Theorem dvdslegcd 11848

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 1021	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> K e. ZZ )
2	1	zred 9286	. . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> K e. RR )
3		simpll2 1022	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) /\ ( K || M /\ K || N ) ) -> M e. ZZ )
4		simpll3 1023	. . . . 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 7957	. . . . . . 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 9174	. . . . . . 7 |- ZZ C_ RR
9		gcdsupex 11841	. . . . . . 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 3193	. . . . . . 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 6942	. . . . 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 1219	. . . 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 3968	. . . . . . . . 9 |- ( n = K -> ( n || M <-> K || M ) )
16		breq1 3968	. . . . . . . . 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 2869	. . . . . . 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 6943	. . . . . 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 1219	. . . . 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 7996	. . 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 11845	. . . 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 1219	. . 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 3992	. 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 963   = wceq 1335   e. wcel 2128   A.wral 2435   E.wrex 2436   {crab 2439   C_ wss 3102   class class class wbr 3965  (class class class)co 5824   supcsup 6926   RRcr 7731   0cc0 7732   < clt 7912   <_ cle 7913   ZZcz 9167   || cdvds 11683   gcd cgcd 11829

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-sup 6928  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-fz 9913  df-fzo 10042  df-fl 10169  df-mod 10222  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-dvds 11684  df-gcd 11830

This theorem is referenced by:  nndvdslegcd  11849  gcd0id  11863  gcdneg  11866  gcdaddm  11868  gcdzeq  11906  rpdvds  11976  coprm  12019  phimullem  12100