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Theorem lcmid 12123
Description: The lcm of an integer and itself is its absolute value. (Contributed by Steve Rodriguez, 20-Jan-2020.)
Assertion
Ref Expression
lcmid  |-  ( M  e.  ZZ  ->  ( M lcm  M )  =  ( abs `  M ) )

Proof of Theorem lcmid
StepHypRef Expression
1 lcm0val 12108 . . . 4  |-  ( M  e.  ZZ  ->  ( M lcm  0 )  =  0 )
21adantr 276 . . 3  |-  ( ( M  e.  ZZ  /\  M  =  0 )  ->  ( M lcm  0
)  =  0 )
3 oveq2 5908 . . . . 5  |-  ( M  =  0  ->  ( M lcm  M )  =  ( M lcm  0 ) )
4 fveq2 5537 . . . . . 6  |-  ( M  =  0  ->  ( abs `  M )  =  ( abs `  0
) )
5 abs0 11108 . . . . . 6  |-  ( abs `  0 )  =  0
64, 5eqtrdi 2238 . . . . 5  |-  ( M  =  0  ->  ( abs `  M )  =  0 )
73, 6eqeq12d 2204 . . . 4  |-  ( M  =  0  ->  (
( M lcm  M )  =  ( abs `  M
)  <->  ( M lcm  0
)  =  0 ) )
87adantl 277 . . 3  |-  ( ( M  e.  ZZ  /\  M  =  0 )  ->  ( ( M lcm 
M )  =  ( abs `  M )  <-> 
( M lcm  0 )  =  0 ) )
92, 8mpbird 167 . 2  |-  ( ( M  e.  ZZ  /\  M  =  0 )  ->  ( M lcm  M
)  =  ( abs `  M ) )
10 df-ne 2361 . . 3  |-  ( M  =/=  0  <->  -.  M  =  0 )
11 lcmcl 12115 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( M lcm  M )  e.  NN0 )
1211nn0cnd 9266 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( M lcm  M )  e.  CC )
1312anidms 397 . . . . 5  |-  ( M  e.  ZZ  ->  ( M lcm  M )  e.  CC )
1413adantr 276 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M lcm  M )  e.  CC )
15 zabscl 11136 . . . . . 6  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  ZZ )
1615zcnd 9411 . . . . 5  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  CC )
1716adantr 276 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  CC )
18 zcn 9293 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
1918adantr 276 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  ->  M  e.  CC )
20 simpr 110 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  ->  M  =/=  0 )
2119, 20absne0d 11237 . . . . 5  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  =/=  0 )
22 0zd 9300 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
0  e.  ZZ )
23 zapne 9362 . . . . . 6  |-  ( ( ( abs `  M
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( abs `  M
) #  0  <->  ( abs `  M )  =/=  0
) )
2415, 22, 23syl2an2r 595 . . . . 5  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( ( abs `  M
) #  0  <->  ( abs `  M )  =/=  0
) )
2521, 24mpbird 167 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
) #  0 )
26 lcmgcd 12121 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( M lcm  M
)  x.  ( M  gcd  M ) )  =  ( abs `  ( M  x.  M )
) )
2726anidms 397 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( M lcm  M )  x.  ( M  gcd  M ) )  =  ( abs `  ( M  x.  M ) ) )
28 gcdid 12028 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M  gcd  M )  =  ( abs `  M
) )
2928oveq2d 5916 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( M lcm  M )  x.  ( M  gcd  M ) )  =  ( ( M lcm  M )  x.  ( abs `  M
) ) )
3018, 18absmuld 11244 . . . . . 6  |-  ( M  e.  ZZ  ->  ( abs `  ( M  x.  M ) )  =  ( ( abs `  M
)  x.  ( abs `  M ) ) )
3127, 29, 303eqtr3d 2230 . . . . 5  |-  ( M  e.  ZZ  ->  (
( M lcm  M )  x.  ( abs `  M
) )  =  ( ( abs `  M
)  x.  ( abs `  M ) ) )
3231adantr 276 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( ( M lcm  M
)  x.  ( abs `  M ) )  =  ( ( abs `  M
)  x.  ( abs `  M ) ) )
3314, 17, 17, 25, 32mulcanap2ad 8656 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M lcm  M )  =  ( abs `  M
) )
3410, 33sylan2br 288 . 2  |-  ( ( M  e.  ZZ  /\  -.  M  =  0
)  ->  ( M lcm  M )  =  ( abs `  M ) )
35 0z 9299 . . . 4  |-  0  e.  ZZ
36 zdceq 9363 . . . 4  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  -> DECID  M  =  0 )
3735, 36mpan2 425 . . 3  |-  ( M  e.  ZZ  -> DECID  M  =  0
)
38 exmiddc 837 . . 3  |-  (DECID  M  =  0  ->  ( M  =  0  \/  -.  M  =  0 ) )
3937, 38syl 14 . 2  |-  ( M  e.  ZZ  ->  ( M  =  0  \/  -.  M  =  0
) )
409, 34, 39mpjaodan 799 1  |-  ( M  e.  ZZ  ->  ( M lcm  M )  =  ( abs `  M ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709  DECID wdc 835    = wceq 1364    e. wcel 2160    =/= wne 2360   class class class wbr 4021   ` cfv 5238  (class class class)co 5900   CCcc 7844   0cc0 7846    x. cmul 7851   # cap 8573   ZZcz 9288   abscabs 11047    gcd cgcd 11984   lcm clcm 12103
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 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964  ax-arch 7965  ax-caucvg 7966
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-isom 5247  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-frec 6420  df-sup 7017  df-inf 7018  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-reap 8567  df-ap 8574  df-div 8665  df-inn 8955  df-2 9013  df-3 9014  df-4 9015  df-n0 9212  df-z 9289  df-uz 9564  df-q 9656  df-rp 9690  df-fz 10045  df-fzo 10179  df-fl 10307  df-mod 10360  df-seqfrec 10485  df-exp 10560  df-cj 10892  df-re 10893  df-im 10894  df-rsqrt 11048  df-abs 11049  df-dvds 11836  df-gcd 11985  df-lcm 12104
This theorem is referenced by:  lcmgcdeq  12126
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