Theorem lcmid 12034

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 12019	. . . 4 |- ( M e. ZZ -> ( M lcm 0 ) = 0 )
2	1	adantr 274	. . 3 |- ( ( M e. ZZ /\ M = 0 ) -> ( M lcm 0 ) = 0 )
3		oveq2 5861	. . . . 5 |- ( M = 0 -> ( M lcm M ) = ( M lcm 0 ) )
4		fveq2 5496	. . . . . 6 |- ( M = 0 -> ( abs ` M ) = ( abs ` 0 ) )
5		abs0 11022	. . . . . 6 |- ( abs ` 0 ) = 0
6	4, 5	eqtrdi 2219	. . . . 5 |- ( M = 0 -> ( abs ` M ) = 0 )
7	3, 6	eqeq12d 2185	. . . 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 2341	. . 3 |- ( M =/= 0 <-> -. M = 0 )
11		lcmcl 12026	. . . . . . 7 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( M lcm M ) e. NN0 )
12	11	nn0cnd 9190	. . . . . 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 11050	. . . . . 6 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
16	15	zcnd 9335	. . . . 5 |- ( M e. ZZ -> ( abs ` M ) e. CC )
17	16	adantr 274	. . . 4 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. CC )
18		zcn 9217	. . . . . . 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 11151	. . . . 5 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) =/= 0 )
22		0zd 9224	. . . . . 6 |- ( ( M e. ZZ /\ M =/= 0 ) -> 0 e. ZZ )
23		zapne 9286	. . . . . 6 |- ( ( ( abs ` M ) e. ZZ /\ 0 e. ZZ ) -> ( ( abs ` M ) # 0 <-> ( abs ` M ) =/= 0 ) )
24	15, 22, 23	syl2an2r 590	. . . . 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 12032	. . . . . . 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 11941	. . . . . . 7 |- ( M e. ZZ -> ( M gcd M ) = ( abs ` M ) )
29	28	oveq2d 5869	. . . . . 6 |- ( M e. ZZ -> ( ( M lcm M ) x. ( M gcd M ) ) = ( ( M lcm M ) x. ( abs ` M ) ) )
30	18, 18	absmuld 11158	. . . . . 6 |- ( M e. ZZ -> ( abs ` ( M x. M ) ) = ( ( abs ` M ) x. ( abs ` M ) ) )
31	27, 29, 30	3eqtr3d 2211	. . . . 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 8582	. . 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 9223	. . . 4 |- 0 e. ZZ
36		zdceq 9287	. . . 4 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M = 0 )
37	35, 36	mpan2 423	. . 3 |- ( M e. ZZ -> DECID M = 0 )
38		exmiddc 831	. . 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 793	1 |- ( M e. ZZ -> ( M lcm M ) = ( abs ` M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  DECID wdc 829   = wceq 1348   e. wcel 2141   =/= wne 2340   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   x. cmul 7779   # cap 8500   ZZcz 9212   abscabs 10961   gcd cgcd 11897   lcm clcm 12014

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898  df-lcm 12015

This theorem is referenced by:  lcmgcdeq  12037