Theorem dvdsgcd 13765

Description: An integer which divides each of two others also divides their gcd. (Contributed by Paul Chapman, 22-Jun-2011.)

Assertion

Ref	Expression
dvdsgcd	|- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((K||M /\ K||N) -> K||(M gcd N)))

Proof of Theorem dvdsgcd

Step	Hyp	Ref	Expression
1		divides 13664	. . . . 5 |- ((K e. ZZ /\ M e. ZZ) -> (K||M <-> E.x e. ZZ (x x. K) = M))
2		divides 13664	. . . . 5 |- ((K e. ZZ /\ N e. ZZ) -> (K||N <-> E.y e. ZZ (y x. K) = N))
3	1, 2	bi2anan9 694	. . . 4 |- (((K e. ZZ /\ M e. ZZ) /\ (K e. ZZ /\ N e. ZZ)) -> ((K||M /\ K||N) <-> (E.x e. ZZ (x x. K) = M /\ E.y e. ZZ (y x. K) = N)))
4	3	3impdi 1152	. . 3 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((K||M /\ K||N) <-> (E.x e. ZZ (x x. K) = M /\ E.y e. ZZ (y x. K) = N)))
5		reeanv 2249	. . 3 |- (E.x e. ZZ E.y e. ZZ ((x x. K) = M /\ (y x. K) = N) <-> (E.x e. ZZ (x x. K) = M /\ E.y e. ZZ (y x. K) = N))
6	4, 5	syl6bbr 597	. 2 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((K||M /\ K||N) <-> E.x e. ZZ E.y e. ZZ ((x x. K) = M /\ (y x. K) = N)))
7		dvdsmul1 13676	. . . . . . . . . . 11 |- ((K e. ZZ /\ (x gcd y) e. ZZ) -> K||(K x. (x gcd y)))
8		gcdcl 13724	. . . . . . . . . . . 12 |- ((x e. ZZ /\ y e. ZZ) -> (x gcd y) e. NN0)
9		nn0z 7363	. . . . . . . . . . . 12 |- ((x gcd y) e. NN0 -> (x gcd y) e. ZZ)
10	8, 9	syl 12	. . . . . . . . . . 11 |- ((x e. ZZ /\ y e. ZZ) -> (x gcd y) e. ZZ)
11	7, 10	sylan2 500	. . . . . . . . . 10 |- ((K e. ZZ /\ (x e. ZZ /\ y e. ZZ)) -> K||(K x. (x gcd y)))
12		dvdsabsb 13674	. . . . . . . . . . . 12 |- ((K e. ZZ /\ (K x. (x gcd y)) e. ZZ) -> (K||(K x. (x gcd y)) <-> K||(abs` (K x. (x gcd y)))))
13		zmulcl 7389	. . . . . . . . . . . . 13 |- ((K e. ZZ /\ (x gcd y) e. ZZ) -> (K x. (x gcd y)) e. ZZ)
14	13, 10	sylan2 500	. . . . . . . . . . . 12 |- ((K e. ZZ /\ (x e. ZZ /\ y e. ZZ)) -> (K x. (x gcd y)) e. ZZ)
15	12, 14	sylan2 500	. . . . . . . . . . 11 |- ((K e. ZZ /\ (K e. ZZ /\ (x e. ZZ /\ y e. ZZ))) -> (K||(K x. (x gcd y)) <-> K||(abs` (K x. (x gcd y)))))
16	15	anabss5 560	. . . . . . . . . 10 |- ((K e. ZZ /\ (x e. ZZ /\ y e. ZZ)) -> (K||(K x. (x gcd y)) <-> K||(abs` (K x. (x gcd y)))))
17	11, 16	mpbid 212	. . . . . . . . 9 |- ((K e. ZZ /\ (x e. ZZ /\ y e. ZZ)) -> K||(abs` (K x. (x gcd y))))
18	17	3impb 1063	. . . . . . . 8 |- ((K e. ZZ /\ x e. ZZ /\ y e. ZZ) -> K||(abs` (K x. (x gcd y))))
19	18	adantr 425	. . . . . . 7 |- (((K e. ZZ /\ x e. ZZ /\ y e. ZZ) /\ ((x x. K) = M /\ (y x. K) = N)) -> K||(abs` (K x. (x gcd y))))
20		mulcom 6459	. . . . . . . . . . . . . . 15 |- ((K e. CC /\ x e. CC) -> (K x. x) = (x x. K))
21	20	eqeq1d 1892	. . . . . . . . . . . . . 14 |- ((K e. CC /\ x e. CC) -> ((K x. x) = M <-> (x x. K) = M))
22		mulcom 6459	. . . . . . . . . . . . . . 15 |- ((K e. CC /\ y e. CC) -> (K x. y) = (y x. K))
