Theorem lcmneg 10456

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 10447	. . . . . . . 8 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
2		znegcl 8382	. . . . . . . . 9 |- ( N e. ZZ -> -u N e. ZZ )
3		lcm0val 10447	. . . . . . . . 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 2116	. . . . . . 7 |- ( N e. ZZ -> ( N lcm 0 ) = ( -u N lcm 0 ) )
6	5	ad2antlr 472	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( N lcm 0 ) = ( -u N lcm 0 ) )
7		oveq2 5540	. . . . . . . 8 |- ( M = 0 -> ( N lcm M ) = ( N lcm 0 ) )
8		oveq2 5540	. . . . . . . 8 |- ( M = 0 -> ( -u N lcm M ) = ( -u N lcm 0 ) )
9	7, 8	eqeq12d 2095	. . . . . . 7 |- ( M = 0 -> ( ( N lcm M ) = ( -u N lcm M ) <-> ( N lcm 0 ) = ( -u N lcm 0 ) ) )
10	9	adantl 271	. . . . . 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 165	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( N lcm M ) = ( -u N lcm M ) )
12		lcmcom 10446	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( N lcm M ) )
13		lcmcom 10446	. . . . . . . 8 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M lcm -u N ) = ( -u N lcm M ) )
14	2, 13	sylan2 280	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) = ( -u N lcm M ) )
15	12, 14	eqeq12d 2095	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = ( M lcm -u N ) <-> ( N lcm M ) = ( -u N lcm M ) ) )
16	15	adantr 270	. . . . 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 165	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M lcm N ) = ( M lcm -u N ) )
18		neg0 7354	. . . . . . . 8 |- -u 0 = 0
19	18	oveq2i 5543	. . . . . . 7 |- ( M lcm -u 0 ) = ( M lcm 0 )
20	19	eqcomi 2085	. . . . . 6 |- ( M lcm 0 ) = ( M lcm -u 0 )
21		oveq2 5540	. . . . . 6 |- ( N = 0 -> ( M lcm N ) = ( M lcm 0 ) )
22		negeq 7301	. . . . . . 7 |- ( N = 0 -> -u N = -u 0 )
23	22	oveq2d 5548	. . . . . 6 |- ( N = 0 -> ( M lcm -u N ) = ( M lcm -u 0 ) )
24	20, 21, 23	3eqtr4a 2139	. . . . 5 |- ( N = 0 -> ( M lcm N ) = ( M lcm -u N ) )
25	24	adantl 271	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M lcm N ) = ( M lcm -u N ) )
26	17, 25	jaodan 743	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = ( M lcm -u N ) )
27		dvdslcm 10451	. . . . . . . 8 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M || ( M lcm -u N ) /\ -u N || ( M lcm -u N ) ) )
28	2, 27	sylan2 280	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm -u N ) /\ -u N || ( M lcm -u N ) ) )
29		simpr 108	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
30		lcmcl 10454	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M lcm -u N ) e. NN0 )
31	2, 30	sylan2 280	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) e. NN0 )
32	31	nn0zd 8467	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) e. ZZ )
33		negdvdsb 10211	. . . . . . . . 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 403	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N || ( M lcm -u N ) <-> -u N || ( M lcm -u N ) ) )
35	34	anbi2d 451	. . . . . . 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 165	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm -u N ) /\ N || ( M lcm -u N ) ) )
37	36	adantr 270	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M || ( M lcm -u N ) /\ N || ( M lcm -u N ) ) )
38		zcn 8356	. . . . . . . . . . . . 13 |- ( N e. ZZ -> N e. CC )
39	38	negeq0d 7411	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( N = 0 <-> -u N = 0 ) )
40	39	orbi2d 736	. . . . . . . . . . 11 |- ( N e. ZZ -> ( ( M = 0 \/ N = 0 ) <-> ( M = 0 \/ -u N = 0 ) ) )
41	40	notbid 624	. . . . . . . . . 10 |- ( N e. ZZ -> ( -. ( M = 0 \/ N = 0 ) <-> -. ( M = 0 \/ -u N = 0 ) ) )
42	41	biimpa 290	. . . . . . . . 9 |- ( ( N e. ZZ /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ -u N = 0 ) )
43	42	adantll 459	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ -u N = 0 ) )
44		lcmn0cl 10450	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ -u N e. ZZ ) /\ -. ( M = 0 \/ -u N = 0 ) ) -> ( M lcm -u N ) e. NN )
45	2, 44	sylanl2 395	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ -u N = 0 ) ) -> ( M lcm -u N ) e. NN )
46	43, 45	syldan 276	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm -u N ) e. NN )
47		simpl 107	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M e. ZZ /\ N e. ZZ ) )
48		3anass 923	. . . . . . 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 408	. . . . . 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 108	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ N = 0 ) )
51		lcmledvds 10452	. . . . . 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 403	. . . . 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 10451	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
55	54	adantr 270	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
56		simplr 496	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. ZZ )
57		lcmn0cl 10450	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. NN )
58	57	nnzd 8468	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. ZZ )
59		negdvdsb 10211	. . . . . . . 8 |- ( ( N e. ZZ /\ ( M lcm N ) e. ZZ ) -> ( N || ( M lcm N ) <-> -u N || ( M lcm N ) ) )
60	56, 58, 59	syl2anc 403	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( N || ( M lcm N ) <-> -u N || ( M lcm N ) ) )
61	60	anbi2d 451	. . . . . 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 10452	. . . . . . . . . 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 113	. . . . . . . . 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 1204	. . . . . . . 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 1141	. . . . . . 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 62	. . . . . 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 148	. . . . 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 10454	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
70	69	nn0red 8342	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. RR )
71	30	nn0red 8342	. . . . . . 7 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M lcm -u N ) e. RR )
72	2, 71	sylan2 280	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) e. RR )
73	70, 72	letri3d 7226	. . . . 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 270	. . . 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 885	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = ( M lcm -u N ) )
76		lcmmndc 10444	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
77		exmiddc 777	. . . 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 744	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( M lcm -u N ) )
80	79	eqcomd 2086	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) = ( M lcm N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661  DECID wdc 775   /\ w3a 919   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   RRcr 6980   0cc0 6981   <_ cle 7154   -ucneg 7280   NNcn 8039   NN0cn0 8288   ZZcz 8351   || cdvds 10195   lcm clcm 10442

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-isom 4931  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-sup 6397  df-inf 6398  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-q 8705  df-rp 8735  df-fz 9030  df-fzo 9153  df-fl 9274  df-mod 9325  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885  df-dvds 10196  df-lcm 10443

This theorem is referenced by:  neglcm  10457  lcmabs  10458