Theorem coprmdvds 13783

Description: If an integer divides the product of two integers and is coprime to one of them, then it divides the other. (Contributed by Paul Chapman, 22-Jun-2011.)

Assertion

Ref	Expression
coprmdvds	|- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((K||(M x. N) /\ (K gcd M) = 1) -> K||N))

Proof of Theorem coprmdvds

Step	Hyp	Ref	Expression
1		mulcom 6459	. . . . . . . . . 10 |- ((M e. CC /\ N e. CC) -> (M x. N) = (N x. M))
2		zcn 7349	. . . . . . . . . 10 |- (M e. ZZ -> M e. CC)
3		zcn 7349	. . . . . . . . . 10 |- (N e. ZZ -> N e. CC)
4	1, 2, 3	syl2an 503	. . . . . . . . 9 |- ((M e. ZZ /\ N e. ZZ) -> (M x. N) = (N x. M))
5	4	3adant1 894	. . . . . . . 8 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (M x. N) = (N x. M))
6	5	breq2d 3350	. . . . . . 7 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (K||(M x. N) <-> K||(N x. M)))
7		dvdsmul2 13677	. . . . . . . . . 10 |- ((N e. ZZ /\ K e. ZZ) -> K||(N x. K))
8	7	ancoms 484	. . . . . . . . 9 |- ((K e. ZZ /\ N e. ZZ) -> K||(N x. K))
9	8	3adant2 895	. . . . . . . 8 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> K||(N x. K))
10		simp1 876	. . . . . . . . 9 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> K e. ZZ)
11		zmulcl 7389	. . . . . . . . . . 11 |- ((N e. ZZ /\ K e. ZZ) -> (N x. K) e. ZZ)
12	11	ancoms 484	. . . . . . . . . 10 |- ((K e. ZZ /\ N e. ZZ) -> (N x. K) e. ZZ)
13	12	3adant2 895	. . . . . . . . 9 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (N x. K) e. ZZ)
14		zmulcl 7389	. . . . . . . . . . 11 |- ((N e. ZZ /\ M e. ZZ) -> (N x. M) e. ZZ)
15	14	ancoms 484	. . . . . . . . . 10 |- ((M e. ZZ /\ N e. ZZ) -> (N x. M) e. ZZ)
16	15	3adant1 894	. . . . . . . . 9 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (N x. M) e. ZZ)
17		dvdsgcd 13765	. . . . . . . . 9 |- ((K e. ZZ /\ (N x. K) e. ZZ /\ (N x. M) e. ZZ) -> ((K||(N x. K) /\ K||(N x. M)) -> K||((N x. K) gcd (N x. M))))
18	10, 13, 16, 17	syl111anc 1100	. . . . . . . 8 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((K||(N x. K) /\ K||(N x. M)) -> K||((N x. K) gcd (N x. M))))
19	9, 18	mpand 765	. . . . . . 7 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (K||(N x. M) -> K||((N x. K) gcd (N x. M))))
20	6, 19	sylbid 220	. . . . . 6 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (K||(M x. N) -> K||((N x. K) gcd (N x. M))))
21	20	adantr 425	. . . . 5 |- (((K e. ZZ /\ M e. ZZ /\ N e. ZZ) /\ (K gcd M) = 1) -> (K||(M x. N) -> K||((N x. K) gcd (N x. M))))
22		absmulgcd 13764	. . . . . . . . . 10 |- ((N e. ZZ /\ K e. ZZ /\ M e. ZZ) -> ((N x. K) gcd (N x. M)) = (abs` (N x. (K gcd M))))
23	22	3coml 1075	. . . . . . . . 9 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((N x. K) gcd (N x. M)) = (abs` (N x. (K gcd M))))
24	23	adantr 425	. . . . . . . 8 |- (((K e. ZZ /\ M e. ZZ /\ N e. ZZ) /\ (K gcd M) = 1) -> ((N x. K) gcd (N x. M)) = (abs` (N x. (K gcd M))))
25		opreq2 4890	. . . . . . . . . . 11 |- ((K gcd M) = 1 -> (N x. (K gcd M)) = (N x. 1))
26		mulid1 6464	. . . . . . . . . . . 12 |- (N e. CC -> (N x. 1) = N)
27	3, 26	syl 12	. . . . . . . . . . 11 |- (N e. ZZ -> (N x. 1) = N)
28	25, 27	sylan9eqr 1951	. . . . . . . . . 10 |- ((N e. ZZ /\ (K gcd M) = 1) -> (N x. (K gcd M)) = N)
29	28	fveq2d 4685	. . . . . . . . 9 |- ((N e. ZZ /\ (K gcd M) = 1) -> (abs` (N x. (K gcd M))) = (abs` N))
30	29	3ad2antl3 1040	. . . . . . . 8 |- (((K e. ZZ /\ M e. ZZ /\ N e. ZZ) /\ (K gcd M) = 1) -> (abs` (N x. (K gcd M))) = (abs` N))
31	24, 30	eqtrd 1925	. . . . . . 7 |- (((K e. ZZ /\ M e. ZZ /\ N e. ZZ) /\ (K gcd M) = 1) -> ((N x. K) gcd (N x. M)) = (abs` N))
32	31	breq2d 3350	. . . . . 6 |- (((K e. ZZ /\ M e. ZZ /\ N e. ZZ) /\ (K gcd M) = 1) -> (K||((N x. K) gcd (N x. M)) <-> K||(abs` N)))
33		dvdsabsb 13674	. . . . . . . 8 |- ((K e. ZZ /\ N e. ZZ) -> (K||N <-> K||(abs` N)))
34	33	3adant2 895	. . . . . . 7 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (K||N <-> K||(abs` N)))
35	34	adantr 425	. . . . . 6 |- (((K e. ZZ /\ M e. ZZ /\ N e. ZZ) /\ (K gcd M) = 1) -> (K||N <-> K||(abs` N)))
36	32, 35	bitr4d 590	. . . . 5 |- (((K e. ZZ /\ M e. ZZ /\ N e. ZZ) /\ (K gcd M) = 1) -> (K||((N x. K) gcd (N x. M)) <-> K||N))
37	21, 36	sylibd 219	. . . 4 |- (((K e. ZZ /\ M e. ZZ /\ N e. ZZ) /\ (K gcd M) = 1) -> (K||(M x. N) -> K||N))
38	37	ex 402	. . 3 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((K gcd M) = 1 -> (K||(M x. N) -> K||N)))
39	38	com23 36	. 2 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (K||(M x. N) -> ((K gcd M) = 1 -> K||N)))
40	39	imp3a 388	1 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((K||(M x. N) /\ (K gcd M) = 1) -> K||N))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  1c1 6387   x. cmul 6391  ZZcz 6451  abscabs 8000  ||cdivides 13662   gcd cgcd 13713

This theorem is referenced by:  prmdvdsmul 13784

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

Copyright terms: Public domain