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Theorem lcmneg 11138
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
StepHypRef Expression
1 lcm0val 11129 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N lcm  0 )  =  0 )
2 znegcl 8751 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
3 lcm0val 11129 . . . . . . . . 9  |-  ( -u N  e.  ZZ  ->  (
-u N lcm  0 )  =  0 )
42, 3syl 14 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( -u N lcm  0 )  =  0 )
51, 4eqtr4d 2123 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( N lcm  0 )  =  (
-u N lcm  0 ) )
65ad2antlr 473 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  ( N lcm  0 )  =  (
-u N lcm  0 ) )
7 oveq2 5642 . . . . . . . 8  |-  ( M  =  0  ->  ( N lcm  M )  =  ( N lcm  0 ) )
8 oveq2 5642 . . . . . . . 8  |-  ( M  =  0  ->  ( -u N lcm  M )  =  ( -u N lcm  0
) )
97, 8eqeq12d 2102 . . . . . . 7  |-  ( M  =  0  ->  (
( N lcm  M )  =  ( -u N lcm  M )  <->  ( N lcm  0
)  =  ( -u N lcm  0 ) ) )
109adantl 271 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  ( ( N lcm  M )  =  (
-u N lcm  M )  <->  ( N lcm  0 )  =  ( -u N lcm  0
) ) )
116, 10mpbird 165 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  ( N lcm  M )  =  ( -u N lcm  M ) )
12 lcmcom 11128 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  =  ( N lcm  M
) )
13 lcmcom 11128 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  -u N  e.  ZZ )  ->  ( M lcm  -u N
)  =  ( -u N lcm  M ) )
142, 13sylan2 280 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  -u N
)  =  ( -u N lcm  M ) )
1512, 14eqeq12d 2102 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm  N
)  =  ( M lcm  -u N )  <->  ( N lcm  M )  =  ( -u N lcm  M ) ) )
1615adantr 270 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  ( ( M lcm  N )  =  ( M lcm  -u N )  <->  ( N lcm  M )  =  ( -u N lcm  M ) ) )
1711, 16mpbird 165 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  ( M lcm  N )  =  ( M lcm  -u N ) )
18 neg0 7707 . . . . . . . 8  |-  -u 0  =  0
1918oveq2i 5645 . . . . . . 7  |-  ( M lcm  -u 0 )  =  ( M lcm  0 )
2019eqcomi 2092 . . . . . 6  |-  ( M lcm  0 )  =  ( M lcm  -u 0 )
21 oveq2 5642 . . . . . 6  |-  ( N  =  0  ->  ( M lcm  N )  =  ( M lcm  0 ) )
22 negeq 7654 . . . . . . 7  |-  ( N  =  0  ->  -u N  =  -u 0 )
2322oveq2d 5650 . . . . . 6  |-  ( N  =  0  ->  ( M lcm  -u N )  =  ( M lcm  -u 0
) )
2420, 21, 233eqtr4a 2146 . . . . 5  |-  ( N  =  0  ->  ( M lcm  N )  =  ( M lcm  -u N ) )
2524adantl 271 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0 )  ->  ( M lcm  N )  =  ( M lcm  -u N ) )
2617, 25jaodan 746 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  \/  N  =  0 ) )  -> 
( M lcm  N )  =  ( M lcm  -u N
) )
27 dvdslcm 11133 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  -u N  e.  ZZ )  ->  ( M  ||  ( M lcm  -u N )  /\  -u N  ||  ( M lcm  -u N ) ) )
282, 27sylan2 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 11136 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  -u N  e.  ZZ )  ->  ( M lcm  -u N
)  e.  NN0 )
312, 30sylan2 280 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  -u N
)  e.  NN0 )
3231nn0zd 8836 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  -u N
)  e.  ZZ )
33 negdvdsb 10894 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  ( M lcm  -u N )  e.  ZZ )  -> 
( N  ||  ( M lcm  -u N )  <->  -u N  ||  ( M lcm  -u N ) ) )
3429, 32, 33syl2anc 403 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  ||  ( M lcm  -u N )  <->  -u N  ||  ( M lcm  -u N ) ) )
3534anbi2d 452 . . . . . . 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 ) ) ) )
3628, 35mpbird 165 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( M lcm  -u N )  /\  N  ||  ( M lcm  -u N
) ) )
3736adantr 270 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  ||  ( M lcm  -u N )  /\  N  ||  ( M lcm  -u N ) ) )
38 zcn 8725 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
3938negeq0d 7764 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  ( N  =  0  <->  -u N  =  0 ) )
4039orbi2d 739 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
( M  =  0  \/  N  =  0 )  <->  ( M  =  0  \/  -u N  =  0 ) ) )
