Theorem lcmid 12012

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 11997	. . . 4 |- ( M e. ZZ -> ( M lcm 0 ) = 0 )
2	1	adantr 274	. . 3 |- ( ( M e. ZZ /\ M = 0 ) -> ( M lcm 0 ) = 0 )
3		oveq2 5850	. . . . 5 |- ( M = 0 -> ( M lcm M ) = ( M lcm 0 ) )
4		fveq2 5486	. . . . . 6 |- ( M = 0 -> ( abs ` M ) = ( abs ` 0 ) )
5		abs0 11000	. . . . . 6 |- ( abs ` 0 ) = 0
6	4, 5	eqtrdi 2215	. . . . 5 |- ( M = 0 -> ( abs ` M ) = 0 )
7	3, 6	eqeq12d 2180	. . . 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 2337	. . 3 |- ( M =/= 0 <-> -. M = 0 )
11		lcmcl 12004	. . . . . . 7 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( M lcm M ) e. NN0 )
12	11	nn0cnd 9169	. . . . . 6 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( M lcm M ) e. CC )
13	12	anidms 395	. . . . 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 11028	. . . . . 6 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
16	15	zcnd 9314	. . . . 5 |- ( M e. ZZ -> ( abs ` M ) e. CC )
17	16	adantr 274	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. CC )
18		zcn 9196	. . . . . . 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 11129	. . . . 5 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) =/= 0 )
22		0zd 9203	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> 0 e. ZZ )
23		zapne 9265	. . . . . 6 |- ( ( ( abs ` M ) e. ZZ /\ 0 e. ZZ ) -> ( ( abs ` M ) # 0 <-> ( abs ` M ) =/= 0 ) )
24	15, 22, 23	syl2an2r 585	. . . . 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 12010	. . . . . . 7 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( ( M lcm M ) x. ( M gcd M ) ) = ( abs ` ( M x. M ) ) )
27	26	anidms 395	. . . . . 6 |- ( M e. ZZ -> ( ( M lcm M ) x. ( M gcd M ) ) = ( abs ` ( M x. M ) ) )
28		gcdid 11919	. . . . . . 7 |- ( M e. ZZ -> ( M gcd M ) = ( abs ` M ) )
29	28	oveq2d 5858	. . . . . 6 |- ( M e. ZZ -> ( ( M lcm M ) x. ( M gcd M ) ) = ( ( M lcm M ) x. ( abs ` M ) ) )
30	18, 18	absmuld 11136	. . . . . 6 |- ( M e. ZZ -> ( abs ` ( M x. M ) ) = ( ( abs ` M ) x. ( abs ` M ) ) )
31	27, 29, 30	3eqtr3d 2206	. . . . 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 8561	. . 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 9202	. . . 4 |- 0 e. ZZ
36		zdceq 9266	. . . 4 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M = 0 )
37	35, 36	mpan2 422	. . 3 |- ( M e. ZZ -> DECID M = 0 )
38		exmiddc 826	. . 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 788	1 |- ( M e. ZZ -> ( M lcm M ) = ( abs ` M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   0cc0 7753   x. cmul 7758   # cap 8479   ZZcz 9191   abscabs 10939   gcd cgcd 11875   lcm clcm 11992

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876  df-lcm 11993

This theorem is referenced by:  lcmgcdeq  12015