Theorem coprmdvds2 10475

Description: If an integer is divisible by two coprime integers, then it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.)

Assertion

Ref	Expression
coprmdvds2	|- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( M || K /\ N || K ) -> ( M x. N ) || K ) )

Proof of Theorem coprmdvds2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		divides 10197	. . . . . 6 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N || K <-> E. x e. ZZ ( x x. N ) = K ) )
2	1	3adant1 956	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( N || K <-> E. x e. ZZ ( x x. N ) = K ) )
3	2	adantr 270	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( N || K <-> E. x e. ZZ ( x x. N ) = K ) )
4		simprr 498	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> x e. ZZ )
5		simpl2 942	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> N e. ZZ )
6		zcn 8356	. . . . . . . . . . . 12 |- ( x e. ZZ -> x e. CC )
7		zcn 8356	. . . . . . . . . . . 12 |- ( N e. ZZ -> N e. CC )
8		mulcom 7102	. . . . . . . . . . . 12 |- ( ( x e. CC /\ N e. CC ) -> ( x x. N ) = ( N x. x ) )
9	6, 7, 8	syl2an 283	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ N e. ZZ ) -> ( x x. N ) = ( N x. x ) )
10	4, 5, 9	syl2anc 403	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( x x. N ) = ( N x. x ) )
11	10	breq2d 3797	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M || ( x x. N ) <-> M || ( N x. x ) ) )
12		simprl 497	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M gcd N ) = 1 )
13		simpl1 941	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> M e. ZZ )
14		coprmdvds 10474	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. ZZ /\ x e. ZZ ) -> ( ( M || ( N x. x ) /\ ( M gcd N ) = 1 ) -> M || x ) )
15	13, 5, 4, 14	syl3anc 1169	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( ( M || ( N x. x ) /\ ( M gcd N ) = 1 ) -> M || x ) )
16	12, 15	mpan2d 418	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M || ( N x. x ) -> M || x ) )
17	11, 16	sylbid 148	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M || ( x x. N ) -> M || x ) )
18		dvdsmulc 10223	. . . . . . . . 9 |- ( ( M e. ZZ /\ x e. ZZ /\ N e. ZZ ) -> ( M || x -> ( M x. N ) || ( x x. N ) ) )
19	13, 4, 5, 18	syl3anc 1169	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M || x -> ( M x. N ) || ( x x. N ) ) )
20	17, 19	syld 44	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M || ( x x. N ) -> ( M x. N ) || ( x x. N ) ) )
21		breq2 3789	. . . . . . . 8 |- ( ( x x. N ) = K -> ( M || ( x x. N ) <-> M || K ) )
22		breq2 3789	. . . . . . . 8 |- ( ( x x. N ) = K -> ( ( M x. N ) || ( x x. N ) <-> ( M x. N ) || K ) )
23	21, 22	imbi12d 232	. . . . . . 7 |- ( ( x x. N ) = K -> ( ( M || ( x x. N ) -> ( M x. N ) || ( x x. N ) ) <-> ( M || K -> ( M x. N ) || K ) ) )
24	20, 23	syl5ibcom 153	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( ( x x. N ) = K -> ( M || K -> ( M x. N ) || K ) ) )
25	24	anassrs 392	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) /\ x e. ZZ ) -> ( ( x x. N ) = K -> ( M || K -> ( M x. N ) || K ) ) )
26	25	rexlimdva 2477	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( E. x e. ZZ ( x x. N ) = K -> ( M || K -> ( M x. N ) || K ) ) )
27	3, 26	sylbid 148	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( N || K -> ( M || K -> ( M x. N ) || K ) ) )
28	27	com23 77	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( M || K -> ( N || K -> ( M x. N ) || K ) ) )
29	28	impd 251	1 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( M || K /\ N || K ) -> ( M x. N ) || K ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   E.wrex 2349   class class class wbr 3785  (class class class)co 5532   CCcc 6979   1c1 6982   x. cmul 6986   ZZcz 8351   || cdvds 10195   gcd cgcd 10338

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-sup 6397  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-q 8705  df-rp 8735  df-fz 9030  df-fzo 9153  df-fl 9274  df-mod 9325  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885  df-dvds 10196  df-gcd 10339

This theorem is referenced by:  rpmulgcd2  10477