ) )
63623impia 1193 1  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  -> 
( D  ||  N  ->  -.  D  ||  ( N  -  K )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   E!wreu 2816   class class class wbr 4447  (class class class)co 6285   0cc0 9493    < clt 9629    - cmin 9806   NNcn 10537   NN0cn0 10796   ZZcz 10865    || cdivides 13850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fz 11674  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-dvds 13851
This theorem is referenced by:  ndvdsadd  13928
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