Theorem 3lcm2e6woprm 12808

Description: The least common multiple of three and two is six. This proof does not use the property of 2 and 3 being prime. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 27-Aug-2020.)

Assertion

Ref	Expression
3lcm2e6woprm	|- ( 3 lcm 2 ) = 6

Proof of Theorem 3lcm2e6woprm

Step	Hyp	Ref	Expression
1		3cn 9329	. . . 4 |- 3 e. CC
2		2cn 9325	. . . 4 |- 2 e. CC
3	1, 2	mulcli 8295	. . 3 |- ( 3 x. 2 ) e. CC
4		3z 9623	. . . 4 |- 3 e. ZZ
5		2z 9622	. . . 4 |- 2 e. ZZ
6		lcmcl 12794	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> ( 3 lcm 2 ) e. NN0 )
7	6	nn0cnd 9572	. . . 4 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> ( 3 lcm 2 ) e. CC )
8	4, 5, 7	mp2an 426	. . 3 |- ( 3 lcm 2 ) e. CC
9	4, 5	pm3.2i 272	. . . . 5 |- ( 3 e. ZZ /\ 2 e. ZZ )
10		2ne0 9346	. . . . . . 7 |- 2 =/= 0
11	10	neii 2416	. . . . . 6 |- -. 2 = 0
12	11	intnan 937	. . . . 5 |- -. ( 3 = 0 /\ 2 = 0 )
13		gcdn0cl 12683	. . . . . 6 |- ( ( ( 3 e. ZZ /\ 2 e. ZZ ) /\ -. ( 3 = 0 /\ 2 = 0 ) ) -> ( 3 gcd 2 ) e. NN )
14	13	nncnd 9268	. . . . 5 |- ( ( ( 3 e. ZZ /\ 2 e. ZZ ) /\ -. ( 3 = 0 /\ 2 = 0 ) ) -> ( 3 gcd 2 ) e. CC )
15	9, 12, 14	mp2an 426	. . . 4 |- ( 3 gcd 2 ) e. CC
16	9, 12, 13	mp2an 426	. . . . . 6 |- ( 3 gcd 2 ) e. NN
17	16	nnne0i 9286	. . . . 5 |- ( 3 gcd 2 ) =/= 0
18	16	nnzi 9615	. . . . . 6 |- ( 3 gcd 2 ) e. ZZ
19		0z 9605	. . . . . 6 |- 0 e. ZZ
20		zapne 9669	. . . . . 6 |- ( ( ( 3 gcd 2 ) e. ZZ /\ 0 e. ZZ ) -> ( ( 3 gcd 2 ) # 0 <-> ( 3 gcd 2 ) =/= 0 ) )
21	18, 19, 20	mp2an 426	. . . . 5 |- ( ( 3 gcd 2 ) # 0 <-> ( 3 gcd 2 ) =/= 0 )
22	17, 21	mpbir 146	. . . 4 |- ( 3 gcd 2 ) # 0
23	15, 22	pm3.2i 272	. . 3 |- ( ( 3 gcd 2 ) e. CC /\ ( 3 gcd 2 ) # 0 )
24		3nn 9417	. . . . . . 7 |- 3 e. NN
25		2nn 9416	. . . . . . 7 |- 2 e. NN
26	24, 25	pm3.2i 272	. . . . . 6 |- ( 3 e. NN /\ 2 e. NN )
27		lcmgcdnn 12804	. . . . . . 7 |- ( ( 3 e. NN /\ 2 e. NN ) -> ( ( 3 lcm 2 ) x. ( 3 gcd 2 ) ) = ( 3 x. 2 ) )
28	27	eqcomd 2240	. . . . . 6 |- ( ( 3 e. NN /\ 2 e. NN ) -> ( 3 x. 2 ) = ( ( 3 lcm 2 ) x. ( 3 gcd 2 ) ) )
29	26, 28	mp1i 10	. . . . 5 |- ( ( ( 3 x. 2 ) e. CC /\ ( 3 lcm 2 ) e. CC /\ ( ( 3 gcd 2 ) e. CC /\ ( 3 gcd 2 ) # 0 ) ) -> ( 3 x. 2 ) = ( ( 3 lcm 2 ) x. ( 3 gcd 2 ) ) )
30		divmulap3 8968	. . . . 5 |- ( ( ( 3 x. 2 ) e. CC /\ ( 3 lcm 2 ) e. CC /\ ( ( 3 gcd 2 ) e. CC /\ ( 3 gcd 2 ) # 0 ) ) -> ( ( ( 3 x. 2 ) / ( 3 gcd 2 ) ) = ( 3 lcm 2 ) <-> ( 3 x. 2 ) = ( ( 3 lcm 2 ) x. ( 3 gcd 2 ) ) ) )
