Theorem lcmneg 12830

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 12821	. . . . . . . 8 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
2		znegcl 9654	. . . . . . . . 9 |- ( N e. ZZ -> -u N e. ZZ )
3		lcm0val 12821	. . . . . . . . 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 2274	. . . . . . 7 |- ( N e. ZZ -> ( N lcm 0 ) = ( -u N lcm 0 ) )
6	5	ad2antlr 493	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( N lcm 0 ) = ( -u N lcm 0 ) )
7		oveq2 6083	. . . . . . . 8 |- ( M = 0 -> ( N lcm M ) = ( N lcm 0 ) )
8		oveq2 6083	. . . . . . . 8 |- ( M = 0 -> ( -u N lcm M ) = ( -u N lcm 0 ) )
9	7, 8	eqeq12d 2253	. . . . . . 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 12820	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( N lcm M ) )
13		lcmcom 12820	. . . . . . . 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 2253	. . . . . 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 8562	. . . . . . . 8 |- -u 0 = 0
19	18	oveq2i 6086	. . . . . . 7 |- ( M lcm -u 0 ) = ( M lcm 0 )
20	19	eqcomi 2242	. . . . . 6 |- ( M lcm 0 ) = ( M lcm -u 0 )
21		oveq2 6083	. . . . . 6 |- ( N = 0 -> ( M lcm N ) = ( M lcm 0 ) )
22		negeq 8509	. . . . . . 7 |- ( N = 0 -> -u N = -u 0 )
23	22	oveq2d 6091	. . . . . 6 |- ( N = 0 -> ( M lcm -u N ) = ( M lcm -u 0 ) )
24	20, 21, 23	3eqtr4a 2297	. . . . 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 809	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = ( M lcm -u N ) )
27		dvdslcm 12825	. . . . . . . 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 12828	. . . . . . . . . . 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 9745	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) e. ZZ )
33		negdvdsb 12552	. . . . . . . . 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 415	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N || ( M lcm -u N ) <-> -u N || ( M lcm -u N ) ) )
35	34	anbi2d 468	. . . . . . 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 9628	. . . . . . . . . . . . 13 |- ( N e. ZZ -> N e. CC )
39	38	negeq0d 8619	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( N = 0 <-> -u N = 0 ) )
40	39	orbi2d 802	. . . . . . . . . . 11 |- ( N e. ZZ -> ( ( M = 0 \/ N = 0 ) <-> ( M = 0 \/ -u N = 0 ) ) )
41	40	notbid 677	. . . . . . . . . 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 480	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ -u N = 0 ) )
44		lcmn0cl 12824	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ -u N e. ZZ ) /\ -. ( M = 0 \/ -u N = 0 ) ) -> ( M lcm -u N ) e. NN )
45	2, 44	sylanl2 407	. . . . . . . 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 1013	. . . . . . 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 421	. . . . . 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 12826	. . . . . 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 415	. . . . 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 12825	. . . . . 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 533	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. ZZ )
57		lcmn0cl 12824	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. NN )
58	57	nnzd 9746	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. ZZ )
59		negdvdsb 12552	. . . . . . . 8 |- ( ( N e. ZZ /\ ( M lcm N ) e. ZZ ) -> ( N || ( M lcm N ) <-> -u N || ( M lcm N ) ) )
60	56, 58, 59	syl2anc 415	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( N || ( M lcm N ) <-> -u N || ( M lcm N ) ) )
61	60	anbi2d 468	. . . . . 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 12826	. . . . . . . . . 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 1313	. . . . . . . 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 1237	. . . . . . 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 12828	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
70	69	nn0red 9600	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. RR )
71	30	nn0red 9600	. . . . . . 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 8431	. . . . 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 957	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = ( M lcm -u N ) )
76		lcmmndc 12818	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
77		exmiddc 848	. . . 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 810	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( M lcm -u N ) )
80	79	eqcomd 2244	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 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   RRcr 8168   0cc0 8169   <_ cle 8351   -ucneg 8488   NNcn 9283   NN0cn0 9542   ZZcz 9623   || 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:  neglcm  12831  lcmabs  12832