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Theorem lcmid 11668
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 11653 . . . 4  |-  ( M  e.  ZZ  ->  ( M lcm  0 )  =  0 )
21adantr 272 . . 3  |-  ( ( M  e.  ZZ  /\  M  =  0 )  ->  ( M lcm  0
)  =  0 )
3 oveq2 5748 . . . . 5  |-  ( M  =  0  ->  ( M lcm  M )  =  ( M lcm  0 ) )
4 fveq2 5387 . . . . . 6  |-  ( M  =  0  ->  ( abs `  M )  =  ( abs `  0
) )
5 abs0 10781 . . . . . 6  |-  ( abs `  0 )  =  0
64, 5syl6eq 2164 . . . . 5  |-  ( M  =  0  ->  ( abs `  M )  =  0 )
73, 6eqeq12d 2130 . . . 4  |-  ( M  =  0  ->  (
( M lcm  M )  =  ( abs `  M
)  <->  ( M lcm  0
)  =  0 ) )
87adantl 273 . . 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 2284 . . 3  |-  ( M  =/=  0  <->  -.  M  =  0 )
11 lcmcl 11660 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( M lcm  M )  e.  NN0 )
1211nn0cnd 8986 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( M lcm  M )  e.  CC )
1312anidms 392 . . . . 5  |-  ( M  e.  ZZ  ->  ( M lcm  M )  e.  CC )
1413adantr 272 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M lcm  M )  e.  CC )
15 zabscl 10809 . . . . . 6  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  ZZ )
1615zcnd 9128 . . . . 5  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  CC )
1716adantr 272 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  CC )
18 zcn 9013 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
1918adantr 272 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  ->  M  e.  CC )
20 simpr 109 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  ->  M  =/=  0 )
2119, 20absne0d 10910 . . . . 5  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  =/=  0 )
22 0zd 9020 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
0  e.  ZZ )
23 zapne 9079 . . . . . 6  |-  ( ( ( abs `  M
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( abs `  M
) #  0  <->  ( abs `  M )  =/=  0
) )
2415, 22, 23syl2an2r 567 . . . . 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 11666 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( M lcm  M
)  x.  ( M  gcd  M ) )  =  ( abs `  ( M  x.  M )
) )
2726anidms 392 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( M lcm  M )  x.  ( M  gcd  M ) )  =  ( abs `  ( M  x.  M ) ) )
28 gcdid 11581 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M  gcd  M )  =  ( abs `  M
) )
2928oveq2d 5756 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( M lcm  M )  x.  ( M  gcd  M ) )  =  ( ( M lcm  M )  x.  ( abs `  M
) ) )
3018, 18absmuld 10917 . . . . . 6  |-  ( M  e.  ZZ  ->  ( abs `  ( M  x.  M ) )  =  ( ( abs `  M
)  x.  ( abs `  M ) ) )
3127, 29, 303eqtr3d 2156 . . . . 5  |-  ( M  e.  ZZ  ->  (
( M lcm  M )  x.  ( abs `  M
) )  =  ( ( abs `  M
)  x.  ( abs `  M ) ) )
3231adantr 272 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( ( M lcm  M
)  x.  ( abs `  M ) )  =  ( ( abs `  M
)  x.  ( abs `  M ) ) )
3314, 17, 17, 25, 32mulcanap2ad 8388 . . 3  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M lcm  M )  =  ( abs `  M
) )
3410, 33sylan2br 284 . 2  |-  ( ( M  e.  ZZ  /\  -.  M  =  0
)  ->  ( M lcm  M )  =  ( abs `  M ) )
35 0z 9019 . . . 4  |-  0  e.  ZZ
36 zdceq 9080 . . . 4  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  -> DECID  M  =  0 )
3735, 36mpan2 419 . . 3  |-  ( M  e.  ZZ  -> DECID  M  =  0
)
38 exmiddc 804 . . 3  |-  (DECID  M  =  0  ->  ( M  =  0  \/  -.  M  =  0 ) )
3937, 38syl 14 . 2  |-  ( M  e.  ZZ  ->  ( M  =  0  \/  -.  M  =  0
) )
409, 34, 39mpjaodan 770 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 680  DECID wdc 802    = wceq 1314    e. wcel 1463    =/= wne 2283   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   CCcc 7582   0cc0 7584    x. cmul 7589   # cap 8306   ZZcz 9008   abscabs 10720    gcd cgcd 11542   lcm clcm 11648
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-sup 6837  df-inf 6838  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-fz 9742  df-fzo 9871  df-fl 9994  df-mod 10047  df-seqfrec 10170  df-exp 10244  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-dvds 11401  df-gcd 11543  df-lcm 11649
This theorem is referenced by:  lcmgcdeq  11671
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