Theorem coprmdvds2 12530

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 12215	. . . . . 6 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N || K <-> E. x e. ZZ ( x x. N ) = K ) )
2	1	3adant1 1018	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( N || K <-> E. x e. ZZ ( x x. N ) = K ) )
3	2	adantr 276	. . . 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 531	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> x e. ZZ )
5		simpl2 1004	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> N e. ZZ )
6		zcn 9412	. . . . . . . . . . . 12 |- ( x e. ZZ -> x e. CC )
7		zcn 9412	. . . . . . . . . . . 12 |- ( N e. ZZ -> N e. CC )
8		mulcom 8089	. . . . . . . . . . . 12 |- ( ( x e. CC /\ N e. CC ) -> ( x x. N ) = ( N x. x ) )
9	6, 7, 8	syl2an 289	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ N e. ZZ ) -> ( x x. N ) = ( N x. x ) )
10	4, 5, 9	syl2anc 411	. . . . . . . . . 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 4071	. . . . . . . . 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 529	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M gcd N ) = 1 )
13		simpl1 1003	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> M e. ZZ )
14		coprmdvds 12529	. . . . . . . . . . 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 1250	. . . . . . . . . 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 428	. . . . . . . . 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 150	. . . . . . . 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 12245	. . . . . . . . 9 |- ( ( M e. ZZ /\ x e. ZZ /\ N e. ZZ ) -> ( M || x -> ( M x. N ) || ( x x. N ) ) )
19	13, 4, 5, 18	syl3anc 1250	. . . . . . . 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 45	. . . . . . 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 4063	. . . . . . . 8 |- ( ( x x. N ) = K -> ( M || ( x x. N ) <-> M || K ) )
22		breq2 4063	. . . . . . . 8 |- ( ( x x. N ) = K -> ( ( M x. N ) || ( x x. N ) <-> ( M x. N ) || K ) )
23	21, 22	imbi12d 234	. . . . . . 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 155	. . . . . 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 400	. . . . 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 2625	. . . 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 150	. . 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 78	. 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 254	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 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   E.wrex 2487   class class class wbr 4059  (class class class)co 5967   CCcc 7958   1c1 7961   x. cmul 7965   ZZcz 9407   || cdvds 12213   gcd cgcd 12389

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-sup 7112  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-gcd 12390

This theorem is referenced by:  rpmulgcd2  12532  crth  12661