Theorem lcmledvds 11927

Description: A positive integer which both operands of the lcm operator divide bounds it. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)

Assertion

Ref	Expression
lcmledvds	|- ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) <_ K ) )

Proof of Theorem lcmledvds

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcmn0val 11923	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) )
2	1	3adantl1 1138	. . . 4 |- ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) )
3	2	adantr 274	. . 3 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> ( M lcm N ) = inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) )
4		1zzd 9177	. . . 4 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> 1 e. ZZ )
5		nnuz 9457	. . . . 5 |- NN = ( ZZ>= ` 1 )
6		rabeq 2704	. . . . 5 |- ( NN = ( ZZ>= ` 1 ) -> { n e. NN | ( M || n /\ N || n ) } = { n e. ( ZZ>= ` 1 ) | ( M || n /\ N || n ) } )
7	5, 6	ax-mp 5	. . . 4 |- { n e. NN | ( M || n /\ N || n ) } = { n e. ( ZZ>= ` 1 ) | ( M || n /\ N || n ) }
8		simpll1 1021	. . . . 5 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> K e. NN )
9		simpr 109	. . . . 5 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> ( M || K /\ N || K ) )
10		breq2 3969	. . . . . . 7 |- ( n = K -> ( M || n <-> M || K ) )
11		breq2 3969	. . . . . . 7 |- ( n = K -> ( N || n <-> N || K ) )
12	10, 11	anbi12d 465	. . . . . 6 |- ( n = K -> ( ( M || n /\ N || n ) <-> ( M || K /\ N || K ) ) )
13	12	elrab 2868	. . . . 5 |- ( K e. { n e. NN | ( M || n /\ N || n ) } <-> ( K e. NN /\ ( M || K /\ N || K ) ) )
14	8, 9, 13	sylanbrc 414	. . . 4 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> K e. { n e. NN | ( M || n /\ N || n ) } )
15		simpll2 1022	. . . . . . 7 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> M e. ZZ )
16		elfzelz 9910	. . . . . . . 8 |- ( n e. ( 1 ... K ) -> n e. ZZ )
17	16	adantl 275	. . . . . . 7 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> n e. ZZ )
18		zdvdsdc 11689	. . . . . . 7 |- ( ( M e. ZZ /\ n e. ZZ ) -> DECID M || n )
19	15, 17, 18	syl2anc 409	. . . . . 6 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> DECID M || n )
20		simpll3 1023	. . . . . . 7 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> N e. ZZ )
21		zdvdsdc 11689	. . . . . . 7 |- ( ( N e. ZZ /\ n e. ZZ ) -> DECID N || n )
22	20, 17, 21	syl2anc 409	. . . . . 6 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> DECID N || n )
23		dcan 919	. . . . . 6 |- (DECID M || n -> (DECID N || n -> DECID ( M || n /\ N || n ) ) )
24	19, 22, 23	sylc 62	. . . . 5 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> DECID ( M || n /\ N || n ) )
25	24	adantlr 469	. . . 4 |- ( ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) /\ n e. ( 1 ... K ) ) -> DECID ( M || n /\ N || n ) )
26	4, 7, 14, 25	infssuzledc 11818	. . 3 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) <_ K )
27	3, 26	eqbrtrd 3986	. 2 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> ( M lcm N ) <_ K )
28	27	ex 114	1 |- ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) <_ K ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698  DECID wdc 820   /\ w3a 963   = wceq 1335   e. wcel 2128   {crab 2439   class class class wbr 3965   ` cfv 5167  (class class class)co 5818  infcinf 6919   RRcr 7714   0cc0 7715   1c1 7716   < clt 7895   <_ cle 7896   NNcn 8816   ZZcz 9150   ZZ>=cuz 9422   ...cfz 9894   || cdvds 11665   lcm clcm 11917

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 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834  ax-caucvg 7835

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 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-isom 5176  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-frec 6332  df-sup 6920  df-inf 6921  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-2 8875  df-3 8876  df-4 8877  df-n0 9074  df-z 9151  df-uz 9423  df-q 9511  df-rp 9543  df-fz 9895  df-fzo 10024  df-fl 10151  df-mod 10204  df-seqfrec 10327  df-exp 10401  df-cj 10724  df-re 10725  df-im 10726  df-rsqrt 10880  df-abs 10881  df-dvds 11666  df-lcm 11918

This theorem is referenced by:  lcmneg  11931