Theorem lcmneg 12212

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 12203	. . . . . . . 8 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
2		znegcl 9348	. . . . . . . . 9 |- ( N e. ZZ -> -u N e. ZZ )
3		lcm0val 12203	. . . . . . . . 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 2229	. . . . . . 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 5926	. . . . . . . 8 |- ( M = 0 -> ( N lcm M ) = ( N lcm 0 ) )
8		oveq2 5926	. . . . . . . 8 |- ( M = 0 -> ( -u N lcm M ) = ( -u N lcm 0 ) )
9	7, 8	eqeq12d 2208	. . . . . . 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 12202	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( N lcm M ) )
13		lcmcom 12202	. . . . . . . 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 2208	. . . . . 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 8265	. . . . . . . 8 |- -u 0 = 0
19	18	oveq2i 5929	. . . . . . 7 |- ( M lcm -u 0 ) = ( M lcm 0 )
20	19	eqcomi 2197	. . . . . 6 |- ( M lcm 0 ) = ( M lcm -u 0 )
21		oveq2 5926	. . . . . 6 |- ( N = 0 -> ( M lcm N ) = ( M lcm 0 ) )
22		negeq 8212	. . . . . . 7 |- ( N = 0 -> -u N = -u 0 )
23	22	oveq2d 5934	. . . . . 6 |- ( N = 0 -> ( M lcm -u N ) = ( M lcm -u 0 ) )
24	20, 21, 23	3eqtr4a 2252	. . . . 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 798	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = ( M lcm -u N ) )
27		dvdslcm 12207	. . . . . . . 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 12210	. . . . . . . . . . 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 9437	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) e. ZZ )
33		negdvdsb 11950	. . . . . . . . 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 9322	. . . . . . . . . . . . 13 |- ( N e. ZZ -> N e. CC )
39	38	negeq0d 8322	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( N = 0 <-> -u N = 0 ) )
40	39	orbi2d 791	. . . . . . . . . . 11 |- ( N e. ZZ -> ( ( M = 0 \/ N = 0 ) <-> ( M = 0 \/ -u N = 0 ) ) )
41	40	notbid 668	. . . . . . . . . 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 12206	. . . . . . . . 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 984	. . . . . . 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 12208	. . . . . 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 12207	. . . . . 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 12206	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. NN )
58	57	nnzd 9438	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. ZZ )
59		negdvdsb 11950	. . . . . . . 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 12208	. . . . . . . . . 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 1284	. . . . . . . 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 1208	. . . . . . 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 12210	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
70	69	nn0red 9294	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. RR )
71	30	nn0red 9294	. . . . . . 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 8135	. . . . 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 946	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = ( M lcm -u N ) )
76		lcmmndc 12200	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
77		exmiddc 837	. . . 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 799	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( M lcm -u N ) )
80	79	eqcomd 2199	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 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   RRcr 7871   0cc0 7872   <_ cle 8055   -ucneg 8191   NNcn 8982   NN0cn0 9240   ZZcz 9317   || cdvds 11930   lcm clcm 12198

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-lcm 12199

This theorem is referenced by:  neglcm  12213  lcmabs  12214