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Theorem absmulgcd 13764
Description: Distribute absolute value of multiplication over gcd. (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 13724 . . . . 5 |- ((M e. ZZ /\ N e. ZZ) -> (M gcd N) e. NN0)
2 nn0re 7317 . . . . . 6 |- ((M gcd N) e. NN0 -> (M gcd N) e. RR)
3 nn0ge0 7326 . . . . . 6 |- ((M gcd N) e. NN0 -> 0 <_ (M gcd N))
4 absid 8113 . . . . . 6 |- (((M gcd N) e. RR /\ 0 <_ (M gcd N)) -> (abs` (M gcd N)) = (M gcd N))
52, 3, 4syl11anc 524 . . . . 5 |- ((M gcd N) e. NN0 -> (abs` (M gcd N)) = (M gcd N))
61, 5syl 12 . . . 4 |- ((M e. ZZ /\ N e. ZZ) -> (abs` (M gcd N)) = (M gcd N))
76opreq2d 4898 . . 3 |- ((M e. ZZ /\ N e. ZZ) -> ((abs` K) x. (abs` (M gcd N))) = ((abs` K) x. (M gcd N)))
873adant1 894 . 2 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((abs` K) x. (abs` (M gcd N))) = ((abs` K) x. (M gcd N)))
9 absmul 8109 . . . 4 |- ((K e. CC /\ (M gcd N) e. CC) -> (abs` (K x. (M gcd N))) = ((abs` K) x. (abs` (M gcd N))))
10 zcn 7349 . . . 4 |- (K e. ZZ -> K e. CC)
11 nn0cn 7318 . . . . 5 |- ((M gcd N) e. NN0 -> (M gcd N) e. CC)
121, 11syl 12 . . . 4 |- ((M e. ZZ /\ N e. ZZ) -> (M gcd N) e. CC)
139, 10, 12syl2an 503 . . 3 |- ((K e. ZZ /\ (M e. ZZ /\ N e. ZZ)) -> (abs` (K x. (M gcd N))) = ((abs` K) x. (abs` (M gcd N))))
14133impb 1063 . 2 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (abs` (K x. (M gcd N))) = ((abs` K) x. (abs` (M gcd N))))
15 absmul 8109 . . . . . . 7 |- ((K e. CC /\ M e. CC) -> (abs` (K x. M)) = ((abs` K) x. (abs` M)))
16 absmul 8109 . . . . . . 7 |- ((K e. CC /\ N e. CC) -> (abs` (K x. N)) = ((abs` K) x. (abs` N)))
1715, 16opreqan12d 4902 . . . . . 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 1152 . . . . 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))))
19 zcn 7349 . . . . 5 |- (M e. ZZ -> M e. CC)
20 zcn 7349 . . . . 5 |- (N e. ZZ -> N e. CC)
2118, 10, 19, 20syl3an 1139 . . . 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))))
22 gcdabs 13737 . . . . . 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)))
23 zmulcl 7389 . . . . . 6 |- ((K e. ZZ /\ M e. ZZ) -> (K x. M) e. ZZ)
24 zmulcl 7389 . . . . . 6 |- ((K e. ZZ /\ N e. ZZ) -> (K x. N) e. ZZ)
2522, 23, 24syl2an 503 . . . . 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)))
26253impdi 1152 . . . 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)))
27 mulgcd 13763 . . . . 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))))
28 nn0abscl 8131 . . . . 5 |- (K e. ZZ -> (abs` K) e. NN0)
29 nn0abscl 8131 . . . . . 6 |- (M e. ZZ -> (abs` M) e. NN0)
30 nn0z 7363 . . . . . 6 |- ((abs` M) e. NN0 -> (abs` M) e. ZZ)
3129, 30syl 12 . . . . 5 |- (M e. ZZ -> (abs` M) e. ZZ)
32 nn0abscl 8131 . . . . . 6 |- (N e. ZZ -> (abs` N) e. NN0)
33 nn0z 7363 . . . . . 6 |- ((abs` N) e. NN0 -> (abs` N) e. ZZ)
3432, 33syl 12 . . . . 5 |- (N e. ZZ -> (abs` N) e. ZZ)
3527, 28, 31, 34syl3an 1139 . . . 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))))
3621, 26, 353eqtr3d 1934 . . 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))))
37 gcdabs 13737 . . . . 5 |- ((M e. ZZ /\ N e. ZZ) -> ((abs` M) gcd (abs` N)) = (M gcd N))
38373adant1 894 . . . 4 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((abs` M) gcd (abs` N)) = (M gcd N))
3938opreq2d 4898 . . 3 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((abs` K) x. ((abs` M) gcd (abs` N))) = ((abs` K) x. (M gcd N)))
4036, 39eqtrd 1925 . 2 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((K x. M) gcd (K x. N)) = ((abs` K) x. (M gcd N)))
418, 14, 403eqtr4rd 1939 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 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   x. cmul 6391   <_ cle 6448  NN0cn0 6450  ZZcz 6451  abscabs 8000   gcd cgcd 13713
This theorem is referenced by:  dvdsgcd 13765  coprmdvds 13783
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-mod 7500  df-uz 7587  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-divides 13663  df-gcd 13714
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