Theorem lcmid 11155

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 11140	. . . 4 |- ( M e. ZZ -> ( M lcm 0 ) = 0 )
2	1	adantr 270	. . 3 |- ( ( M e. ZZ /\ M = 0 ) -> ( M lcm 0 ) = 0 )
3		oveq2 5642	. . . . 5 |- ( M = 0 -> ( M lcm M ) = ( M lcm 0 ) )
4		fveq2 5289	. . . . . 6 |- ( M = 0 -> ( abs ` M ) = ( abs ` 0 ) )
5		abs0 10456	. . . . . 6 |- ( abs ` 0 ) = 0
6	4, 5	syl6eq 2136	. . . . 5 |- ( M = 0 -> ( abs ` M ) = 0 )
7	3, 6	eqeq12d 2102	. . . 4 |- ( M = 0 -> ( ( M lcm M ) = ( abs ` M ) <-> ( M lcm 0 ) = 0 ) )
8	7	adantl 271	. . 3 |- ( ( M e. ZZ /\ M = 0 ) -> ( ( M lcm M ) = ( abs ` M ) <-> ( M lcm 0 ) = 0 ) )
9	2, 8	mpbird 165	. 2 |- ( ( M e. ZZ /\ M = 0 ) -> ( M lcm M ) = ( abs ` M ) )
10		df-ne 2256	. . 3 |- ( M =/= 0 <-> -. M = 0 )
11		lcmcl 11147	. . . . . . 7 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( M lcm M ) e. NN0 )
12	11	nn0cnd 8698	. . . . . 6 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( M lcm M ) e. CC )
13	12	anidms 389	. . . . 5 |- ( M e. ZZ -> ( M lcm M ) e. CC )
14	13	adantr 270	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( M lcm M ) e. CC )
15		zabscl 10484	. . . . . 6 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
16	15	zcnd 8839	. . . . 5 |- ( M e. ZZ -> ( abs ` M ) e. CC )
17	16	adantr 270	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. CC )
18		zcn 8725	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
19	18	adantr 270	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> M e. CC )
20		simpr 108	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> M =/= 0 )
21	19, 20	absne0d 10585	. . . . 5 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) =/= 0 )
22		0zd 8732	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> 0 e. ZZ )
23		zapne 8791	. . . . . 6 |- ( ( ( abs ` M ) e. ZZ /\ 0 e. ZZ ) -> ( ( abs ` M ) # 0 <-> ( abs ` M ) =/= 0 ) )
24	15, 22, 23	syl2an2r 562	. . . . 5 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( ( abs ` M ) # 0 <-> ( abs ` M ) =/= 0 ) )
25	21, 24	mpbird 165	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) # 0 )
26		lcmgcd 11153	. . . . . . 7 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( ( M lcm M ) x. ( M gcd M ) ) = ( abs ` ( M x. M ) ) )
27	26	anidms 389	. . . . . 6 |- ( M e. ZZ -> ( ( M lcm M ) x. ( M gcd M ) ) = ( abs ` ( M x. M ) ) )
28		gcdid 11070	. . . . . . 7 |- ( M e. ZZ -> ( M gcd M ) = ( abs ` M ) )
29	28	oveq2d 5650	. . . . . 6 |- ( M e. ZZ -> ( ( M lcm M ) x. ( M gcd M ) ) = ( ( M lcm M ) x. ( abs ` M ) ) )
30	18, 18	absmuld 10592	. . . . . 6 |- ( M e. ZZ -> ( abs ` ( M x. M ) ) = ( ( abs ` M ) x. ( abs ` M ) ) )
31	27, 29, 30	3eqtr3d 2128	. . . . 5 |- ( M e. ZZ -> ( ( M lcm M ) x. ( abs ` M ) ) = ( ( abs ` M ) x. ( abs ` M ) ) )
32	31	adantr 270	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( ( M lcm M ) x. ( abs ` M ) ) = ( ( abs ` M ) x. ( abs ` M ) ) )
33	14, 17, 17, 25, 32	mulcanap2ad 8107	. . 3 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( M lcm M ) = ( abs ` M ) )
34	10, 33	sylan2br 282	. 2 |- ( ( M e. ZZ /\ -. M = 0 ) -> ( M lcm M ) = ( abs ` M ) )
35		0z 8731	. . . 4 |- 0 e. ZZ
36		zdceq 8792	. . . 4 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M = 0 )
37	35, 36	mpan2 416	. . 3 |- ( M e. ZZ -> DECID M = 0 )
38		exmiddc 782	. . 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 747	1 |- ( M e. ZZ -> ( M lcm M ) = ( abs ` M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 664  DECID wdc 780   = wceq 1289   e. wcel 1438   =/= wne 2255   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   CCcc 7327   0cc0 7329   x. cmul 7334   # cap 8034   ZZcz 8720   abscabs 10395   gcd cgcd 11031   lcm clcm 11135

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-sup 6658  df-inf 6659  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10890  df-gcd 11032  df-lcm 11136

This theorem is referenced by:  lcmgcdeq  11158