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Theorem absmulgcd 12053
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 12002 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
2 nn0re 9216 . . . . . 6 |- ( ( M gcd N ) e. NN0 -> ( M gcd N ) e. RR )
3 nn0ge0 9232 . . . . . 6 |- ( ( M gcd N ) e. NN0 -> 0 <_ ( M gcd N ) )
42, 3absidd 11211 . . . . 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 5913 . . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` K ) x. ( abs ` ( M gcd N ) ) ) = ( ( abs ` K ) x. ( M gcd N ) ) )
763adant1 1017 . 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 9289 . . . 4 |- ( K e. ZZ -> K e. CC )
91nn0cnd 9262 . . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
10 absmul 11113 . . . 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 1201 . 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 9289 . . . . 5 |- ( M e. ZZ -> M e. CC )
14 zcn 9289 . . . . 5 |- ( N e. ZZ -> N e. CC )
15 absmul 11113 . . . . . . 7 |- ( ( K e. CC /\ M e. CC ) -> ( abs ` ( K x. M ) ) = ( ( abs ` K ) x. ( abs ` M ) ) )
16 absmul 11113 . . . . . . 7 |- ( ( K e. CC /\ N e. CC ) -> ( abs ` ( K x. N ) ) = ( ( abs ` K ) x. ( abs ` N ) ) )
1715, 16oveqan12d 5916 . . . . . 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 1304 . . . . 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 1291 . . . 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 9337 . . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K x. M ) e. ZZ )
21 zmulcl 9337 . . . . . 6 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
22 gcdabs 12024 . . . . . 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 1304 . . . 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 11129 . . . . 5 |- ( K e. ZZ -> ( abs ` K ) e. NN0 )
26 zabscl 11130 . . . . 5 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
27 zabscl 11130 . . . . 5 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
28 mulgcd 12052 . . . . 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 1291 . . . 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 2230 . . 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 12024 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
32313adant1 1017 . . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
3332oveq2d 5913 . . 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 2222 . 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 2233 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 980   = wceq 1364   e. wcel 2160   ` cfv 5235  (class class class)co 5897   CCcc 7840   x. cmul 7847   NN0cn0 9207   ZZcz 9284   abscabs 11041   gcd cgcd 11978
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-sup 7014  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-fz 10041  df-fzo 10175  df-fl 10303  df-mod 10356  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-dvds 11830  df-gcd 11979
This theorem is referenced by: (None)
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