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Theorem absmulgcd 12794
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 12743 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
2 nn0re 9572 . . . . . 6 |- ( ( M gcd N ) e. NN0 -> ( M gcd N ) e. RR )
3 nn0ge0 9588 . . . . . 6 |- ( ( M gcd N ) e. NN0 -> 0 <_ ( M gcd N ) )
42, 3absidd 11933 . . . . 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 6101 . . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` K ) x. ( abs ` ( M gcd N ) ) ) = ( ( abs ` K ) x. ( M gcd N ) ) )
763adant1 1046 . 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 9649 . . . 4 |- ( K e. ZZ -> K e. CC )
91nn0cnd 9622 . . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
10 absmul 11835 . . . 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 289 . . 3 |- ( ( K e. ZZ /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( abs ` ( K x. ( M gcd N ) ) ) = ( ( abs ` K ) x. ( abs ` ( M gcd N ) ) ) )
12113impb 1230 . 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 9649 . . . . 5 |- ( M e. ZZ -> M e. CC )
14 zcn 9649 . . . . 5 |- ( N e. ZZ -> N e. CC )
15 absmul 11835 . . . . . . 7 |- ( ( K e. CC /\ M e. CC ) -> ( abs ` ( K x. M ) ) = ( ( abs ` K ) x. ( abs ` M ) ) )
16 absmul 11835 . . . . . . 7 |- ( ( K e. CC /\ N e. CC ) -> ( abs ` ( K x. N ) ) = ( ( abs ` K ) x. ( abs ` N ) ) )
1715, 16oveqan12d 6104 . . . . . 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 1334 . . . . 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 1320 . . . 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 9698 . . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K x. M ) e. ZZ )
21 zmulcl 9698 . . . . . 6 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
22 gcdabs 12765 . . . . . 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 289 . . . . 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 1334 . . . 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 11851 . . . . 5 |- ( K e. ZZ -> ( abs ` K ) e. NN0 )
26 zabscl 11852 . . . . 5 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
27 zabscl 11852 . . . . 5 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
28 mulgcd 12793 . . . . 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 1320 . . . 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 2279 . . 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 12765 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
32313adant1 1046 . . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
3332oveq2d 6101 . . 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 2271 . 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 2282 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   CCcc 8177   x. cmul 8184   NN0cn0 9563   ZZcz 9644   abscabs 11763   gcd cgcd 12730
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731
This theorem is used by: (None)
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