Theorem lcmid 12597

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 12582	. . . 4 |- ( M e. ZZ -> ( M lcm 0 ) = 0 )
2	1	adantr 276	. . 3 |- ( ( M e. ZZ /\ M = 0 ) -> ( M lcm 0 ) = 0 )
3		oveq2 6008	. . . . 5 |- ( M = 0 -> ( M lcm M ) = ( M lcm 0 ) )
4		fveq2 5626	. . . . . 6 |- ( M = 0 -> ( abs ` M ) = ( abs ` 0 ) )
5		abs0 11564	. . . . . 6 |- ( abs ` 0 ) = 0
6	4, 5	eqtrdi 2278	. . . . 5 |- ( M = 0 -> ( abs ` M ) = 0 )
7	3, 6	eqeq12d 2244	. . . 4 |- ( M = 0 -> ( ( M lcm M ) = ( abs ` M ) <-> ( M lcm 0 ) = 0 ) )
8	7	adantl 277	. . 3 |- ( ( M e. ZZ /\ M = 0 ) -> ( ( M lcm M ) = ( abs ` M ) <-> ( M lcm 0 ) = 0 ) )
9	2, 8	mpbird 167	. 2 |- ( ( M e. ZZ /\ M = 0 ) -> ( M lcm M ) = ( abs ` M ) )
10		df-ne 2401	. . 3 |- ( M =/= 0 <-> -. M = 0 )
11		lcmcl 12589	. . . . . . 7 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( M lcm M ) e. NN0 )
12	11	nn0cnd 9420	. . . . . 6 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( M lcm M ) e. CC )
13	12	anidms 397	. . . . 5 |- ( M e. ZZ -> ( M lcm M ) e. CC )
14	13	adantr 276	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( M lcm M ) e. CC )
15		zabscl 11592	. . . . . 6 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
16	15	zcnd 9566	. . . . 5 |- ( M e. ZZ -> ( abs ` M ) e. CC )
17	16	adantr 276	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. CC )
18		zcn 9447	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
19	18	adantr 276	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> M e. CC )
20		simpr 110	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> M =/= 0 )
21	19, 20	absne0d 11693	. . . . 5 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) =/= 0 )
22		0zd 9454	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> 0 e. ZZ )
23		zapne 9517	. . . . . 6 |- ( ( ( abs ` M ) e. ZZ /\ 0 e. ZZ ) -> ( ( abs ` M ) # 0 <-> ( abs ` M ) =/= 0 ) )
24	15, 22, 23	syl2an2r 597	. . . . 5 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( ( abs ` M ) # 0 <-> ( abs ` M ) =/= 0 ) )
25	21, 24	mpbird 167	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) # 0 )
26		lcmgcd 12595	. . . . . . 7 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( ( M lcm M ) x. ( M gcd M ) ) = ( abs ` ( M x. M ) ) )
27	26	anidms 397	. . . . . 6 |- ( M e. ZZ -> ( ( M lcm M ) x. ( M gcd M ) ) = ( abs ` ( M x. M ) ) )
28		gcdid 12502	. . . . . . 7 |- ( M e. ZZ -> ( M gcd M ) = ( abs ` M ) )
29	28	oveq2d 6016	. . . . . 6 |- ( M e. ZZ -> ( ( M lcm M ) x. ( M gcd M ) ) = ( ( M lcm M ) x. ( abs ` M ) ) )
30	18, 18	absmuld 11700	. . . . . 6 |- ( M e. ZZ -> ( abs ` ( M x. M ) ) = ( ( abs ` M ) x. ( abs ` M ) ) )
31	27, 29, 30	3eqtr3d 2270	. . . . 5 |- ( M e. ZZ -> ( ( M lcm M ) x. ( abs ` M ) ) = ( ( abs ` M ) x. ( abs ` M ) ) )
32	31	adantr 276	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( ( M lcm M ) x. ( abs ` M ) ) = ( ( abs ` M ) x. ( abs ` M ) ) )
33	14, 17, 17, 25, 32	mulcanap2ad 8807	. . 3 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( M lcm M ) = ( abs ` M ) )
34	10, 33	sylan2br 288	. 2 |- ( ( M e. ZZ /\ -. M = 0 ) -> ( M lcm M ) = ( abs ` M ) )
35		0z 9453	. . . 4 |- 0 e. ZZ
36		zdceq 9518	. . . 4 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M = 0 )
37	35, 36	mpan2 425	. . 3 |- ( M e. ZZ -> DECID M = 0 )
38		exmiddc 841	. . 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 803	1 |- ( M e. ZZ -> ( M lcm M ) = ( abs ` M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   CCcc 7993   0cc0 7995   x. cmul 8000   # cap 8724   ZZcz 9442   abscabs 11503   gcd cgcd 12469   lcm clcm 12577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-fl 10485  df-mod 10540  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-dvds 12294  df-gcd 12470  df-lcm 12578

This theorem is referenced by:  lcmgcdeq  12600