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Theorem dvdsmulc 12163
Description: Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvdsmulc |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
Proof of Theorem dvdsmulc
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 3simpc 999 . . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M e. ZZ /\ N e. ZZ ) )
2 zmulcl 9428 . . . . . 6 |- ( ( M e. ZZ /\ K e. ZZ ) -> ( M x. K ) e. ZZ )
323adant2 1019 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M x. K ) e. ZZ )
4 zmulcl 9428 . . . . . 6 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N x. K ) e. ZZ )
543adant1 1018 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( N x. K ) e. ZZ )
63, 5jca 306 . . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( ( M x. K ) e. ZZ /\ ( N x. K ) e. ZZ ) )
763comr 1214 . . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M x. K ) e. ZZ /\ ( N x. K ) e. ZZ ) )
8 simpr 110 . . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> x e. ZZ )
9 zcn 9379 . . . . . . . . 9 |- ( x e. ZZ -> x e. CC )
10 zcn 9379 . . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
11 zcn 9379 . . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
12 mulass 8058 . . . . . . . . 9 |- ( ( x e. CC /\ M e. CC /\ K e. CC ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
139, 10, 11, 12syl3an 1292 . . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ K e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
14133com13 1211 . . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
15143expa 1206 . . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
16153adantl3 1158 . . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) x. K ) = ( x x. ( M x. K ) ) )
17 oveq1 5953 . . . . 5 |- ( ( x x. M ) = N -> ( ( x x. M ) x. K ) = ( N x. K ) )
1816, 17sylan9req 2259 . . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) /\ ( x x. M ) = N ) -> ( x x. ( M x. K ) ) = ( N x. K ) )
1918ex 115 . . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ x e. ZZ ) -> ( ( x x. M ) = N -> ( x x. ( M x. K ) ) = ( N x. K ) ) )
201, 7, 8, 19dvds1lem 12146 . 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
21203coml 1213 1 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( M || N -> ( M x. K ) || ( N x. K ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2176   class class class wbr 4045  (class class class)co 5946   CCcc 7925   x. cmul 7932   ZZcz 9374   || cdvds 12131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038
This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-sub 8247  df-neg 8248  df-inn 9039  df-n0 9298  df-z 9375  df-dvds 12132
This theorem is referenced by:  dvdsmulcr  12165  coprmdvds2  12448  mulgcddvds  12449  rpmulgcd2  12450  pcpremul  12649  znrrg  14455  mpodvdsmulf1o  15495
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