4140notbid 627 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( -.  ( M  =  0  \/  N  =  0 )  <->  -.  ( M  =  0  \/  -u N  =  0 ) ) )
4241biimpa 290 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  -.  ( M  =  0  \/  -u N  =  0 ) )
4342adantll 460 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  -.  ( M  =  0  \/  -u N  =  0 ) )
44 lcmn0cl 11132 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  -u N  e.  ZZ )  /\  -.  ( M  =  0  \/  -u N  =  0 ) )  ->  ( M lcm  -u N
)  e.  NN )
452, 44sylanl2 395 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  -u N  =  0 ) )  ->  ( M lcm  -u N
)  e.  NN )
4643, 45syldan 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 928 . . . . . . 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 ) ) )
4946, 47, 48sylanbrc 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 11134 . . . . . 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 ) ) )
5249, 50, 51syl2anc 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 ) ) )
5337, 52mpd 13 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N
)  <_  ( M lcm  -u N ) )
54 dvdslcm 11133 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm  N ) ) )
5554adantr 270 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm 
N ) ) )
56 simplr 497 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  e.  ZZ )
57 lcmn0cl 11132 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N
)  e.  NN )
5857nnzd 8837 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N
)  e.  ZZ )
59 negdvdsb 10894 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  ( M lcm  N )  e.  ZZ )  ->  ( N  ||  ( M lcm  N
)  <->  -u N  ||  ( M lcm  N ) ) )
6056, 58, 59syl2anc 403 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( N  ||  ( M lcm  N )  <->  -u N  ||  ( M lcm 
N ) ) )
6160anbi2d 452 . . . . . 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 11134 . . . . . . . . . 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 ) ) )
6362ex 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 ) ) ) )
642, 63syl3an3 1209 . . . . . . . 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 ) ) ) )
65643expib 1146 . . . . . . 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 ) ) ) ) )
6657, 47, 43, 65syl3c 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 ) ) )
6761, 66sylbid 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 ) ) )
6855, 67mpd 13 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  -u N
)  <_  ( M lcm  N ) )
69 lcmcl 11136 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
7069nn0red 8697 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  RR )
7130nn0red 8697 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  -u N  e.  ZZ )  ->  ( M lcm  -u N
)  e.  RR )
722, 71sylan2 280 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  -u N
)  e.  RR )
7370, 72letri3d 7579 . . . . 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
) ) ) )
7473adantr 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
) ) ) )
7553, 68, 74mpbir2and 890 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N
)  =  ( M lcm  -u N ) )
76 lcmmndc 11126 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  \/  N  =  0 ) )
77 exmiddc 782 . . . 4  |-  (DECID  ( M  =  0  \/  N  =  0 )  -> 
( ( M  =  0  \/  N  =  0 )  \/  -.  ( M  =  0  \/  N  =  0
) ) )
7876, 77syl 14 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  =  0  \/  N  =  0 )  \/  -.  ( M  =  0  \/  N  =  0
) ) )
7926, 75, 78mpjaodan 747 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  =  ( M lcm  -u N
) )
8079eqcomd 2093 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  -u N
)  =  ( M lcm 
N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  DECID wdc 780    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   RRcr 7328   0cc0 7329    <_ cle 7502   -ucneg 7633   NNcn 8394   NN0cn0 8643   ZZcz 8720    || cdvds 10878   lcm clcm 11124
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-sup 6658  df-inf 6659  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10879  df-lcm 11125
This theorem is referenced by:  neglcm  11139  lcmabs  11140
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