Theorem lcmcllem 12823

Description: Lemma for lcmn0cl 12824 and dvdslcm 12825. (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 12822	. 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 9650	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> 1 e. ZZ )
3		nnuz 9937	. . . 4 |- NN = ( ZZ>= ` 1 )
4	3	rabeqi 2814	. . 3 |- { n e. NN | ( M || n /\ N || n ) } = { n e. ( ZZ>= ` 1 ) | ( M || n /\ N || n ) }
5		breq2 4129	. . . . 5 |- ( n = ( abs ` ( M x. N ) ) -> ( M || n <-> M || ( abs ` ( M x. N ) ) ) )
6		breq2 4129	. . . . 5 |- ( n = ( abs ` ( M x. N ) ) -> ( N || n <-> N || ( abs ` ( M x. N ) ) ) )
7	5, 6	anbi12d 477	. . . 4 |- ( n = ( abs ` ( M x. N ) ) -> ( ( M || n /\ N || n ) <-> ( M || ( abs ` ( M x. N ) ) /\ N || ( abs ` ( M x. N ) ) ) ) )
8		simpll 531	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M e. ZZ )
9		simplr 533	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. ZZ )
10	8, 9	zmulcld 9753	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M x. N ) e. ZZ )
11	8	zcnd 9748	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M e. CC )
12	9	zcnd 9748	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. CC )
13		ioran 764	. . . . . . . . . . . 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 2499	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M =/= 0 )
18		0zd 9635	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> 0 e. ZZ )
19		zapne 9698	. . . . . . . . 9 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
20	8, 18, 19	syl2anc 415	. . . . . . . 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 2499	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N =/= 0 )
24		zapne 9698	. . . . . . . . 9 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N # 0 <-> N =/= 0 ) )
25	9, 18, 24	syl2anc 415	. . . . . . . 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 8976	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M x. N ) # 0 )
28		zapne 9698	. . . . . . 7 |- ( ( ( M x. N ) e. ZZ /\ 0 e. ZZ ) -> ( ( M x. N ) # 0 <-> ( M x. N ) =/= 0 ) )
29	10, 18, 28	syl2anc 415	. . . . . 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 11844	. . . . 5 |- ( ( ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) -> ( abs ` ( M x. N ) ) e. NN )
32	10, 30, 31	syl2anc 415	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) e. NN )
33		dvdsmul1 12558	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> M || ( M x. N ) )
34		zmulcl 9677	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
35		dvdsabsb 12555	. . . . . . . 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 12559	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> N || ( M x. N ) )
39		dvdsabsb 12555	. . . . . . . . 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 589	. . . . . . 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 2984	. . 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 539	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> M e. ZZ )
47		elfzelz 10407	. . . . . 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 12557	. . . . 5 |- ( ( M e. ZZ /\ n e. ZZ ) -> DECID M || n )
50	46, 48, 49	syl2anc 415	. . . 4 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID M || n )
51		simpllr 540	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> N e. ZZ )
52		zdvdsdc 12557	. . . . 5 |- ( ( N e. ZZ /\ n e. ZZ ) -> DECID N || n )
53	51, 48, 52	syl2anc 415	. . . 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 945	. . 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 10646	. 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 2315	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 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   class class class wbr 4125   ` cfv 5372  (class class class)co 6075  infcinf 7313   RRcr 8168   0cc0 8169   1c1 8170   x. cmul 8174   < clt 8350   # cap 8899   NNcn 9283   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   abscabs 11741   || cdvds 12532   lcm clcm 12816

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-dvds 12533  df-lcm 12817

This theorem is referenced by:  lcmn0cl  12824  dvdslcm  12825