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Theorem lcmid 11961
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 11946 . . . 4  |-  ( M  e.  ZZ  ->  ( M lcm  0 )  =  0 )
21adantr 274 . . 3  |-  ( ( M  e.  ZZ  /\  M  =  0 )  ->  ( M lcm  0
)  =  0 )
3 oveq2 5833 . . . . 5  |-  ( M  =  0  ->  ( M lcm  M )  =  ( M lcm  0 ) )
4 fveq2 5469 . . . . . 6  |-  ( M  =  0  ->  ( abs `  M )  =  ( abs `  0
) )
5 abs0 10962 . . . . . 6  |-  ( abs `  0 )  =  0
64, 5eqtrdi 2206 . . . . 5  |-  ( M  =  0  ->  ( abs `  M )  =  0 )
73, 6eqeq12d 2172 . . . 4  |-  ( M  =  0  ->  (
( M lcm  M )  =  ( abs `  M
)  <->  ( M lcm  0
)  =  0 ) )
87adantl 275 . . 3  |-  ( ( M  e.  ZZ  /\  M  =  0 )  ->  ( ( M lcm 
M )  =  ( abs `  M )  <-> 
( M lcm  0 )  =  0 ) )
92, 8mpbird 166 . 2  |-  ( ( M  e.  ZZ  /\  M  =  0 )  ->  ( M lcm  M
)  =  ( abs `  M ) )
10 df-ne 2328 . . 3  |-  ( M  =/=  0  <->  -.  M  =  0 )
11 lcmcl 11953 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( M lcm  M )  e.  NN0 )
1211nn0cnd 9146 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( M lcm  M )  e.  CC )
1312anidms 395 . . . . 5  |-  ( M  e.  ZZ  ->  ( M lcm  M )  e.  CC )
1413adantr 274 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M lcm  M )  e.  CC )
15 zabscl 10990 . . . . . 6  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  ZZ )
1615zcnd 9288 . . . . 5  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  CC )
1716adantr 274 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  CC )
18 zcn 9173 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
1918adantr 274 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  ->  M  e.  CC )
20 simpr 109 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  ->  M  =/=  0 )
2119, 20absne0d 11091 . . . . 5  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  =/=  0 )
22 0zd 9180 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
0  e.  ZZ )
23 zapne 9239 . . . . . 6  |-  ( ( ( abs `  M
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( abs `  M
) #  0  <->  ( abs `  M )  =/=  0
) )
2415, 22, 23syl2an2r 585 . . . . 5  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( ( abs `  M
) #  0  <->  ( abs `  M )  =/=  0
) )
2521, 24mpbird 166 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
) #  0 )
26 lcmgcd 11959 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( M lcm  M
)  x.  ( M  gcd  M ) )  =  ( abs `  ( M  x.  M )
) )
2726anidms 395 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( M lcm  M )  x.  ( M  gcd  M ) )  =  ( abs `  ( M  x.  M ) ) )
28 gcdid 11874 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M  gcd  M )  =  ( abs `  M
) )
2928oveq2d 5841 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( M lcm  M )  x.  ( M  gcd  M ) )  =  ( ( M lcm  M )  x.  ( abs `  M
) ) )
3018, 18absmuld 11098 . . . . . 6  |-  ( M  e.  ZZ  ->  ( abs `  ( M  x.  M ) )  =  ( ( abs `  M
)  x.  ( abs `  M ) ) )
3127, 29, 303eqtr3d 2198 . . . . 5  |-  ( M  e.  ZZ  ->  (
( M lcm  M )  x.  ( abs `  M
) )  =  ( ( abs `  M
)  x.  ( abs `  M ) ) )
3231adantr 274 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( ( M lcm  M
)  x.  ( abs `  M ) )  =  ( ( abs `  M
)  x.  ( abs `  M ) ) )
3314, 17, 17, 25, 32mulcanap2ad 8539 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M lcm  M )  =  ( abs `  M
) )
3410, 33sylan2br 286 . 2  |-  ( ( M  e.  ZZ  /\  -.  M  =  0
)  ->  ( M lcm  M )  =  ( abs `  M ) )
35 0z 9179 . . . 4  |-  0  e.  ZZ
36 zdceq 9240 . . . 4  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  -> DECID  M  =  0 )
3735, 36mpan2 422 . . 3  |-  ( M  e.  ZZ  -> DECID  M  =  0
)
38 exmiddc 822 . . 3  |-  (DECID  M  =  0  ->  ( M  =  0  \/  -.  M  =  0 ) )
3937, 38syl 14 . 2  |-  ( M  e.  ZZ  ->  ( M  =  0  \/  -.  M  =  0
) )
409, 34, 39mpjaodan 788 1  |-  ( M  e.  ZZ  ->  ( M lcm  M )  =  ( abs `  M ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698  DECID wdc 820    = wceq 1335    e. wcel 2128    =/= wne 2327   class class class wbr 3966   ` cfv 5171  (class class class)co 5825   CCcc 7731   0cc0 7733    x. cmul 7738   # cap 8457   ZZcz 9168   abscabs 10901    gcd cgcd 11833   lcm clcm 11941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-mulrcl 7832  ax-addcom 7833  ax-mulcom 7834  ax-addass 7835  ax-mulass 7836  ax-distr 7837  ax-i2m1 7838  ax-0lt1 7839  ax-1rid 7840  ax-0id 7841  ax-rnegex 7842  ax-precex 7843  ax-cnre 7844  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-apti 7848  ax-pre-ltadd 7849  ax-pre-mulgt0 7850  ax-pre-mulext 7851  ax-arch 7852  ax-caucvg 7853
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-ilim 4330  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-isom 5180  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-frec 6339  df-sup 6929  df-inf 6930  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-sub 8049  df-neg 8050  df-reap 8451  df-ap 8458  df-div 8547  df-inn 8835  df-2 8893  df-3 8894  df-4 8895  df-n0 9092  df-z 9169  df-uz 9441  df-q 9530  df-rp 9562  df-fz 9914  df-fzo 10046  df-fl 10173  df-mod 10226  df-seqfrec 10349  df-exp 10423  df-cj 10746  df-re 10747  df-im 10748  df-rsqrt 10902  df-abs 10903  df-dvds 11688  df-gcd 11834  df-lcm 11942
This theorem is referenced by:  lcmgcdeq  11964
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