Theorem lcmabs 12584

Description: The lcm of two integers is the same as that of their absolute values. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Assertion

Ref	Expression
lcmabs	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )

Proof of Theorem lcmabs

Step	Hyp	Ref	Expression
1		zq 9809	. . 3 |- ( M e. ZZ -> M e. QQ )
2		zq 9809	. . 3 |- ( N e. ZZ -> N e. QQ )
3		qabsor 11572	. . . 4 |- ( M e. QQ -> ( ( abs ` M ) = M \/ ( abs ` M ) = -u M ) )
4		qabsor 11572	. . . 4 |- ( N e. QQ -> ( ( abs ` N ) = N \/ ( abs ` N ) = -u N ) )
5	3, 4	anim12i 338	. . 3 |- ( ( M e. QQ /\ N e. QQ ) -> ( ( ( abs ` M ) = M \/ ( abs ` M ) = -u M ) /\ ( ( abs ` N ) = N \/ ( abs ` N ) = -u N ) ) )
6	1, 2, 5	syl2an 289	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` M ) = M \/ ( abs ` M ) = -u M ) /\ ( ( abs ` N ) = N \/ ( abs ` N ) = -u N ) ) )
7		oveq12 6003	. . . 4 |- ( ( ( abs ` M ) = M /\ ( abs ` N ) = N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
8	7	a1i 9	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` M ) = M /\ ( abs ` N ) = N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) ) )
9		oveq12 6003	. . . . 5 |- ( ( ( abs ` M ) = -u M /\ ( abs ` N ) = N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( -u M lcm N ) )
10		neglcm 12583	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M lcm N ) = ( M lcm N ) )
11	9, 10	sylan9eqr 2284	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( ( abs ` M ) = -u M /\ ( abs ` N ) = N ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
12	11	ex 115	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` M ) = -u M /\ ( abs ` N ) = N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) ) )
13		oveq12 6003	. . . . 5 |- ( ( ( abs ` M ) = M /\ ( abs ` N ) = -u N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm -u N ) )
14		lcmneg 12582	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) = ( M lcm N ) )
15	13, 14	sylan9eqr 2284	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( ( abs ` M ) = M /\ ( abs ` N ) = -u N ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
16	15	ex 115	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` M ) = M /\ ( abs ` N ) = -u N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) ) )
17		oveq12 6003	. . . . 5 |- ( ( ( abs ` M ) = -u M /\ ( abs ` N ) = -u N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( -u M lcm -u N ) )
18		znegcl 9465	. . . . . . 7 |- ( M e. ZZ -> -u M e. ZZ )
19		lcmneg 12582	. . . . . . 7 |- ( ( -u M e. ZZ /\ N e. ZZ ) -> ( -u M lcm -u N ) = ( -u M lcm N ) )
20	18, 19	sylan 283	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M lcm -u N ) = ( -u M lcm N ) )
21	20, 10	eqtrd 2262	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M lcm -u N ) = ( M lcm N ) )
22	17, 21	sylan9eqr 2284	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( ( abs ` M ) = -u M /\ ( abs ` N ) = -u N ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
23	22	ex 115	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` M ) = -u M /\ ( abs ` N ) = -u N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) ) )
24	8, 12, 16, 23	ccased 971	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( ( abs ` M ) = M \/ ( abs ` M ) = -u M ) /\ ( ( abs ` N ) = N \/ ( abs ` N ) = -u N ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) ) )
25	6, 24	mpd 13	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   ` cfv 5314  (class class class)co 5994   -ucneg 8306   ZZcz 9434   QQcq 9802   abscabs 11494   lcm clcm 12568

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 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

This theorem depends on definitions:  df-bi 117  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 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-isom 5323  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-sup 7139  df-inf 7140  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-rp 9838  df-fz 10193  df-fzo 10327  df-fl 10477  df-mod 10532  df-seqfrec 10657  df-exp 10748  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-dvds 12285  df-lcm 12569

This theorem is referenced by:  lcmgcd  12586  lcmdvds  12587  lcmgcdeq  12591