Theorem dvdslegcd 11897

Description: An integer which divides both operands of the gcd operator is bounded by it. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
dvdslegcd	|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( K || M /\ K || N ) -> K <_ ( M gcd N ) ) )

Proof of Theorem dvdslegcd

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

Step	Hyp	Ref	Expression
1		simpll1 1026	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> K e. ZZ )
2	1	zred 9313	. . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> K e. RR )
3		simpll2 1027	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> M e. ZZ )
4		simpll3 1028	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> N e. ZZ )
5		simplr 520	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> -. ( M = 0 /\ N = 0 ) )
6		lttri3 7978	. . . . . . 7 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
7	6	adantl 275	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
8		zssre 9198	. . . . . . 7 |- ZZ C_ RR
9		gcdsupex 11890	. . . . . . 7 |- ( ( ( 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 ) ) )
10		ssrexv 3207	. . . . . . 7 |- ( 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 ) ) ) )
11	8, 9, 10	mpsyl 65	. . . . . 6 |- ( ( ( 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 ) ) )
12	7, 11	supclti 6963	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) e. RR )
13	3, 4, 5, 12	syl21anc 1227	. . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) e. RR )
14		simpr 109	. . . . . 6 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> ( K || M /\ K || N ) )
15		breq1 3985	. . . . . . . . 9 |- ( n = K -> ( n || M <-> K || M ) )
16		breq1 3985	. . . . . . . . 9 |- ( n = K -> ( n || N <-> K || N ) )
17	15, 16	anbi12d 465	. . . . . . . 8 |- ( n = K -> ( ( n || M /\ n || N ) <-> ( K || M /\ K || N ) ) )
18	17	elrab3 2883	. . . . . . 7 |- ( K e. ZZ -> ( K e. { n e. ZZ | ( n || M /\ n || N ) } <-> ( K || M /\ K || N ) ) )
19	1, 18	syl 14	. . . . . 6 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> ( K e. { n e. ZZ | ( n || M /\ n || N ) } <-> ( K || M /\ K || N ) ) )
20	14, 19	mpbird 166	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> K e. { n e. ZZ | ( n || M /\ n || N ) } )
21	7, 11	supubti 6964	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( K e. { n e. ZZ | ( n || M /\ n || N ) } -> -. sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) < K ) )
22	3, 4, 5, 21	syl21anc 1227	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> ( K e. { n e. ZZ | ( n || M /\ n || N ) } -> -. sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) < K ) )
23	20, 22	mpd 13	. . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> -. sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) < K )
24	2, 13, 23	nltled 8019	. . 3 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> K <_ sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) )
25		gcdn0val 11894	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) )
26	3, 4, 5, 25	syl21anc 1227	. . 3 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> ( M gcd N ) = sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) )
27	24, 26	breqtrrd 4010	. 2 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> K <_ ( M gcd N ) )
28	27	ex 114	1 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( K || M /\ K || N ) -> K <_ ( M gcd N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   {crab 2448   C_ wss 3116   class class class wbr 3982  (class class class)co 5842   supcsup 6947   RRcr 7752   0cc0 7753   < clt 7933   <_ cle 7934   ZZcz 9191   || cdvds 11727   gcd cgcd 11875

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876

This theorem is referenced by:  nndvdslegcd  11898  gcd0id  11912  gcdneg  11915  gcdaddm  11917  gcdzeq  11955  rpdvds  12031  coprm  12076  phimullem  12157  pockthlem  12286  2sqlem8a  13598  2sqlem8  13599