Theorem lcmledvds 11751

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 11747	. . . . 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 1137	. . . 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 9081	. . . 4 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> 1 e. ZZ )
5		nnuz 9361	. . . . 5 |- NN = ( ZZ>= ` 1 )
6		rabeq 2678	. . . . 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 1020	. . . . 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 3933	. . . . . . 7 |- ( n = K -> ( M || n <-> M || K ) )
11		breq2 3933	. . . . . . 7 |- ( n = K -> ( N || n <-> N || K ) )
12	10, 11	anbi12d 464	. . . . . 6 |- ( n = K -> ( ( M || n /\ N || n ) <-> ( M || K /\ N || K ) ) )
13	12	elrab 2840	. . . . 5 |- ( K e. { n e. NN | ( M || n /\ N || n ) } <-> ( K e. NN /\ ( M || K /\ N || K ) ) )
14	8, 9, 13	sylanbrc 413	. . . 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 1021	. . . . . . 7 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> M e. ZZ )
16		elfzelz 9806	. . . . . . . 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 11514	. . . . . . 7 |- ( ( M e. ZZ /\ n e. ZZ ) -> DECID M || n )
19	15, 17, 18	syl2anc 408	. . . . . 6 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> DECID M || n )
20		simpll3 1022	. . . . . . 7 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> N e. ZZ )
21		zdvdsdc 11514	. . . . . . 7 |- ( ( N e. ZZ /\ n e. ZZ ) -> DECID N || n )
22	20, 17, 21	syl2anc 408	. . . . . 6 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> DECID N || n )
23		dcan 918	. . . . . 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 468	. . . 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 11643	. . 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 3950	. 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 697  DECID wdc 819   /\ w3a 962   = wceq 1331   e. wcel 1480   {crab 2420   class class class wbr 3929   ` cfv 5123  (class class class)co 5774  infcinf 6870   RRcr 7619   0cc0 7620   1c1 7621   < clt 7800   <_ cle 7801   NNcn 8720   ZZcz 9054   ZZ>=cuz 9326   ...cfz 9790   || cdvds 11493   lcm clcm 11741

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-inf 6872  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494  df-lcm 11742

This theorem is referenced by:  lcmneg  11755