Theorem coprmdvds2 12004

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 11715	. . . . . 6 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N || K <-> E. x e. ZZ ( x x. N ) = K ) )
2	1	3adant1 1004	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( N || K <-> E. x e. ZZ ( x x. N ) = K ) )
3	2	adantr 274	. . . 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 522	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> x e. ZZ )
5		simpl2 990	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> N e. ZZ )
6		zcn 9187	. . . . . . . . . . . 12 |- ( x e. ZZ -> x e. CC )
7		zcn 9187	. . . . . . . . . . . 12 |- ( N e. ZZ -> N e. CC )
8		mulcom 7873	. . . . . . . . . . . 12 |- ( ( x e. CC /\ N e. CC ) -> ( x x. N ) = ( N x. x ) )
9	6, 7, 8	syl2an 287	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ N e. ZZ ) -> ( x x. N ) = ( N x. x ) )
10	4, 5, 9	syl2anc 409	. . . . . . . . . 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 3988	. . . . . . . . 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 521	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> ( M gcd N ) = 1 )
13		simpl1 989	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( ( M gcd N ) = 1 /\ x e. ZZ ) ) -> M e. ZZ )
14		coprmdvds 12003	. . . . . . . . . . 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 1227	. . . . . . . . . 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 425	. . . . . . . . 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 149	. . . . . . . 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 11745	. . . . . . . . 9 |- ( ( M e. ZZ /\ x e. ZZ /\ N e. ZZ ) -> ( M || x -> ( M x. N ) || ( x x. N ) ) )
19	13, 4, 5, 18	syl3anc 1227	. . . . . . . 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 3980	. . . . . . . 8 |- ( ( x x. N ) = K -> ( M || ( x x. N ) <-> M || K ) )
22		breq2 3980	. . . . . . . 8 |- ( ( x x. N ) = K -> ( ( M x. N ) || ( x x. N ) <-> ( M x. N ) || K ) )
23	21, 22	imbi12d 233	. . . . . . 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 154	. . . . . 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 398	. . . . 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 2581	. . . 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 149	. . 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 252	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 103   <-> wb 104   /\ w3a 967   = wceq 1342   e. wcel 2135   E.wrex 2443   class class class wbr 3976  (class class class)co 5836   CCcc 7742   1c1 7745   x. cmul 7749   ZZcz 9182   || cdvds 11713   gcd cgcd 11860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862  ax-arch 7863  ax-caucvg 7864

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-sup 6940  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-3 8908  df-4 8909  df-n0 9106  df-z 9183  df-uz 9458  df-q 9549  df-rp 9581  df-fz 9936  df-fzo 10068  df-fl 10195  df-mod 10248  df-seqfrec 10371  df-exp 10445  df-cj 10770  df-re 10771  df-im 10772  df-rsqrt 10926  df-abs 10927  df-dvds 11714  df-gcd 11861

This theorem is referenced by:  rpmulgcd2  12006  crth  12133