Theorem lcmneg 12057

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 12048	. . . . . . . 8 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
2		znegcl 9273	. . . . . . . . 9 |- ( N e. ZZ -> -u N e. ZZ )
3		lcm0val 12048	. . . . . . . . 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 2213	. . . . . . 7 |- ( N e. ZZ -> ( N lcm 0 ) = ( -u N lcm 0 ) )
6	5	ad2antlr 489	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( N lcm 0 ) = ( -u N lcm 0 ) )
7		oveq2 5877	. . . . . . . 8 |- ( M = 0 -> ( N lcm M ) = ( N lcm 0 ) )
8		oveq2 5877	. . . . . . . 8 |- ( M = 0 -> ( -u N lcm M ) = ( -u N lcm 0 ) )
9	7, 8	eqeq12d 2192	. . . . . . 7 |- ( M = 0 -> ( ( N lcm M ) = ( -u N lcm M ) <-> ( N lcm 0 ) = ( -u N lcm 0 ) ) )
10	9	adantl 277	. . . . . 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 167	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( N lcm M ) = ( -u N lcm M ) )
12		lcmcom 12047	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( N lcm M ) )
13		lcmcom 12047	. . . . . . . 8 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M lcm -u N ) = ( -u N lcm M ) )
14	2, 13	sylan2 286	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) = ( -u N lcm M ) )
15	12, 14	eqeq12d 2192	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = ( M lcm -u N ) <-> ( N lcm M ) = ( -u N lcm M ) ) )
16	15	adantr 276	. . . . 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 167	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M lcm N ) = ( M lcm -u N ) )
18		neg0 8193	. . . . . . . 8 |- -u 0 = 0
19	18	oveq2i 5880	. . . . . . 7 |- ( M lcm -u 0 ) = ( M lcm 0 )
20	19	eqcomi 2181	. . . . . 6 |- ( M lcm 0 ) = ( M lcm -u 0 )
21		oveq2 5877	. . . . . 6 |- ( N = 0 -> ( M lcm N ) = ( M lcm 0 ) )
22		negeq 8140	. . . . . . 7 |- ( N = 0 -> -u N = -u 0 )
23	22	oveq2d 5885	. . . . . 6 |- ( N = 0 -> ( M lcm -u N ) = ( M lcm -u 0 ) )
24	20, 21, 23	3eqtr4a 2236	. . . . 5 |- ( N = 0 -> ( M lcm N ) = ( M lcm -u N ) )
25	24	adantl 277	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M lcm N ) = ( M lcm -u N ) )
26	17, 25	jaodan 797	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = ( M lcm -u N ) )
27		dvdslcm 12052	. . . . . . . 8 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M || ( M lcm -u N ) /\ -u N || ( M lcm -u N ) ) )
28	2, 27	sylan2 286	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm -u N ) /\ -u N || ( M lcm -u N ) ) )
29		simpr 110	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
30		lcmcl 12055	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M lcm -u N ) e. NN0 )
31	2, 30	sylan2 286	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) e. NN0 )
32	31	nn0zd 9362	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) e. ZZ )
33		negdvdsb 11798	. . . . . . . . 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 411	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N || ( M lcm -u N ) <-> -u N || ( M lcm -u N ) ) )
35	34	anbi2d 464	. . . . . . 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 167	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm -u N ) /\ N || ( M lcm -u N ) ) )
37	36	adantr 276	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M || ( M lcm -u N ) /\ N || ( M lcm -u N ) ) )
38		zcn 9247	. . . . . . . . . . . . 13 |- ( N e. ZZ -> N e. CC )
39	38	negeq0d 8250	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( N = 0 <-> -u N = 0 ) )
40	39	orbi2d 790	. . . . . . . . . . 11 |- ( N e. ZZ -> ( ( M = 0 \/ N = 0 ) <-> ( M = 0 \/ -u N = 0 ) ) )
41	40	notbid 667	. . . . . . . . . 10 |- ( N e. ZZ -> ( -. ( M = 0 \/ N = 0 ) <-> -. ( M = 0 \/ -u N = 0 ) ) )
42	41	biimpa 296	. . . . . . . . 9 |- ( ( N e. ZZ /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ -u N = 0 ) )
43	42	adantll 476	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ -u N = 0 ) )
44		lcmn0cl 12051	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ -u N e. ZZ ) /\ -. ( M = 0 \/ -u N = 0 ) ) -> ( M lcm -u N ) e. NN )
45	2, 44	sylanl2 403	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ -u N = 0 ) ) -> ( M lcm -u N ) e. NN )
46	43, 45	syldan 282	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm -u N ) e. NN )
47		simpl 109	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M e. ZZ /\ N e. ZZ ) )
48		3anass 982	. . . . . . 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 417	. . . . . 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 110	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ N = 0 ) )
51		lcmledvds 12053	. . . . . 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 411	. . . . 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 12052	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
55	54	adantr 276	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
56		simplr 528	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. ZZ )
57		lcmn0cl 12051	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. NN )
58	57	nnzd 9363	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. ZZ )
59		negdvdsb 11798	. . . . . . . 8 |- ( ( N e. ZZ /\ ( M lcm N ) e. ZZ ) -> ( N || ( M lcm N ) <-> -u N || ( M lcm N ) ) )
60	56, 58, 59	syl2anc 411	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( N || ( M lcm N ) <-> -u N || ( M lcm N ) ) )
61	60	anbi2d 464	. . . . . 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 12053	. . . . . . . . . 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 115	. . . . . . . . 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 1273	. . . . . . . 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 1206	. . . . . . 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 150	. . . . 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 12055	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
70	69	nn0red 9219	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. RR )
71	30	nn0red 9219	. . . . . . 7 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M lcm -u N ) e. RR )
72	2, 71	sylan2 286	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) e. RR )
73	70, 72	letri3d 8063	. . . . 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 276	. . . 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 944	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = ( M lcm -u N ) )
76		lcmmndc 12045	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
77		exmiddc 836	. . . 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 798	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( M lcm -u N ) )
80	79	eqcomd 2183	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) = ( M lcm N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4000  (class class class)co 5869   RRcr 7801   0cc0 7802   <_ cle 7983   -ucneg 8119   NNcn 8908   NN0cn0 9165   ZZcz 9242   || cdvds 11778   lcm clcm 12043

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-lcm 12044

This theorem is referenced by:  neglcm  12058  lcmabs  12059