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Theorem muldvds1 12866
Description: If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
muldvds1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  ||  N  ->  K 
||  N ) )

Proof of Theorem muldvds1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 zmulcl 10316 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
21anim1i 552 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  e.  ZZ  /\  N  e.  ZZ ) )
323impa 1148 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  N  e.  ZZ )
)
4 3simpb 955 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  N  e.  ZZ ) )
5 zmulcl 10316 . . . 4  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( x  x.  M
)  e.  ZZ )
65ancoms 440 . . 3  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( x  x.  M
)  e.  ZZ )
763ad2antl2 1120 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( x  x.  M )  e.  ZZ )
8 zcn 10279 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 10279 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
10 zcn 10279 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  CC )
11 mulass 9070 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  K  e.  CC  /\  M  e.  CC )  ->  (
( x  x.  K
)  x.  M )  =  ( x  x.  ( K  x.  M
) ) )
12 mul32 9225 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  K  e.  CC  /\  M  e.  CC )  ->  (
( x  x.  K
)  x.  M )  =  ( ( x  x.  M )  x.  K ) )
1311, 12eqtr3d 2469 . . . . . . . 8  |-  ( ( x  e.  CC  /\  K  e.  CC  /\  M  e.  CC )  ->  (
x  x.  ( K  x.  M ) )  =  ( ( x  x.  M )  x.  K ) )
148, 9, 10, 13syl3an 1226 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
x  x.  ( K  x.  M ) )  =  ( ( x  x.  M )  x.  K ) )
15143coml 1160 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  x  e.  ZZ )  ->  (
x  x.  ( K  x.  M ) )  =  ( ( x  x.  M )  x.  K ) )
16153expa 1153 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  x  e.  ZZ )  ->  ( x  x.  ( K  x.  M
) )  =  ( ( x  x.  M
)  x.  K ) )
17163adantl3 1115 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( x  x.  ( K  x.  M
) )  =  ( ( x  x.  M
)  x.  K ) )
1817eqeq1d 2443 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  ( K  x.  M ) )  =  N  <->  ( ( x  x.  M )  x.  K )  =  N ) )
1918biimpd 199 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  ( K  x.  M ) )  =  N  ->  ( (
x  x.  M )  x.  K )  =  N ) )
203, 4, 7, 19dvds1lem 12853 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  ||  N  ->  K 
||  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   CCcc 8980    x. cmul 8987   ZZcz 10274    || cdivides 12844
This theorem is referenced by:  3dvds  12904  odmulg  15184  jm2.20nn  27059  jm2.27c  27069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-dvds 12845
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