Theorem gcddvds 12723

Description: The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

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

Proof of Theorem gcddvds

Dummy variables n x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0z 9638	. . . . . 6 |- 0 e. ZZ
2		dvds0 12556	. . . . . 6 |- ( 0 e. ZZ -> 0 || 0 )
3	1, 2	ax-mp 5	. . . . 5 |- 0 || 0
4		breq2 4132	. . . . . . 7 |- ( M = 0 -> ( 0 || M <-> 0 || 0 ) )
5		breq2 4132	. . . . . . 7 |- ( N = 0 -> ( 0 || N <-> 0 || 0 ) )
6	4, 5	bi2anan9 614	. . . . . 6 |- ( ( M = 0 /\ N = 0 ) -> ( ( 0 || M /\ 0 || N ) <-> ( 0 || 0 /\ 0 || 0 ) ) )
7		anidm 400	. . . . . 6 |- ( ( 0 || 0 /\ 0 || 0 ) <-> 0 || 0 )
8	6, 7	bitrdi 196	. . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( ( 0 || M /\ 0 || N ) <-> 0 || 0 ) )
9	3, 8	mpbiri 168	. . . 4 |- ( ( M = 0 /\ N = 0 ) -> ( 0 || M /\ 0 || N ) )
10		oveq12 6088	. . . . . . 7 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = ( 0 gcd 0 ) )
11		gcd0val 12720	. . . . . . 7 |- ( 0 gcd 0 ) = 0
12	10, 11	eqtrdi 2287	. . . . . 6 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = 0 )
13	12	breq1d 4138	. . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( ( M gcd N ) || M <-> 0 || M ) )
14	12	breq1d 4138	. . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( ( M gcd N ) || N <-> 0 || N ) )
15	13, 14	anbi12d 477	. . . 4 |- ( ( M = 0 /\ N = 0 ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || N ) <-> ( 0 || M /\ 0 || N ) ) )
16	9, 15	mpbird 167	. . 3 |- ( ( M = 0 /\ N = 0 ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
17	16	adantl 277	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
18		gcdn0val 12721	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) )
19		zssre 9634	. . . . . 6 |- ZZ C_ RR
20		gcdsupex 12717	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> E. x e. ZZ ( A. y e. { n e. ZZ | ( n || M /\ n || N ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ( n || M /\ n || N ) } y < z ) ) )
21		ssrexv 3313	. . . . . 6 |- ( ZZ C_ RR -> ( E. x e. ZZ ( A. y e. { n e. ZZ | ( n || M /\ n || N ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ( n || M /\ n || N ) } y < z ) ) -> E. x e. RR ( A. y e. { n e. ZZ | ( n || M /\ n || N ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ( n || M /\ n || N ) } y < z ) ) ) )
22	19, 20, 21	mpsyl 65	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> E. x e. RR ( A. y e. { n e. ZZ | ( n || M /\ n || N ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ( n || M /\ n || N ) } y < z ) ) )
23		ssrab2 3333	. . . . . 6 |- { n e. ZZ | ( n || M /\ n || N ) } C_ ZZ
24	23	a1i 9	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> { n e. ZZ | ( n || M /\ n || N ) } C_ ZZ )
25	22, 24	suprzclex 9727	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) e. { n e. ZZ | ( n || M /\ n || N ) } )
26	18, 25	eqeltrd 2315	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. { n e. ZZ | ( n || M /\ n || N ) } )
27		gcdn0cl 12722	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. NN )
28	27	nnzd 9750	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. ZZ )
29		breq1 4131	. . . . . 6 |- ( n = ( M gcd N ) -> ( n || M <-> ( M gcd N ) || M ) )
30		breq1 4131	. . . . . 6 |- ( n = ( M gcd N ) -> ( n || N <-> ( M gcd N ) || N ) )
31	29, 30	anbi12d 477	. . . . 5 |- ( n = ( M gcd N ) -> ( ( n || M /\ n || N ) <-> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) ) )
32	31	elrab3 2983	. . . 4 |- ( ( M gcd N ) e. ZZ -> ( ( M gcd N ) e. { n e. ZZ | ( n || M /\ n || N ) } <-> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) ) )
33	28, 32	syl 14	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( M gcd N ) e. { n e. ZZ | ( n || M /\ n || N ) } <-> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) ) )
34	26, 33	mpbid 147	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
35		gcdmndc 12715	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
36		exmiddc 848	. . 3 |- (DECID ( M = 0 /\ N = 0 ) -> ( ( M = 0 /\ N = 0 ) \/ -. ( M = 0 /\ N = 0 ) ) )
37	35, 36	syl 14	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ N = 0 ) \/ -. ( M = 0 /\ N = 0 ) ) )
38	17, 34, 37	mpjaodan 810	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   C_ wss 3220   class class class wbr 4128  (class class class)co 6079   supcsup 7316   RRcr 8172   0cc0 8173   < clt 8354   ZZcz 9627   || cdvds 12537   gcd cgcd 12713

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-sup 7318  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-fl 10688  df-mod 10743  df-seqfrec 10868  df-exp 10959  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-dvds 12538  df-gcd 12714

This theorem is referenced by:  zeqzmulgcd  12730  divgcdz  12731  divgcdnn  12735  gcd0id  12739  gcdneg  12742  gcdaddm  12744  gcd1  12747  dvdsgcdb  12773  dfgcd2  12774  mulgcd  12776  gcdzeq  12782  dvdsmulgcd  12785  sqgcd  12789  dvdssqlem  12790  bezoutr  12792  gcddvdslcm  12834  lcmgcdlem  12838  lcmgcdeq  12844  coprmgcdb  12849  ncoprmgcdne1b  12850  mulgcddvds  12855  rpmulgcd2  12856  qredeu  12858  rpdvds  12860  divgcdcoprm0  12862  divgcdodd  12904  coprm  12905  rpexp  12914  divnumden  12957  phimullem  12986  hashgcdlem  12999  hashgcdeq  13001  phisum  13002  pythagtriplem4  13030  pythagtriplem19  13044  pcgcd1  13090  pc2dvds  13092  pockthlem  13118  znunit  14977  znrrg  14978  mpodvdsmulf1o  16087  2sqlem8  16225