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Theorem absmulgcd 11983
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 11932 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
2 nn0re 9156 . . . . . 6 |- ( ( M gcd N ) e. NN0 -> ( M gcd N ) e. RR )
3 nn0ge0 9172 . . . . . 6 |- ( ( M gcd N ) e. NN0 -> 0 <_ ( M gcd N ) )
42, 3absidd 11142 . . . . 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 5881 . . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` K ) x. ( abs ` ( M gcd N ) ) ) = ( ( abs ` K ) x. ( M gcd N ) ) )
763adant1 1015 . 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 9229 . . . 4 |- ( K e. ZZ -> K e. CC )
91nn0cnd 9202 . . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
10 absmul 11044 . . . 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 1199 . 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 9229 . . . . 5 |- ( M e. ZZ -> M e. CC )
14 zcn 9229 . . . . 5 |- ( N e. ZZ -> N e. CC )
15 absmul 11044 . . . . . . 7 |- ( ( K e. CC /\ M e. CC ) -> ( abs ` ( K x. M ) ) = ( ( abs ` K ) x. ( abs ` M ) ) )
16 absmul 11044 . . . . . . 7 |- ( ( K e. CC /\ N e. CC ) -> ( abs ` ( K x. N ) ) = ( ( abs ` K ) x. ( abs ` N ) ) )
1715, 16oveqan12d 5884 . . . . . 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 1293 . . . . 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 1280 . . . 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 9277 . . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K x. M ) e. ZZ )
21 zmulcl 9277 . . . . . 6 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
22 gcdabs 11954 . . . . . 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 1293 . . . 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 11060 . . . . 5 |- ( K e. ZZ -> ( abs ` K ) e. NN0 )
26 zabscl 11061 . . . . 5 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
27 zabscl 11061 . . . . 5 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
28 mulgcd 11982 . . . . 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 1280 . . . 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 2216 . . 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 11954 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
32313adant1 1015 . . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
3332oveq2d 5881 . . 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 2208 . 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 2219 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 978   = wceq 1353   e. wcel 2146   ` cfv 5208  (class class class)co 5865   CCcc 7784   x. cmul 7791   NN0cn0 9147   ZZcz 9224   abscabs 10972   gcd cgcd 11908
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-fl 10238  df-mod 10291  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-dvds 11761  df-gcd 11909
This theorem is referenced by: (None)
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