Theorem lcmeq0 12088

Description: The lcm of two integers is zero iff either is zero. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Assertion

Ref	Expression
lcmeq0	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = 0 <-> ( M = 0 \/ N = 0 ) ) )

Proof of Theorem lcmeq0

Step	Hyp	Ref	Expression
1		lcmmndc 12079	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
2		lcmn0cl 12085	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. NN )
3	2	nnne0d 8981	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) =/= 0 )
4	3	neneqd 2380	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M lcm N ) = 0 )
5	4	ex 115	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 \/ N = 0 ) -> -. ( M lcm N ) = 0 ) )
6		condc 854	. . 3 |- (DECID ( M = 0 \/ N = 0 ) -> ( ( -. ( M = 0 \/ N = 0 ) -> -. ( M lcm N ) = 0 ) -> ( ( M lcm N ) = 0 -> ( M = 0 \/ N = 0 ) ) ) )
7	1, 5, 6	sylc 62	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = 0 -> ( M = 0 \/ N = 0 ) ) )
8		oveq1 5897	. . . . . 6 |- ( M = 0 -> ( M lcm N ) = ( 0 lcm N ) )
9		0z 9281	. . . . . . . 8 |- 0 e. ZZ
10		lcmcom 12081	. . . . . . . 8 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N lcm 0 ) = ( 0 lcm N ) )
11	9, 10	mpan2 425	. . . . . . 7 |- ( N e. ZZ -> ( N lcm 0 ) = ( 0 lcm N ) )
12		lcm0val 12082	. . . . . . 7 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
13	11, 12	eqtr3d 2223	. . . . . 6 |- ( N e. ZZ -> ( 0 lcm N ) = 0 )
14	8, 13	sylan9eqr 2243	. . . . 5 |- ( ( N e. ZZ /\ M = 0 ) -> ( M lcm N ) = 0 )
15	14	adantll 476	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M lcm N ) = 0 )
16		oveq2 5898	. . . . . 6 |- ( N = 0 -> ( M lcm N ) = ( M lcm 0 ) )
17		lcm0val 12082	. . . . . 6 |- ( M e. ZZ -> ( M lcm 0 ) = 0 )
18	16, 17	sylan9eqr 2243	. . . . 5 |- ( ( M e. ZZ /\ N = 0 ) -> ( M lcm N ) = 0 )
19	18	adantlr 477	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M lcm N ) = 0 )
20	15, 19	jaodan 798	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = 0 )
21	20	ex 115	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 \/ N = 0 ) -> ( M lcm N ) = 0 ) )
22	7, 21	impbid 129	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = 0 <-> ( M = 0 \/ N = 0 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1363   e. wcel 2159  (class class class)co 5890   0cc0 7828   ZZcz 9270   lcm clcm 12077

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947  ax-caucvg 7948

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-isom 5239  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-sup 7000  df-inf 7001  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-n0 9194  df-z 9271  df-uz 9546  df-q 9637  df-rp 9671  df-fz 10026  df-fzo 10160  df-fl 10287  df-mod 10340  df-seqfrec 10463  df-exp 10537  df-cj 10868  df-re 10869  df-im 10870  df-rsqrt 11024  df-abs 11025  df-dvds 11812  df-lcm 12078

This theorem is referenced by:  lcmass  12102