23	22	eqeq1d 1892	. . . . . . . . . . . . . 14 |- ((K e. CC /\ y e. CC) -> ((K x. y) = N <-> (y x. K) = N))
24	21, 23	bi2anan9 694	. . . . . . . . . . . . 13 |- (((K e. CC /\ x e. CC) /\ (K e. CC /\ y e. CC)) -> (((K x. x) = M /\ (K x. y) = N) <-> ((x x. K) = M /\ (y x. K) = N)))
25	24	3impdi 1152	. . . . . . . . . . . 12 |- ((K e. CC /\ x e. CC /\ y e. CC) -> (((K x. x) = M /\ (K x. y) = N) <-> ((x x. K) = M /\ (y x. K) = N)))
26		zcn 7349	. . . . . . . . . . . 12 |- (K e. ZZ -> K e. CC)
27		zcn 7349	. . . . . . . . . . . 12 |- (x e. ZZ -> x e. CC)
28		zcn 7349	. . . . . . . . . . . 12 |- (y e. ZZ -> y e. CC)
29	25, 26, 27, 28	syl3an 1139	. . . . . . . . . . 11 |- ((K e. ZZ /\ x e. ZZ /\ y e. ZZ) -> (((K x. x) = M /\ (K x. y) = N) <-> ((x x. K) = M /\ (y x. K) = N)))
30	29	biimpar 461	. . . . . . . . . 10 |- (((K e. ZZ /\ x e. ZZ /\ y e. ZZ) /\ ((x x. K) = M /\ (y x. K) = N)) -> ((K x. x) = M /\ (K x. y) = N))
31		opreq12 4891	. . . . . . . . . 10 |- (((K x. x) = M /\ (K x. y) = N) -> ((K x. x) gcd (K x. y)) = (M gcd N))
32	30, 31	syl 12	. . . . . . . . 9 |- (((K e. ZZ /\ x e. ZZ /\ y e. ZZ) /\ ((x x. K) = M /\ (y x. K) = N)) -> ((K x. x) gcd (K x. y)) = (M gcd N))
33		absmulgcd 13764	. . . . . . . . . 10 |- ((K e. ZZ /\ x e. ZZ /\ y e. ZZ) -> ((K x. x) gcd (K x. y)) = (abs` (K x. (x gcd y))))
34	33	adantr 425	. . . . . . . . 9 |- (((K e. ZZ /\ x e. ZZ /\ y e. ZZ) /\ ((x x. K) = M /\ (y x. K) = N)) -> ((K x. x) gcd (K x. y)) = (abs` (K x. (x gcd y))))
35	32, 34	eqtr3d 1927	. . . . . . . 8 |- (((K e. ZZ /\ x e. ZZ /\ y e. ZZ) /\ ((x x. K) = M /\ (y x. K) = N)) -> (M gcd N) = (abs` (K x. (x gcd y))))
36	35	breq2d 3350	. . . . . . 7 |- (((K e. ZZ /\ x e. ZZ /\ y e. ZZ) /\ ((x x. K) = M /\ (y x. K) = N)) -> (K||(M gcd N) <-> K||(abs` (K x. (x gcd y)))))
37	19, 36	mpbird 213	. . . . . 6 |- (((K e. ZZ /\ x e. ZZ /\ y e. ZZ) /\ ((x x. K) = M /\ (y x. K) = N)) -> K||(M gcd N))
38	37	ex 402	. . . . 5 |- ((K e. ZZ /\ x e. ZZ /\ y e. ZZ) -> (((x x. K) = M /\ (y x. K) = N) -> K||(M gcd N)))
39	38	3expib 1070	. . . 4 |- (K e. ZZ -> ((x e. ZZ /\ y e. ZZ) -> (((x x. K) = M /\ (y x. K) = N) -> K||(M gcd N))))
40	39	r19.23advv 2218	. . 3 |- (K e. ZZ -> (E.x e. ZZ E.y e. ZZ ((x x. K) = M /\ (y x. K) = N) -> K||(M gcd N)))
41	40	3ad2ant1 897	. 2 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (E.x e. ZZ E.y e. ZZ ((x x. K) = M /\ (y x. K) = N) -> K||(M gcd N)))
42	6, 41	sylbid 220	1 |- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> ((K||M /\ K||N) -> K||(M gcd N)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384   x. cmul 6391  NN0cn0 6450  ZZcz 6451  abscabs 8000  ||cdivides 13662   gcd cgcd 13713

This theorem is referenced by:  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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