Theorem lcmledvds 12002

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 11998	. . . . 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 1143	. . . 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 9218	. . . 4 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> 1 e. ZZ )
5		nnuz 9501	. . . . 5 |- NN = ( ZZ>= ` 1 )
6		rabeq 2718	. . . . 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 1026	. . . . 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 3986	. . . . . . 7 |- ( n = K -> ( M || n <-> M || K ) )
11		breq2 3986	. . . . . . 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 2882	. . . . 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 1027	. . . . . . 7 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> M e. ZZ )
16		elfzelz 9960	. . . . . . . 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 11752	. . . . . . 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 1028	. . . . . . 7 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> N e. ZZ )
21		zdvdsdc 11752	. . . . . . 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		dcan2 924	. . . . . 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 11883	. . 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 4004	. 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 824   /\ w3a 968   = wceq 1343   e. wcel 2136   {crab 2448   class class class wbr 3982   ` cfv 5188  (class class class)co 5842  infcinf 6948   RRcr 7752   0cc0 7753   1c1 7754   < clt 7933   <_ cle 7934   NNcn 8857   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944   || cdvds 11727   lcm clcm 11992

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-lcm 11993

This theorem is referenced by:  lcmneg  12006