Theorem gcddvds 12742

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 9657	. . . . . 6 |- 0 e. ZZ
2		dvds0 12575	. . . . . 6 |- ( 0 e. ZZ -> 0 || 0 )
3	1, 2	ax-mp 5	. . . . 5 |- 0 || 0
4		breq2 4134	. . . . . . 7 |- ( M = 0 -> ( 0 || M <-> 0 || 0 ) )
5		breq2 4134	. . . . . . 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 6094	. . . . . . 7 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = ( 0 gcd 0 ) )
11		gcd0val 12739	. . . . . . 7 |- ( 0 gcd 0 ) = 0
12	10, 11	eqtrdi 2287	. . . . . 6 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = 0 )
13	12	breq1d 4140	. . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( ( M gcd N ) || M <-> 0 || M ) )
14	12	breq1d 4140	. . . . 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 12740	. . . 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 9653	. . . . . 6 |- ZZ C_ RR
20		gcdsupex 12736	. . . . . 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 9746	. . . 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 12741	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. NN )
28	27	nnzd 9769	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. ZZ )
29		breq1 4133	. . . . . 6 |- ( n = ( M gcd N ) -> ( n || M <-> ( M gcd N ) || M ) )
30		breq1 4133	. . . . . 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 12734	. . 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 ) )

This proof depends on syntax axioms:   -. 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 4130  (class class class)co 6085   supcsup 7322   RRcr 8178   0cc0 8179   < clt 8360   ZZcz 9646   || cdvds 12556   gcd cgcd 12732

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-fl 10707  df-mod 10762  df-seqfrec 10887  df-exp 10978  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-dvds 12557  df-gcd 12733

This theorem is used by:  zeqzmulgcd  12749  divgcdz  12750  divgcdnn  12754  gcd0id  12758  gcdneg  12761  gcdaddm  12763  gcd1  12766  dvdsgcdb  12792  dfgcd2  12793  mulgcd  12795  gcdzeq  12801  dvdsmulgcd  12804  sqgcd  12808  dvdssqlem  12809  bezoutr  12811  gcddvdslcm  12853  lcmgcdlem  12857  lcmgcdeq  12863  coprmgcdb  12868  ncoprmgcdne1b  12869  mulgcddvds  12874  rpmulgcd2  12875  qredeu  12877  rpdvds  12879  divgcdcoprm0  12881  divgcdodd  12923  coprm  12924  rpexp  12933  divnumden  12976  phimullem  13005  hashgcdlem  13018  hashgcdeq  13020  phisum  13021  pythagtriplem4  13049  pythagtriplem19  13063  pcgcd1  13109  pc2dvds  13111  pockthlem  13137  znunit  14996  znrrg  14997  mpodvdsmulf1o  16110  2sqlem8  16254