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Theorem absmulgcd 11950
Description: Distribute absolute value of multiplication over gcd. Theorem 1.4(c) in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
absmulgcd |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( abs ` ( K x. ( M gcd N ) ) ) )
Proof of Theorem absmulgcd
StepHypRef Expression
1 gcdcl 11899 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
2 nn0re 9123 . . . . . 6 |- ( ( M gcd N ) e. NN0 -> ( M gcd N ) e. RR )
3 nn0ge0 9139 . . . . . 6 |- ( ( M gcd N ) e. NN0 -> 0 <_ ( M gcd N ) )
42, 3absidd 11109 . . . . 5 |- ( ( M gcd N ) e. NN0 -> ( abs ` ( M gcd N ) ) = ( M gcd N ) )
51, 4syl 14 . . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( M gcd N ) ) = ( M gcd N ) )
65oveq2d 5858 . . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` K ) x. ( abs ` ( M gcd N ) ) ) = ( ( abs ` K ) x. ( M gcd N ) ) )
763adant1 1005 . 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` K ) x. ( abs ` ( M gcd N ) ) ) = ( ( abs ` K ) x. ( M gcd N ) ) )
8 zcn 9196 . . . 4 |- ( K e. ZZ -> K e. CC )
91nn0cnd 9169 . . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
10 absmul 11011 . . . 4 |- ( ( K e. CC /\ ( M gcd N ) e. CC ) -> ( abs ` ( K x. ( M gcd N ) ) ) = ( ( abs ` K ) x. ( abs ` ( M gcd N ) ) ) )
118, 9, 10syl2an 287 . . 3 |- ( ( K e. ZZ /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( abs ` ( K x. ( M gcd N ) ) ) = ( ( abs ` K ) x. ( abs ` ( M gcd N ) ) ) )
12113impb 1189 . 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( abs ` ( K x. ( M gcd N ) ) ) = ( ( abs ` K ) x. ( abs ` ( M gcd N ) ) ) )
13 zcn 9196 . . . . 5 |- ( M e. ZZ -> M e. CC )
14 zcn 9196 . . . . 5 |- ( N e. ZZ -> N e. CC )
15 absmul 11011 . . . . . . 7 |- ( ( K e. CC /\ M e. CC ) -> ( abs ` ( K x. M ) ) = ( ( abs ` K ) x. ( abs ` M ) ) )
16 absmul 11011 . . . . . . 7 |- ( ( K e. CC /\ N e. CC ) -> ( abs ` ( K x. N ) ) = ( ( abs ` K ) x. ( abs ` N ) ) )
1715, 16oveqan12d 5861 . . . . . 6 |- ( ( ( K e. CC /\ M e. CC ) /\ ( K e. CC /\ N e. CC ) ) -> ( ( abs ` ( K x. M ) ) gcd ( abs ` ( K x. N ) ) ) = ( ( ( abs ` K ) x. ( abs ` M ) ) gcd ( ( abs ` K ) x. ( abs ` N ) ) ) )
18173impdi 1283 . . . . 5 |- ( ( K e. CC /\ M e. CC /\ N e. CC ) -> ( ( abs ` ( K x. M ) ) gcd ( abs ` ( K x. N ) ) ) = ( ( ( abs ` K ) x. ( abs ` M ) ) gcd ( ( abs ` K ) x. ( abs ` N ) ) ) )
198, 13, 14, 18syl3an 1270 . . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` ( K x. M ) ) gcd ( abs ` ( K x. N ) ) ) = ( ( ( abs ` K ) x. ( abs ` M ) ) gcd ( ( abs ` K ) x. ( abs ` N ) ) ) )
20 zmulcl 9244 . . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K x. M ) e. ZZ )
21 zmulcl 9244 . . . . . 6 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
22 gcdabs 11921 . . . . . 6 |- ( ( ( K x. M ) e. ZZ /\ ( K x. N ) e. ZZ ) -> ( ( abs ` ( K x. M ) ) gcd ( abs ` ( K x. N ) ) ) = ( ( K x. M ) gcd ( K x. N ) ) )
2320, 21, 22syl2an 287 . . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ ( K e. ZZ /\ N e. ZZ ) ) -> ( ( abs ` ( K x. M ) ) gcd ( abs ` ( K x. N ) ) ) = ( ( K x. M ) gcd ( K x. N ) ) )
24233impdi 1283 . . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` ( K x. M ) ) gcd ( abs ` ( K x. N ) ) ) = ( ( K x. M ) gcd ( K x. N ) ) )
25 nn0abscl 11027 . . . . 5 |- ( K e. ZZ -> ( abs ` K ) e. NN0 )
26 zabscl 11028 . . . . 5 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
27 zabscl 11028 . . . . 5 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
28 mulgcd 11949 . . . . 5 |- ( ( ( abs ` K ) e. NN0 /\ ( abs ` M ) e. ZZ /\ ( abs ` N ) e. ZZ ) -> ( ( ( abs ` K ) x. ( abs ` M ) ) gcd ( ( abs ` K ) x. ( abs ` N ) ) ) = ( ( abs ` K ) x. ( ( abs ` M ) gcd ( abs ` N ) ) ) )
2925, 26, 27, 28syl3an 1270 . . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` K ) x. ( abs ` M ) ) gcd ( ( abs ` K ) x. ( abs ` N ) ) ) = ( ( abs ` K ) x. ( ( abs ` M ) gcd ( abs ` N ) ) ) )
3019, 24, 293eqtr3d 2206 . . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( ( abs ` K ) x. ( ( abs ` M ) gcd ( abs ` N ) ) ) )
31 gcdabs 11921 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
32313adant1 1005 . . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
3332oveq2d 5858 . . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` K ) x. ( ( abs ` M ) gcd ( abs ` N ) ) ) = ( ( abs ` K ) x. ( M gcd N ) ) )
3430, 33eqtrd 2198 . 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( ( abs ` K ) x. ( M gcd N ) ) )
357, 12, 343eqtr4rd 2209 1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( abs ` ( K x. ( M gcd N ) ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   ` cfv 5188  (class class class)co 5842   CCcc 7751   x. cmul 7758   NN0cn0 9114   ZZcz 9191   abscabs 10939   gcd cgcd 11875
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876
This theorem is referenced by: (None)
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