31	29, 30	mpbird 167	. . . 4 |- ( ( ( 3 x. 2 ) e. CC /\ ( 3 lcm 2 ) e. CC /\ ( ( 3 gcd 2 ) e. CC /\ ( 3 gcd 2 ) # 0 ) ) -> ( ( 3 x. 2 ) / ( 3 gcd 2 ) ) = ( 3 lcm 2 ) )
32	31	eqcomd 2240	. . 3 |- ( ( ( 3 x. 2 ) e. CC /\ ( 3 lcm 2 ) e. CC /\ ( ( 3 gcd 2 ) e. CC /\ ( 3 gcd 2 ) # 0 ) ) -> ( 3 lcm 2 ) = ( ( 3 x. 2 ) / ( 3 gcd 2 ) ) )
33	3, 8, 23, 32	mp3an 1374	. 2 |- ( 3 lcm 2 ) = ( ( 3 x. 2 ) / ( 3 gcd 2 ) )
34		gcdcom 12694	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> ( 3 gcd 2 ) = ( 2 gcd 3 ) )
35	4, 5, 34	mp2an 426	. . . 4 |- ( 3 gcd 2 ) = ( 2 gcd 3 )
36		1z 9620	. . . . . . . . 9 |- 1 e. ZZ
37		gcdid 12707	. . . . . . . . 9 |- ( 1 e. ZZ -> ( 1 gcd 1 ) = ( abs ` 1 ) )
38	36, 37	ax-mp 5	. . . . . . . 8 |- ( 1 gcd 1 ) = ( abs ` 1 )
39		abs1 11782	. . . . . . . 8 |- ( abs ` 1 ) = 1
40	38, 39	eqtr2i 2256	. . . . . . 7 |- 1 = ( 1 gcd 1 )
41		gcdadd 12706	. . . . . . . 8 |- ( ( 1 e. ZZ /\ 1 e. ZZ ) -> ( 1 gcd 1 ) = ( 1 gcd ( 1 + 1 ) ) )
42	36, 36, 41	mp2an 426	. . . . . . 7 |- ( 1 gcd 1 ) = ( 1 gcd ( 1 + 1 ) )
43		1p1e2 9371	. . . . . . . 8 |- ( 1 + 1 ) = 2
44	43	oveq2i 6069	. . . . . . 7 |- ( 1 gcd ( 1 + 1 ) ) = ( 1 gcd 2 )
45	40, 42, 44	3eqtri 2259	. . . . . 6 |- 1 = ( 1 gcd 2 )
46		gcdcom 12694	. . . . . . 7 |- ( ( 1 e. ZZ /\ 2 e. ZZ ) -> ( 1 gcd 2 ) = ( 2 gcd 1 ) )
47	36, 5, 46	mp2an 426	. . . . . 6 |- ( 1 gcd 2 ) = ( 2 gcd 1 )
48		gcdadd 12706	. . . . . . 7 |- ( ( 2 e. ZZ /\ 1 e. ZZ ) -> ( 2 gcd 1 ) = ( 2 gcd ( 1 + 2 ) ) )
49	5, 36, 48	mp2an 426	. . . . . 6 |- ( 2 gcd 1 ) = ( 2 gcd ( 1 + 2 ) )
50	45, 47, 49	3eqtri 2259	. . . . 5 |- 1 = ( 2 gcd ( 1 + 2 ) )
51		1p2e3 9389	. . . . . 6 |- ( 1 + 2 ) = 3
52	51	oveq2i 6069	. . . . 5 |- ( 2 gcd ( 1 + 2 ) ) = ( 2 gcd 3 )
53	50, 52	eqtr2i 2256	. . . 4 |- ( 2 gcd 3 ) = 1
54	35, 53	eqtri 2255	. . 3 |- ( 3 gcd 2 ) = 1
55	54	oveq2i 6069	. 2 |- ( ( 3 x. 2 ) / ( 3 gcd 2 ) ) = ( ( 3 x. 2 ) / 1 )
56		3t2e6 9411	. . . 4 |- ( 3 x. 2 ) = 6
57	56	oveq1i 6068	. . 3 |- ( ( 3 x. 2 ) / 1 ) = ( 6 / 1 )
58		6cn 9336	. . . 4 |- 6 e. CC
59	58	div1i 9031	. . 3 |- ( 6 / 1 ) = 6
60	57, 59	eqtri 2255	. 2 |- ( ( 3 x. 2 ) / 1 ) = 6
61	33, 55, 60	3eqtri 2259	1 |- ( 3 lcm 2 ) = 6

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   # cap 8872   / cdiv 8963   NNcn 9254   2c2 9305   3c3 9306   6c6 9309   ZZcz 9594   abscabs 11707   gcd cgcd 12674   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-stab 839  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-5 9316  df-6 9317  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-gcd 12675  df-lcm 12783

This theorem is referenced by: (None)