Theorem lcmneg 12006

Description: Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Assertion

Ref	Expression
lcmneg	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) = ( M lcm N ) )

Proof of Theorem lcmneg

Step	Hyp	Ref	Expression
1		lcm0val 11997	. . . . . . . 8 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
2		znegcl 9222	. . . . . . . . 9 |- ( N e. ZZ -> -u N e. ZZ )
3		lcm0val 11997	. . . . . . . . 9 |- ( -u N e. ZZ -> ( -u N lcm 0 ) = 0 )
4	2, 3	syl 14	. . . . . . . 8 |- ( N e. ZZ -> ( -u N lcm 0 ) = 0 )
5	1, 4	eqtr4d 2201	. . . . . . 7 |- ( N e. ZZ -> ( N lcm 0 ) = ( -u N lcm 0 ) )
6	5	ad2antlr 481	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( N lcm 0 ) = ( -u N lcm 0 ) )
7		oveq2 5850	. . . . . . . 8 |- ( M = 0 -> ( N lcm M ) = ( N lcm 0 ) )
8		oveq2 5850	. . . . . . . 8 |- ( M = 0 -> ( -u N lcm M ) = ( -u N lcm 0 ) )
9	7, 8	eqeq12d 2180	. . . . . . 7 |- ( M = 0 -> ( ( N lcm M ) = ( -u N lcm M ) <-> ( N lcm 0 ) = ( -u N lcm 0 ) ) )
10	9	adantl 275	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( N lcm M ) = ( -u N lcm M ) <-> ( N lcm 0 ) = ( -u N lcm 0 ) ) )
11	6, 10	mpbird 166	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( N lcm M ) = ( -u N lcm M ) )
12		lcmcom 11996	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( N lcm M ) )
13		lcmcom 11996	. . . . . . . 8 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M lcm -u N ) = ( -u N lcm M ) )
14	2, 13	sylan2 284	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) = ( -u N lcm M ) )
15	12, 14	eqeq12d 2180	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = ( M lcm -u N ) <-> ( N lcm M ) = ( -u N lcm M ) ) )
16	15	adantr 274	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( M lcm N ) = ( M lcm -u N ) <-> ( N lcm M ) = ( -u N lcm M ) ) )
17	11, 16	mpbird 166	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M lcm N ) = ( M lcm -u N ) )
18		neg0 8144	. . . . . . . 8 |- -u 0 = 0
19	18	oveq2i 5853	. . . . . . 7 |- ( M lcm -u 0 ) = ( M lcm 0 )
20	19	eqcomi 2169	. . . . . 6 |- ( M lcm 0 ) = ( M lcm -u 0 )
21		oveq2 5850	. . . . . 6 |- ( N = 0 -> ( M lcm N ) = ( M lcm 0 ) )
22		negeq 8091	. . . . . . 7 |- ( N = 0 -> -u N = -u 0 )
23	22	oveq2d 5858	. . . . . 6 |- ( N = 0 -> ( M lcm -u N ) = ( M lcm -u 0 ) )
24	20, 21, 23	3eqtr4a 2225	. . . . 5 |- ( N = 0 -> ( M lcm N ) = ( M lcm -u N ) )
25	24	adantl 275	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M lcm N ) = ( M lcm -u N ) )
26	17, 25	jaodan 787	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = ( M lcm -u N ) )
27		dvdslcm 12001	. . . . . . . 8 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M || ( M lcm -u N ) /\ -u N || ( M lcm -u N ) ) )
28	2, 27	sylan2 284	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm -u N ) /\ -u N || ( M lcm -u N ) ) )
29		simpr 109	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
30		lcmcl 12004	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M lcm -u N ) e. NN0 )
31	2, 30	sylan2 284	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) e. NN0 )
32	31	nn0zd 9311	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) e. ZZ )
33		negdvdsb 11747	. . . . . . . . 9 |- ( ( N e. ZZ /\ ( M lcm -u N ) e. ZZ ) -> ( N || ( M lcm -u N ) <-> -u N || ( M lcm -u N ) ) )
34	29, 32, 33	syl2anc 409	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N || ( M lcm -u N ) <-> -u N || ( M lcm -u N ) ) )
35	34	anbi2d 460	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M || ( M lcm -u N ) /\ N || ( M lcm -u N ) ) <-> ( M || ( M lcm -u N ) /\ -u N || ( M lcm -u N ) ) ) )
36	28, 35	mpbird 166	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm -u N ) /\ N || ( M lcm -u N ) ) )
37	36	adantr 274	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M || ( M lcm -u N ) /\ N || ( M lcm -u N ) ) )
38		zcn 9196	. . . . . . . . . . . . 13 |- ( N e. ZZ -> N e. CC )
39	38	negeq0d 8201	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( N = 0 <-> -u N = 0 ) )
40	39	orbi2d 780	. . . . . . . . . . 11 |- ( N e. ZZ -> ( ( M = 0 \/ N = 0 ) <-> ( M = 0 \/ -u N = 0 ) ) )
41	40	notbid 657	. . . . . . . . . 10 |- ( N e. ZZ -> ( -. ( M = 0 \/ N = 0 ) <-> -. ( M = 0 \/ -u N = 0 ) ) )
42	41	biimpa 294	. . . . . . . . 9 |- ( ( N e. ZZ /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ -u N = 0 ) )
43	42	adantll 468	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ -u N = 0 ) )
44		lcmn0cl 12000	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ -u N e. ZZ ) /\ -. ( M = 0 \/ -u N = 0 ) ) -> ( M lcm -u N ) e. NN )
