Theorem lcmcllem 12629

Description: Lemma for lcmn0cl 12630 and dvdslcm 12631. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)

Assertion

Ref	Expression
lcmcllem	|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. { n e. NN | ( M || n /\ N || n ) } )

Distinct variable groups:   n, M   n, N

Proof of Theorem lcmcllem

Step	Hyp	Ref	Expression
1		lcmn0val 12628	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) )
2		1zzd 9496	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> 1 e. ZZ )
3		nnuz 9782	. . . 4 |- NN = ( ZZ>= ` 1 )
4	3	rabeqi 2793	. . 3 |- { n e. NN | ( M || n /\ N || n ) } = { n e. ( ZZ>= ` 1 ) | ( M || n /\ N || n ) }
5		breq2 4090	. . . . 5 |- ( n = ( abs ` ( M x. N ) ) -> ( M || n <-> M || ( abs ` ( M x. N ) ) ) )
6		breq2 4090	. . . . 5 |- ( n = ( abs ` ( M x. N ) ) -> ( N || n <-> N || ( abs ` ( M x. N ) ) ) )
7	5, 6	anbi12d 473	. . . 4 |- ( n = ( abs ` ( M x. N ) ) -> ( ( M || n /\ N || n ) <-> ( M || ( abs ` ( M x. N ) ) /\ N || ( abs ` ( M x. N ) ) ) ) )
8		simpll 527	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M e. ZZ )
9		simplr 528	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. ZZ )
10	8, 9	zmulcld 9598	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M x. N ) e. ZZ )
11	8	zcnd 9593	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M e. CC )
12	9	zcnd 9593	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. CC )
13		ioran 757	. . . . . . . . . . . 12 |- ( -. ( M = 0 \/ N = 0 ) <-> ( -. M = 0 /\ -. N = 0 ) )
14	13	biimpi 120	. . . . . . . . . . 11 |- ( -. ( M = 0 \/ N = 0 ) -> ( -. M = 0 /\ -. N = 0 ) )
15	14	adantl 277	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( -. M = 0 /\ -. N = 0 ) )
16	15	simpld 112	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. M = 0 )
17	16	neneqad 2479	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M =/= 0 )
18		0zd 9481	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> 0 e. ZZ )
19		zapne 9544	. . . . . . . . 9 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
20	8, 18, 19	syl2anc 411	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M # 0 <-> M =/= 0 ) )
21	17, 20	mpbird 167	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M # 0 )
22	15	simprd 114	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. N = 0 )
23	22	neneqad 2479	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N =/= 0 )
24		zapne 9544	. . . . . . . . 9 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N # 0 <-> N =/= 0 ) )
25	9, 18, 24	syl2anc 411	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( N # 0 <-> N =/= 0 ) )
26	23, 25	mpbird 167	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N # 0 )
27	11, 12, 21, 26	mulap0d 8828	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M x. N ) # 0 )
28		zapne 9544	. . . . . . 7 |- ( ( ( M x. N ) e. ZZ /\ 0 e. ZZ ) -> ( ( M x. N ) # 0 <-> ( M x. N ) =/= 0 ) )
29	10, 18, 28	syl2anc 411	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M x. N ) # 0 <-> ( M x. N ) =/= 0 ) )
30	27, 29	mpbid 147	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M x. N ) =/= 0 )
31		nnabscl 11651	. . . . 5 |- ( ( ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) -> ( abs ` ( M x. N ) ) e. NN )
32	10, 30, 31	syl2anc 411	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) e. NN )
33		dvdsmul1 12364	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> M || ( M x. N ) )
34		zmulcl 9523	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
35		dvdsabsb 12361	. . . . . . . 8 |- ( ( M e. ZZ /\ ( M x. N ) e. ZZ ) -> ( M || ( M x. N ) <-> M || ( abs ` ( M x. N ) ) ) )
36	34, 35	syldan 282	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M x. N ) <-> M || ( abs ` ( M x. N ) ) ) )
37	33, 36	mpbid 147	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> M || ( abs ` ( M x. N ) ) )
38		dvdsmul2 12365	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> N || ( M x. N ) )
39		dvdsabsb 12361	. . . . . . . . 9 |- ( ( N e. ZZ /\ ( M x. N ) e. ZZ ) -> ( N || ( M x. N ) <-> N || ( abs ` ( M x. N ) ) ) )
40	34, 39	sylan2 286	. . . . . . . 8 |- ( ( N e. ZZ /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( N || ( M x. N ) <-> N || ( abs ` ( M x. N ) ) ) )
41	40	anabss7 583	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N || ( M x. N ) <-> N || ( abs ` ( M x. N ) ) ) )
42	38, 41	mpbid 147	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> N || ( abs ` ( M x. N ) ) )
43	37, 42	jca 306	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( abs ` ( M x. N ) ) /\ N || ( abs ` ( M x. N ) ) ) )
44	43	adantr 276	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M || ( abs ` ( M x. N ) ) /\ N || ( abs ` ( M x. N ) ) ) )
45	7, 32, 44	elrabd 2962	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) e. { n e. NN | ( M || n /\ N || n ) } )
46		simplll 533	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> M e. ZZ )
47		elfzelz 10250	. . . . . 6 |- ( n e. ( 1 ... ( abs ` ( M x. N ) ) ) -> n e. ZZ )
48	47	adantl 277	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> n e. ZZ )
49		zdvdsdc 12363	. . . . 5 |- ( ( M e. ZZ /\ n e. ZZ ) -> DECID M || n )
50	46, 48, 49	syl2anc 411	. . . 4 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID M || n )
51		simpllr 534	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> N e. ZZ )
52		zdvdsdc 12363	. . . . 5 |- ( ( N e. ZZ /\ n e. ZZ ) -> DECID N || n )
53	51, 48, 52	syl2anc 411	. . . 4 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID N || n )
54	50, 53	dcand 938	. . 3 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID ( M || n /\ N || n ) )
55	2, 4, 45, 54	infssuzcldc 10485	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) e. { n e. NN | ( M || n /\ N || n ) } )
56	1, 55	eqeltrd 2306	1 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. { n e. NN | ( M || n /\ N || n ) } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   {crab 2512   class class class wbr 4086   ` cfv 5324  (class class class)co 6013  infcinf 7173   RRcr 8021   0cc0 8022   1c1 8023   x. cmul 8027   < clt 8204   # cap 8751   NNcn 9133   ZZcz 9469   ZZ>=cuz 9745   ...cfz 10233   abscabs 11548   || cdvds 12338   lcm clcm 12622

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-dvds 12339  df-lcm 12623

This theorem is referenced by:  lcmn0cl  12630  dvdslcm  12631