Theorem dvdslegcd 12660

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 1063	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> K e. ZZ )
2	1	zred 9700	. . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> K e. RR )
3		simpll2 1064	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> M e. ZZ )
4		simpll3 1065	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> N e. ZZ )
5		simplr 529	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> -. ( M = 0 /\ N = 0 ) )
6		lttri3 8353	. . . . . . 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 9584	. . . . . . 7 |- ZZ C_ RR
9		gcdsupex 12653	. . . . . . 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 3303	. . . . . . 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 7289	. . . . 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 1273	. . . 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 4112	. . . . . . . . 9 |- ( n = K -> ( n || M <-> K || M ) )
16		breq1 4112	. . . . . . . . 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 2974	. . . . . . 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 7290	. . . . . 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 1273	. . . . 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 8394	. . 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 12657	. . . 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 1273	. . 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 4137	. 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 1005   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   {crab 2524   C_ wss 3211   class class class wbr 4109  (class class class)co 6050   supcsup 7273   RRcr 8126   0cc0 8127   < clt 8308   <_ cle 8309   ZZcz 9577   || cdvds 12473   gcd cgcd 12649

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-sup 7275  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474  df-gcd 12650

This theorem is referenced by:  nndvdslegcd  12661  gcd0id  12675  gcdneg  12678  gcdaddm  12680  gcdzeq  12718  rpdvds  12796  coprm  12841  phimullem  12922  pockthlem  13054  2sqlem8a  15995  2sqlem8  15996