Theorem lcmabs 12769

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 9957	. . 3 |- ( M e. ZZ -> M e. QQ )
2		zq 9957	. . 3 |- ( N e. ZZ -> N e. QQ )
3		qabsor 11756	. . . 4 |- ( M e. QQ -> ( ( abs ` M ) = M \/ ( abs ` M ) = -u M ) )
4		qabsor 11756	. . . 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 6058	. . . 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 6058	. . . . 5 |- ( ( ( abs ` M ) = -u M /\ ( abs ` N ) = N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( -u M lcm N ) )
10		neglcm 12768	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M lcm N ) = ( M lcm N ) )
11	9, 10	sylan9eqr 2287	. . . 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 6058	. . . . 5 |- ( ( ( abs ` M ) = M /\ ( abs ` N ) = -u N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm -u N ) )
14		lcmneg 12767	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm -u N ) = ( M lcm N ) )
15	13, 14	sylan9eqr 2287	. . . 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 6058	. . . . 5 |- ( ( ( abs ` M ) = -u M /\ ( abs ` N ) = -u N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( -u M lcm -u N ) )
18		znegcl 9607	. . . . . . 7 |- ( M e. ZZ -> -u M e. ZZ )
19		lcmneg 12767	. . . . . . 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 2265	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M lcm -u N ) = ( M lcm N ) )
22	17, 21	sylan9eqr 2287	. . . 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 2203   ` cfv 5351  (class class class)co 6049   -ucneg 8444   ZZcz 9576   QQcq 9950   abscabs 11678   lcm clcm 12753

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-sup 7274  df-inf 7275  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-fz 10342  df-fzo 10476  df-fl 10629  df-mod 10684  df-seqfrec 10809  df-exp 10900  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-dvds 12470  df-lcm 12754

This theorem is referenced by:  lcmgcd  12771  lcmdvds  12772  lcmgcdeq  12776