Theorem lcmabs 12798

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 9976	. . 3 |- ( M e. ZZ -> M e. QQ )
2		zq 9976	. . 3 |- ( N e. ZZ -> N e. QQ )
3		qabsor 11785	. . . 4 |- ( M e. QQ -> ( ( abs ` M ) = M \/ ( abs ` M ) = -u M ) )
4		qabsor 11785	. . . 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 6067	. . . 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 6067	. . . . 5 |- ( ( ( abs ` M ) = -u M /\ ( abs ` N ) = N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( -u M lcm N ) )
10		neglcm 12797	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M lcm N ) = ( M lcm N ) )
11	9, 10	sylan9eqr 2289	. . . 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 6067	. . . . 5 |- ( ( ( abs ` M ) = M /\ ( abs ` N ) = -u N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm -u N ) )
14		lcmneg 12796	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) = ( M lcm N ) )
15	13, 14	sylan9eqr 2289	. . . 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 6067	. . . . 5 |- ( ( ( abs ` M ) = -u M /\ ( abs ` N ) = -u N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( -u M lcm -u N ) )
18		znegcl 9625	. . . . . . 7 |- ( M e. ZZ -> -u M e. ZZ )
19		lcmneg 12796	. . . . . . 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 2267	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M lcm -u N ) = ( M lcm N ) )
22	17, 21	sylan9eqr 2289	. . . 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 974	. 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 716   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   -ucneg 8461   ZZcz 9594   QQcq 9969   abscabs 11707   lcm clcm 12782

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-lcm 12783

This theorem is referenced by:  lcmgcd  12800  lcmdvds  12801  lcmgcdeq  12805