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Theorem dvds2sub 13877
Description: If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvds2sub  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  N )  ->  K  ||  ( M  -  N )
) )

Proof of Theorem dvds2sub
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpa 993 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  M  e.  ZZ ) )
2 3simpb 994 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  N  e.  ZZ ) )
3 zsubcl 10905 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N
)  e.  ZZ )
43anim2i 569 . . 3  |-  ( ( K  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( K  e.  ZZ  /\  ( M  -  N )  e.  ZZ ) )
543impb 1192 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  ( M  -  N )  e.  ZZ ) )
6 zsubcl 10905 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  -  y
)  e.  ZZ )
76adantl 466 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  -  y )  e.  ZZ )
8 zcn 10869 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 10869 . . . . . . . 8  |-  ( y  e.  ZZ  ->  y  e.  CC )
10 zcn 10869 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
11 subdir 9991 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  K  e.  CC )  ->  (
( x  -  y
)  x.  K )  =  ( ( x  x.  K )  -  ( y  x.  K
) ) )
128, 9, 10, 11syl3an 1270 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ  /\  K  e.  ZZ )  ->  (
( x  -  y
)  x.  K )  =  ( ( x  x.  K )  -  ( y  x.  K
) ) )
13123comr 1204 . . . . . 6  |-  ( ( K  e.  ZZ  /\  x  e.  ZZ  /\  y  e.  ZZ )  ->  (
( x  -  y
)  x.  K )  =  ( ( x  x.  K )  -  ( y  x.  K
) ) )
14133expb 1197 . . . . 5  |-  ( ( K  e.  ZZ  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
x  -  y )  x.  K )  =  ( ( x  x.  K )  -  (
y  x.  K ) ) )
15 oveq12 6293 . . . . 5  |-  ( ( ( x  x.  K
)  =  M  /\  ( y  x.  K
)  =  N )  ->  ( ( x  x.  K )  -  ( y  x.  K
) )  =  ( M  -  N ) )
1614, 15sylan9eq 2528 . . . 4  |-  ( ( ( K  e.  ZZ  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  /\  ( ( x  x.  K )  =  M  /\  (
y  x.  K )  =  N ) )  ->  ( ( x  -  y )  x.  K )  =  ( M  -  N ) )
1716ex 434 . . 3  |-  ( ( K  e.  ZZ  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
( x  x.  K
)  =  M  /\  ( y  x.  K
)  =  N )  ->  ( ( x  -  y )  x.  K )  =  ( M  -  N ) ) )
18173ad2antl1 1158 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
( x  x.  K
)  =  M  /\  ( y  x.  K
)  =  N )  ->  ( ( x  -  y )  x.  K )  =  ( M  -  N ) ) )
191, 2, 5, 7, 18dvds2lem 13857 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  N )  ->  K  ||  ( M  -  N )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447  (class class class)co 6284   CCcc 9490    x. cmul 9497    - cmin 9805   ZZcz 10864    || cdivides 13847
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 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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-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 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-dvds 13848
This theorem is referenced by:  dvdssub2  13882  divalglem9  13918  prmdiv  14174  prmdiveq  14175  4sqlem10  14324  4sqlem14  14335  jm2.20nn  30571
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