Theorem dvdslegcd 12101

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 1038	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> K e. ZZ )
2	1	zred 9439	. . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> K e. RR )
3		simpll2 1039	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> M e. ZZ )
4		simpll3 1040	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> N e. ZZ )
5		simplr 528	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> -. ( M = 0 /\ N = 0 ) )
6		lttri3 8099	. . . . . . 7 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
7	6	adantl 277	. . . . . 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 9324	. . . . . . 7 |- ZZ C_ RR
9		gcdsupex 12094	. . . . . . 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 3244	. . . . . . 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 7057	. . . . 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 1248	. . . 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 110	. . . . . 6 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> ( K || M /\ K || N ) )
15		breq1 4032	. . . . . . . . 9 |- ( n = K -> ( n || M <-> K || M ) )
16		breq1 4032	. . . . . . . . 9 |- ( n = K -> ( n || N <-> K || N ) )
17	15, 16	anbi12d 473	. . . . . . . 8 |- ( n = K -> ( ( n || M /\ n || N ) <-> ( K || M /\ K || N ) ) )
18	17	elrab3 2917	. . . . . . 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 167	. . . . 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 7058	. . . . . 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 1248	. . . . 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 8140	. . 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 12098	. . . 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 1248	. . 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 4057	. 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 115	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   {crab 2476   C_ wss 3153   class class class wbr 4029  (class class class)co 5918   supcsup 7041   RRcr 7871   0cc0 7872   < clt 8054   <_ cle 8055   ZZcz 9317   || cdvds 11930   gcd cgcd 12079

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080

This theorem is referenced by:  nndvdslegcd  12102  gcd0id  12116  gcdneg  12119  gcdaddm  12121  gcdzeq  12159  rpdvds  12237  coprm  12282  phimullem  12363  pockthlem  12494  2sqlem8a  15209  2sqlem8  15210