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Theorem ndvdssub 13632
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 10607 . . . . . . . 8  |-  ( K  e.  NN  ->  K  e.  NN0 )
2 nnne0 10375 . . . . . . . 8  |-  ( K  e.  NN  ->  K  =/=  0 )
31, 2jca 532 . . . . . . 7  |-  ( K  e.  NN  ->  ( K  e.  NN0  /\  K  =/=  0 ) )
4 df-ne 2622 . . . . . . . . . . . 12  |-  ( K  =/=  0  <->  -.  K  =  0 )
54anbi2i 694 . . . . . . . . . . 11  |-  ( ( K  <  D  /\  K  =/=  0 )  <->  ( K  <  D  /\  -.  K  =  0 ) )
6 divalg2 13630 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E! r  e.  NN0  ( r  <  D  /\  D  ||  ( N  -  r ) ) )
7 breq1 4316 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( r  =  x  ->  (
r  <  D  <->  x  <  D ) )
8 oveq2 6120 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( r  =  x  ->  ( N  -  r )  =  ( N  -  x ) )
98breq2d 4325 . . . . . . . . . . . . . . . . . . . . . 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 3174 . . . . . . . . . . . . . . . . . . . 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 10372 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( D  e.  NN  ->  0  <  D )
15143ad2ant2 1010 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  0  <  D )
16 zcn 10672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( N  e.  ZZ  ->  N  e.  CC )
1716subid1d 9729 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  ZZ  ->  ( N  -  0 )  =  N )
1817breq2d 4325 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  e.  ZZ  ->  ( D  ||  ( N  - 
0 )  <->  D  ||  N
) )
1918biimpar 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( N  e.  ZZ  /\  D  ||  N )  ->  D  ||  ( N  - 
0 ) )
20193adant2 1007 . . . . . . . . . . . . . . . . . . . . . . . . 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 1187 . . . . . . . . . . . . . . . . . . . . . . 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 10615 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  NN0
26 breq1 4316 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( x  =  0  ->  (
x  <  D  <->  0  <  D ) )
27 oveq2 6120 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( x  =  0  ->  ( N  -  x )  =  ( N  - 
0 ) )
2827breq2d 4325 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2452 . . . . . . . . . . . . . . . . . . . . . . . 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 3090 . . . . . . . . . . . . . . . . . . . . . 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 2812 . . . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  D  ||  N )  ->  A. r  e.  NN0  ( ( r  <  D  /\  D  ||  ( N  -  r
) )  ->  r  =  0 ) )
44 breq1 4316 . . . . . . . . . . . . . . . 16  |-  ( r  =  K  ->  (
r  <  D  <->  K  <  D ) )
45 oveq2 6120 . . . . . . . . . . . . . . . . 17  |-  ( r  =  K  ->  ( N  -  r )  =  ( N  -  K ) )
4645breq2d 4325 . . . . . . . . . . . . . . . 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 2449 . . . . . . . . . . . . . . 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 3090 . . . . . . . . . . . . 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 1189 . . 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 1184 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737   E!wreu 2738   class class class wbr 4313  (class class class)co 6112   0cc0 9303    < clt 9439    - cmin 9616   NNcn 10343   NN0cn0 10600   ZZcz 10667    || cdivides 13556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-fz 11459  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-dvds 13557
This theorem is referenced by:  ndvdsadd  13633
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