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Theorem ndvdssub 11554
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 8952 . . . . . . . 8  |-  ( K  e.  NN  ->  K  e.  NN0 )
2 nnne0 8716 . . . . . . . 8  |-  ( K  e.  NN  ->  K  =/=  0 )
31, 2jca 304 . . . . . . 7  |-  ( K  e.  NN  ->  ( K  e.  NN0  /\  K  =/=  0 ) )
4 df-ne 2286 . . . . . . . . . . . 12  |-  ( K  =/=  0  <->  -.  K  =  0 )
54anbi2i 452 . . . . . . . . . . 11  |-  ( ( K  <  D  /\  K  =/=  0 )  <->  ( K  <  D  /\  -.  K  =  0 ) )
6 divalg2 11550 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E! r  e.  NN0  ( r  <  D  /\  D  ||  ( N  -  r ) ) )
7 breq1 3902 . . . . . . . . . . . . . . . . . . . . 21  |-  ( r  =  x  ->  (
r  <  D  <->  x  <  D ) )
8 oveq2 5750 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( r  =  x  ->  ( N  -  r )  =  ( N  -  x ) )
98breq2d 3911 . . . . . . . . . . . . . . . . . . . . 21  |-  ( r  =  x  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  x )
) )
107, 9anbi12d 464 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  =  x  ->  (
( r  <  D  /\  D  ||  ( N  -  r ) )  <-> 
( x  <  D  /\  D  ||  ( N  -  x ) ) ) )
1110reu4 2851 . . . . . . . . . . . . . . . . . . 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 121 . . . . . . . . . . . . . . . . . 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 8713 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( D  e.  NN  ->  0  <  D )
14133ad2ant2 988 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  0  <  D )
15 zcn 9027 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( N  e.  ZZ  ->  N  e.  CC )
1615subid1d 8030 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  ZZ  ->  ( N  -  0 )  =  N )
1716breq2d 3911 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  e.  ZZ  ->  ( D  ||  ( N  - 
0 )  <->  D  ||  N
) )
1817biimpar 295 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( N  e.  ZZ  /\  D  ||  N )  ->  D  ||  ( N  - 
0 ) )
19183adant2 985 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  D  ||  ( N  -  0 ) )
2014, 19jca 304 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  (
0  <  D  /\  D  ||  ( N  - 
0 ) ) )
21203expa 1166 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  ZZ  /\  D  e.  NN )  /\  D  ||  N
)  ->  ( 0  <  D  /\  D  ||  ( N  -  0 ) ) )
2221anim2i 339 . . . . . . . . . . . . . . . . . . . . . 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 266 . . . . . . . . . . . . . . . . . . . . 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 8960 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  NN0
25 breq1 3902 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( x  =  0  ->  (
x  <  D  <->  0  <  D ) )
26 oveq2 5750 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( x  =  0  ->  ( N  -  x )  =  ( N  - 
0 ) )
2726breq2d 3911 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( x  =  0  ->  ( D  ||  ( N  -  x )  <->  D  ||  ( N  -  0 ) ) )
2825, 27anbi12d 464 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( x  =  0  ->  (
( x  <  D  /\  D  ||  ( N  -  x ) )  <-> 
( 0  <  D  /\  D  ||  ( N  -  0 ) ) ) )
2928anbi2d 459 . . . . . . . . . . . . . . . . . . . . . . . 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 2127 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  0  ->  (
r  =  x  <->  r  = 
0 ) )
3129, 30imbi12d 233 . . . . . . . . . . . . . . . . . . . . . . 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 2759 . . . . . . . . . . . . . . . . . . . . . 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 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 ) )
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 256 . . . . . . . . . . . . . . . . . . 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 2472 . . . . . . . . . . . . . . . . . 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 383 . . . . . . . . . . . . . . . . 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 2486 . . . . . . . . . . . . . . . . 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 121 . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . 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 49 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  ||  N  ->  A. r  e.  NN0  ( ( r  < 
D  /\  D  ||  ( N  -  r )
)  ->  r  = 
0 ) ) )
42413impia 1163 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  A. r  e.  NN0  ( ( r  <  D  /\  D  ||  ( N  -  r
) )  ->  r  =  0 ) )
43 breq1 3902 . . . . . . . . . . . . . . . 16  |-  ( r  =  K  ->  (
r  <  D  <->  K  <  D ) )
44 oveq2 5750 . . . . . . . . . . . . . . . . 17  |-  ( r  =  K  ->  ( N  -  r )  =  ( N  -  K ) )
4544breq2d 3911 . . . . . . . . . . . . . . . 16  |-  ( r  =  K  ->  ( D  ||  ( N  -  r )  <->  D  ||  ( N  -  K )
) )
4643, 45anbi12d 464 . . . . . . . . . . . . . . 15  |-  ( r  =  K  ->  (
( r  <  D  /\  D  ||  ( N  -  r ) )  <-> 
( K  <  D  /\  D  ||  ( N  -  K ) ) ) )
47 eqeq1 2124 . . . . . . . . . . . . . . 15  |-  ( r  =  K  ->  (
r  =  0  <->  K  =  0 ) )
4846, 47imbi12d 233 . . . . . . . . . . . . . 14  |-  ( r  =  K  ->  (
( ( r  < 
D  /\  D  ||  ( N  -  r )
)  ->  r  = 
0 )  <->  ( ( K  <  D  /\  D  ||  ( N  -  K
) )  ->  K  =  0 ) ) )
4948rspcv 2759 . . . . . . . . . . . . 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 pm3.37 663 . . . . . . . . . . . 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 164 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN0  ->  (
( K  <  D  /\  K  =/=  0
)  ->  -.  D  ||  ( N  -  K
) ) ) )
5453exp4a 363 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN0  ->  ( K  <  D  ->  ( K  =/=  0  ->  -.  D  ||  ( N  -  K ) ) ) ) )
5554com23 78 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  <  D  ->  ( K  e.  NN0  ->  ( K  =/=  0  ->  -.  D  ||  ( N  -  K ) ) ) ) )
5655imp4a 346 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  <  D  ->  (
( K  e.  NN0  /\  K  =/=  0 )  ->  -.  D  ||  ( N  -  K )
) ) )
573, 56syl7 69 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  <  D  ->  ( K  e.  NN  ->  -.  D  ||  ( N  -  K ) ) ) )
5857com23 78 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  ( K  e.  NN  ->  ( K  <  D  ->  -.  D  ||  ( N  -  K ) ) ) )
5958impd 252 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  (
( K  e.  NN  /\  K  <  D )  ->  -.  D  ||  ( N  -  K )
) )
60593expia 1168 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  ||  N  ->  ( ( K  e.  NN  /\  K  < 
D )  ->  -.  D  ||  ( N  -  K ) ) ) )
6160com23 78 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( K  e.  NN  /\  K  < 
D )  ->  ( D  ||  N  ->  -.  D  ||  ( N  -  K ) ) ) )
62613impia 1163 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 103    /\ w3a 947    = wceq 1316    e. wcel 1465    =/= wne 2285   A.wral 2393   E.wrex 2394   E!wreu 2395   class class class wbr 3899  (class class class)co 5742   0cc0 7588    < clt 7768    - cmin 7901   NNcn 8688   NN0cn0 8945   ZZcz 9022    || cdvds 11420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fl 10011  df-mod 10064  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-dvds 11421
This theorem is referenced by:  ndvdsadd  11555
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