Theorem lcmledvds 12797

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 12793	. . . . 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 1180	. . . 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 276	. . 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 9625	. . . 4 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> 1 e. ZZ )
5		nnuz 9912	. . . . 5 |- NN = ( ZZ>= ` 1 )
6	5	rabeqi 2808	. . . 4 |- { n e. NN | ( M || n /\ N || n ) } = { n e. ( ZZ>= ` 1 ) | ( M || n /\ N || n ) }
7		breq2 4119	. . . . . 6 |- ( n = K -> ( M || n <-> M || K ) )
8		breq2 4119	. . . . . 6 |- ( n = K -> ( N || n <-> N || K ) )
9	7, 8	anbi12d 473	. . . . 5 |- ( n = K -> ( ( M || n /\ N || n ) <-> ( M || K /\ N || K ) ) )
10		simpll1 1063	. . . . 5 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> K e. NN )
11		simpr 110	. . . . 5 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> ( M || K /\ N || K ) )
12	9, 10, 11	elrabd 2978	. . . 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 ) } )
13		simpll2 1064	. . . . . . 7 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> M e. ZZ )
14		elfzelz 10382	. . . . . . . 8 |- ( n e. ( 1 ... K ) -> n e. ZZ )
15	14	adantl 277	. . . . . . 7 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> n e. ZZ )
16		zdvdsdc 12528	. . . . . . 7 |- ( ( M e. ZZ /\ n e. ZZ ) -> DECID M || n )
17	13, 15, 16	syl2anc 411	. . . . . 6 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> DECID M || n )
18		simpll3 1065	. . . . . . 7 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> N e. ZZ )
19		zdvdsdc 12528	. . . . . . 7 |- ( ( N e. ZZ /\ n e. ZZ ) -> DECID N || n )
20	18, 15, 19	syl2anc 411	. . . . . 6 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> DECID N || n )
21	17, 20	dcand 941	. . . . 5 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> DECID ( M || n /\ N || n ) )
22	21	adantlr 477	. . . 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 ) )
23	4, 6, 12, 22	infssuzledc 10620	. . 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 )
24	3, 23	eqbrtrd 4137	. 2 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> ( M lcm N ) <_ K )
25	24	ex 115	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 104   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   {crab 2526   class class class wbr 4115   ` cfv 5358  (class class class)co 6059  infcinf 7288   RRcr 8143   0cc0 8144   1c1 8145   < clt 8325   <_ cle 8326   NNcn 9258   ZZcz 9598   ZZ>=cuz 9875   ...cfz 10365   || cdvds 12503   lcm clcm 12787

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-iinf 4716  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-mulrcl 8243  ax-addcom 8244  ax-mulcom 8245  ax-addass 8246  ax-mulass 8247  ax-distr 8248  ax-i2m1 8249  ax-0lt1 8250  ax-1rid 8251  ax-0id 8252  ax-rnegex 8253  ax-precex 8254  ax-cnre 8255  ax-pre-ltirr 8256  ax-pre-ltwlin 8257  ax-pre-lttrn 8258  ax-pre-apti 8259  ax-pre-ltadd 8260  ax-pre-mulgt0 8261  ax-pre-mulext 8262  ax-arch 8263  ax-caucvg 8264

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4719  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-recs 6550  df-frec 6636  df-sup 7289  df-inf 7290  df-pnf 8327  df-mnf 8328  df-xr 8329  df-ltxr 8330  df-le 8331  df-sub 8464  df-neg 8465  df-reap 8868  df-ap 8875  df-div 8968  df-inn 9259  df-2 9317  df-3 9318  df-4 9319  df-n0 9518  df-z 9599  df-uz 9876  df-q 9974  df-rp 10009  df-fz 10366  df-fzo 10503  df-fl 10658  df-mod 10713  df-seqfrec 10838  df-exp 10929  df-cj 11556  df-re 11557  df-im 11558  df-rsqrt 11713  df-abs 11714  df-dvds 12504  df-lcm 12788

This theorem is referenced by:  lcmneg  12801