Theorem lcmledvds 12248

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 12244	. . . . 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 1155	. . . 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 9355	. . . 4 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ ( M || K /\ N || K ) ) -> 1 e. ZZ )
5		nnuz 9639	. . . . 5 |- NN = ( ZZ>= ` 1 )
6	5	rabeqi 2756	. . . 4 |- { n e. NN | ( M || n /\ N || n ) } = { n e. ( ZZ>= ` 1 ) | ( M || n /\ N || n ) }
7		breq2 4038	. . . . . 6 |- ( n = K -> ( M || n <-> M || K ) )
8		breq2 4038	. . . . . 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 1038	. . . . 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 2922	. . . 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 1039	. . . . . . 7 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> M e. ZZ )
14		elfzelz 10102	. . . . . . . 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 11979	. . . . . . 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 1040	. . . . . . 7 |- ( ( ( ( K e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... K ) ) -> N e. ZZ )
19		zdvdsdc 11979	. . . . . . 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 934	. . . . 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 10326	. . 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 4056	. 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 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2167   {crab 2479   class class class wbr 4034   ` cfv 5259  (class class class)co 5923  infcinf 7050   RRcr 7880   0cc0 7881   1c1 7882   < clt 8063   <_ cle 8064   NNcn 8992   ZZcz 9328   ZZ>=cuz 9603   ...cfz 10085   || cdvds 11954   lcm clcm 12238

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-mulrcl 7980  ax-addcom 7981  ax-mulcom 7982  ax-addass 7983  ax-mulass 7984  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-1rid 7988  ax-0id 7989  ax-rnegex 7990  ax-precex 7991  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-apti 7996  ax-pre-ltadd 7997  ax-pre-mulgt0 7998  ax-pre-mulext 7999  ax-arch 8000  ax-caucvg 8001

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6199  df-2nd 6200  df-recs 6364  df-frec 6450  df-sup 7051  df-inf 7052  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-reap 8604  df-ap 8611  df-div 8702  df-inn 8993  df-2 9051  df-3 9052  df-4 9053  df-n0 9252  df-z 9329  df-uz 9604  df-q 9696  df-rp 9731  df-fz 10086  df-fzo 10220  df-fl 10362  df-mod 10417  df-seqfrec 10542  df-exp 10633  df-cj 11009  df-re 11010  df-im 11011  df-rsqrt 11165  df-abs 11166  df-dvds 11955  df-lcm 12239

This theorem is referenced by:  lcmneg  12252