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Theorem absmulgcd 12554
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 12503 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
2 nn0re 9389 . . . . . 6 |- ( ( M gcd N ) e. NN0 -> ( M gcd N ) e. RR )
3 nn0ge0 9405 . . . . . 6 |- ( ( M gcd N ) e. NN0 -> 0 <_ ( M gcd N ) )
42, 3absidd 11694 . . . . 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 6023 . . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` K ) x. ( abs ` ( M gcd N ) ) ) = ( ( abs ` K ) x. ( M gcd N ) ) )
763adant1 1039 . 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 9462 . . . 4 |- ( K e. ZZ -> K e. CC )
91nn0cnd 9435 . . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
10 absmul 11596 . . . 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 1223 . 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 9462 . . . . 5 |- ( M e. ZZ -> M e. CC )
14 zcn 9462 . . . . 5 |- ( N e. ZZ -> N e. CC )
15 absmul 11596 . . . . . . 7 |- ( ( K e. CC /\ M e. CC ) -> ( abs ` ( K x. M ) ) = ( ( abs ` K ) x. ( abs ` M ) ) )
16 absmul 11596 . . . . . . 7 |- ( ( K e. CC /\ N e. CC ) -> ( abs ` ( K x. N ) ) = ( ( abs ` K ) x. ( abs ` N ) ) )
1715, 16oveqan12d 6026 . . . . . 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 1327 . . . . 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 1313 . . . 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 9511 . . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K x. M ) e. ZZ )
21 zmulcl 9511 . . . . . 6 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
22 gcdabs 12525 . . . . . 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 1327 . . . 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 11612 . . . . 5 |- ( K e. ZZ -> ( abs ` K ) e. NN0 )
26 zabscl 11613 . . . . 5 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
27 zabscl 11613 . . . . 5 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
28 mulgcd 12553 . . . . 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 1313 . . . 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 2270 . . 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 12525 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
32313adant1 1039 . . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
3332oveq2d 6023 . . 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 2262 . 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 2273 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 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6007   CCcc 8008   x. cmul 8015   NN0cn0 9380   ZZcz 9457   abscabs 11524   gcd cgcd 12490
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-sup 7162  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-dvds 12315  df-gcd 12491
This theorem is referenced by: (None)
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