Theorem lcmid 11761

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

Step	Hyp	Ref	Expression
1		lcm0val 11746	. . . 4 |- ( M e. ZZ -> ( M lcm 0 ) = 0 )
2	1	adantr 274	. . 3 |- ( ( M e. ZZ /\ M = 0 ) -> ( M lcm 0 ) = 0 )
3		oveq2 5782	. . . . 5 |- ( M = 0 -> ( M lcm M ) = ( M lcm 0 ) )
4		fveq2 5421	. . . . . 6 |- ( M = 0 -> ( abs ` M ) = ( abs ` 0 ) )
5		abs0 10830	. . . . . 6 |- ( abs ` 0 ) = 0
6	4, 5	syl6eq 2188	. . . . 5 |- ( M = 0 -> ( abs ` M ) = 0 )
7	3, 6	eqeq12d 2154	. . . 4 |- ( M = 0 -> ( ( M lcm M ) = ( abs ` M ) <-> ( M lcm 0 ) = 0 ) )
8	7	adantl 275	. . 3 |- ( ( M e. ZZ /\ M = 0 ) -> ( ( M lcm M ) = ( abs ` M ) <-> ( M lcm 0 ) = 0 ) )
9	2, 8	mpbird 166	. 2 |- ( ( M e. ZZ /\ M = 0 ) -> ( M lcm M ) = ( abs ` M ) )
10		df-ne 2309	. . 3 |- ( M =/= 0 <-> -. M = 0 )
11		lcmcl 11753	. . . . . . 7 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( M lcm M ) e. NN0 )
12	11	nn0cnd 9032	. . . . . 6 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( M lcm M ) e. CC )
13	12	anidms 394	. . . . 5 |- ( M e. ZZ -> ( M lcm M ) e. CC )
14	13	adantr 274	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( M lcm M ) e. CC )
15		zabscl 10858	. . . . . 6 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
16	15	zcnd 9174	. . . . 5 |- ( M e. ZZ -> ( abs ` M ) e. CC )
17	16	adantr 274	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. CC )
18		zcn 9059	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
19	18	adantr 274	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> M e. CC )
20		simpr 109	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> M =/= 0 )
21	19, 20	absne0d 10959	. . . . 5 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) =/= 0 )
22		0zd 9066	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> 0 e. ZZ )
23		zapne 9125	. . . . . 6 |- ( ( ( abs ` M ) e. ZZ /\ 0 e. ZZ ) -> ( ( abs ` M ) # 0 <-> ( abs ` M ) =/= 0 ) )
24	15, 22, 23	syl2an2r 584	. . . . 5 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( ( abs ` M ) # 0 <-> ( abs ` M ) =/= 0 ) )
25	21, 24	mpbird 166	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) # 0 )
26		lcmgcd 11759	. . . . . . 7 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( ( M lcm M ) x. ( M gcd M ) ) = ( abs ` ( M x. M ) ) )
27	26	anidms 394	. . . . . 6 |- ( M e. ZZ -> ( ( M lcm M ) x. ( M gcd M ) ) = ( abs ` ( M x. M ) ) )
28		gcdid 11674	. . . . . . 7 |- ( M e. ZZ -> ( M gcd M ) = ( abs ` M ) )
29	28	oveq2d 5790	. . . . . 6 |- ( M e. ZZ -> ( ( M lcm M ) x. ( M gcd M ) ) = ( ( M lcm M ) x. ( abs ` M ) ) )
30	18, 18	absmuld 10966	. . . . . 6 |- ( M e. ZZ -> ( abs ` ( M x. M ) ) = ( ( abs ` M ) x. ( abs ` M ) ) )
31	27, 29, 30	3eqtr3d 2180	. . . . 5 |- ( M e. ZZ -> ( ( M lcm M ) x. ( abs ` M ) ) = ( ( abs ` M ) x. ( abs ` M ) ) )
32	31	adantr 274	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( ( M lcm M ) x. ( abs ` M ) ) = ( ( abs ` M ) x. ( abs ` M ) ) )
33	14, 17, 17, 25, 32	mulcanap2ad 8425	. . 3 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( M lcm M ) = ( abs ` M ) )
34	10, 33	sylan2br 286	. 2 |- ( ( M e. ZZ /\ -. M = 0 ) -> ( M lcm M ) = ( abs ` M ) )
35		0z 9065	. . . 4 |- 0 e. ZZ
36		zdceq 9126	. . . 4 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M = 0 )
37	35, 36	mpan2 421	. . 3 |- ( M e. ZZ -> DECID M = 0 )
38		exmiddc 821	. . 3 |- (DECID M = 0 -> ( M = 0 \/ -. M = 0 ) )
39	37, 38	syl 14	. 2 |- ( M e. ZZ -> ( M = 0 \/ -. M = 0 ) )
40	9, 34, 39	mpjaodan 787	1 |- ( M e. ZZ -> ( M lcm M ) = ( abs ` M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  DECID wdc 819   = wceq 1331   e. wcel 1480   =/= wne 2308   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7618   0cc0 7620   x. cmul 7625   # cap 8343   ZZcz 9054   abscabs 10769   gcd cgcd 11635   lcm clcm 11741

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-inf 6872  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494  df-gcd 11636  df-lcm 11742

This theorem is referenced by:  lcmgcdeq  11764