45	2, 44	sylanl2 401	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ -u N = 0 ) ) -> ( M lcm -u N ) e. NN )
46	43, 45	syldan 280	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm -u N ) e. NN )
47		simpl 108	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M e. ZZ /\ N e. ZZ ) )
48		3anass 972	. . . . . . 7 |- ( ( ( M lcm -u N ) e. NN /\ M e. ZZ /\ N e. ZZ ) <-> ( ( M lcm -u N ) e. NN /\ ( M e. ZZ /\ N e. ZZ ) ) )
49	46, 47, 48	sylanbrc 414	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M lcm -u N ) e. NN /\ M e. ZZ /\ N e. ZZ ) )
50		simpr 109	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ N = 0 ) )
51		lcmledvds 12002	. . . . . 6 |- ( ( ( ( M lcm -u N ) e. NN /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M || ( M lcm -u N ) /\ N || ( M lcm -u N ) ) -> ( M lcm N ) <_ ( M lcm -u N ) ) )
52	49, 50, 51	syl2anc 409	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M || ( M lcm -u N ) /\ N || ( M lcm -u N ) ) -> ( M lcm N ) <_ ( M lcm -u N ) ) )
53	37, 52	mpd 13	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) <_ ( M lcm -u N ) )
54		dvdslcm 12001	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
55	54	adantr 274	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
56		simplr 520	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. ZZ )
57		lcmn0cl 12000	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. NN )
58	57	nnzd 9312	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. ZZ )
59		negdvdsb 11747	. . . . . . . 8 |- ( ( N e. ZZ /\ ( M lcm N ) e. ZZ ) -> ( N || ( M lcm N ) <-> -u N || ( M lcm N ) ) )
60	56, 58, 59	syl2anc 409	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( N || ( M lcm N ) <-> -u N || ( M lcm N ) ) )
61	60	anbi2d 460	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M || ( M lcm N ) /\ N || ( M lcm N ) ) <-> ( M || ( M lcm N ) /\ -u N || ( M lcm N ) ) ) )
62		lcmledvds 12002	. . . . . . . . . 10 |- ( ( ( ( M lcm N ) e. NN /\ M e. ZZ /\ -u N e. ZZ ) /\ -. ( M = 0 \/ -u N = 0 ) ) -> ( ( M || ( M lcm N ) /\ -u N || ( M lcm N ) ) -> ( M lcm -u N ) <_ ( M lcm N ) ) )
63	62	ex 114	. . . . . . . . 9 |- ( ( ( M lcm N ) e. NN /\ M e. ZZ /\ -u N e. ZZ ) -> ( -. ( M = 0 \/ -u N = 0 ) -> ( ( M || ( M lcm N ) /\ -u N || ( M lcm N ) ) -> ( M lcm -u N ) <_ ( M lcm N ) ) ) )
64	2, 63	syl3an3 1263	. . . . . . . 8 |- ( ( ( M lcm N ) e. NN /\ M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 \/ -u N = 0 ) -> ( ( M || ( M lcm N ) /\ -u N || ( M lcm N ) ) -> ( M lcm -u N ) <_ ( M lcm N ) ) ) )
65	64	3expib 1196	. . . . . . 7 |- ( ( M lcm N ) e. NN -> ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 \/ -u N = 0 ) -> ( ( M || ( M lcm N ) /\ -u N || ( M lcm N ) ) -> ( M lcm -u N ) <_ ( M lcm N ) ) ) ) )
66	57, 47, 43, 65	syl3c 63	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M || ( M lcm N ) /\ -u N || ( M lcm N ) ) -> ( M lcm -u N ) <_ ( M lcm N ) ) )
67	61, 66	sylbid 149	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M || ( M lcm N ) /\ N || ( M lcm N ) ) -> ( M lcm -u N ) <_ ( M lcm N ) ) )
68	55, 67	mpd 13	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm -u N ) <_ ( M lcm N ) )
69		lcmcl 12004	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
70	69	nn0red 9168	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. RR )
71	30	nn0red 9168	. . . . . . 7 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M lcm -u N ) e. RR )
72	2, 71	sylan2 284	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) e. RR )
73	70, 72	letri3d 8014	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = ( M lcm -u N ) <-> ( ( M lcm N ) <_ ( M lcm -u N ) /\ ( M lcm -u N ) <_ ( M lcm N ) ) ) )
74	73	adantr 274	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M lcm N ) = ( M lcm -u N ) <-> ( ( M lcm N ) <_ ( M lcm -u N ) /\ ( M lcm -u N ) <_ ( M lcm N ) ) ) )
75	53, 68, 74	mpbir2and 934	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = ( M lcm -u N ) )
76		lcmmndc 11994	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
77		exmiddc 826	. . . 4 |- (DECID ( M = 0 \/ N = 0 ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
78	76, 77	syl 14	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
79	26, 75, 78	mpjaodan 788	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( M lcm -u N ) )
80	79	eqcomd 2171	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) = ( M lcm N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   <_ cle 7934   -ucneg 8070   NNcn 8857   NN0cn0 9114   ZZcz 9191   || 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:  neglcm  12007  lcmabs  12008