Theorem 3lcm2e6woprm 11560

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 8653	. . . 4 |- 3 e. CC
2		2cn 8649	. . . 4 |- 2 e. CC
3	1, 2	mulcli 7643	. . 3 |- ( 3 x. 2 ) e. CC
4		3z 8935	. . . 4 |- 3 e. ZZ
5		2z 8934	. . . 4 |- 2 e. ZZ
6		lcmcl 11546	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> ( 3 lcm 2 ) e. NN0 )
7	6	nn0cnd 8884	. . . 4 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> ( 3 lcm 2 ) e. CC )
8	4, 5, 7	mp2an 420	. . 3 |- ( 3 lcm 2 ) e. CC
9	4, 5	pm3.2i 268	. . . . 5 |- ( 3 e. ZZ /\ 2 e. ZZ )
10		2ne0 8670	. . . . . . 7 |- 2 =/= 0
11	10	neii 2269	. . . . . 6 |- -. 2 = 0
12	11	intnan 882	. . . . 5 |- -. ( 3 = 0 /\ 2 = 0 )
13		gcdn0cl 11446	. . . . . 6 |- ( ( ( 3 e. ZZ /\ 2 e. ZZ ) /\ -. ( 3 = 0 /\ 2 = 0 ) ) -> ( 3 gcd 2 ) e. NN )
14	13	nncnd 8592	. . . . 5 |- ( ( ( 3 e. ZZ /\ 2 e. ZZ ) /\ -. ( 3 = 0 /\ 2 = 0 ) ) -> ( 3 gcd 2 ) e. CC )
15	9, 12, 14	mp2an 420	. . . 4 |- ( 3 gcd 2 ) e. CC
16	9, 12, 13	mp2an 420	. . . . . 6 |- ( 3 gcd 2 ) e. NN
17	16	nnne0i 8610	. . . . 5 |- ( 3 gcd 2 ) =/= 0
18	16	nnzi 8927	. . . . . 6 |- ( 3 gcd 2 ) e. ZZ
19		0z 8917	. . . . . 6 |- 0 e. ZZ
20		zapne 8977	. . . . . 6 |- ( ( ( 3 gcd 2 ) e. ZZ /\ 0 e. ZZ ) -> ( ( 3 gcd 2 ) # 0 <-> ( 3 gcd 2 ) =/= 0 ) )
21	18, 19, 20	mp2an 420	. . . . 5 |- ( ( 3 gcd 2 ) # 0 <-> ( 3 gcd 2 ) =/= 0 )
22	17, 21	mpbir 145	. . . 4 |- ( 3 gcd 2 ) # 0
23	15, 22	pm3.2i 268	. . 3 |- ( ( 3 gcd 2 ) e. CC /\ ( 3 gcd 2 ) # 0 )
24		3nn 8734	. . . . . . 7 |- 3 e. NN
25		2nn 8733	. . . . . . 7 |- 2 e. NN
26	24, 25	pm3.2i 268	. . . . . 6 |- ( 3 e. NN /\ 2 e. NN )
27		lcmgcdnn 11556	. . . . . . 7 |- ( ( 3 e. NN /\ 2 e. NN ) -> ( ( 3 lcm 2 ) x. ( 3 gcd 2 ) ) = ( 3 x. 2 ) )
28	27	eqcomd 2105	. . . . . 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 8298	. . . . 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 166	. . . 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 2105	. . 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 1283	. 2 |- ( 3 lcm 2 ) = ( ( 3 x. 2 ) / ( 3 gcd 2 ) )
34		gcdcom 11457	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> ( 3 gcd 2 ) = ( 2 gcd 3 ) )
35	4, 5, 34	mp2an 420	. . . 4 |- ( 3 gcd 2 ) = ( 2 gcd 3 )
36		1z 8932	. . . . . . . . 9 |- 1 e. ZZ
37		gcdid 11469	. . . . . . . . 9 |- ( 1 e. ZZ -> ( 1 gcd 1 ) = ( abs ` 1 ) )
38	36, 37	ax-mp 7	. . . . . . . 8 |- ( 1 gcd 1 ) = ( abs ` 1 )
39		abs1 10684	. . . . . . . 8 |- ( abs ` 1 ) = 1
40	38, 39	eqtr2i 2121	. . . . . . 7 |- 1 = ( 1 gcd 1 )
41		gcdadd 11468	. . . . . . . 8 |- ( ( 1 e. ZZ /\ 1 e. ZZ ) -> ( 1 gcd 1 ) = ( 1 gcd ( 1 + 1 ) ) )
42	36, 36, 41	mp2an 420	. . . . . . 7 |- ( 1 gcd 1 ) = ( 1 gcd ( 1 + 1 ) )
43		1p1e2 8695	. . . . . . . 8 |- ( 1 + 1 ) = 2
44	43	oveq2i 5717	. . . . . . 7 |- ( 1 gcd ( 1 + 1 ) ) = ( 1 gcd 2 )
45	40, 42, 44	3eqtri 2124	. . . . . 6 |- 1 = ( 1 gcd 2 )
46		gcdcom 11457	. . . . . . 7 |- ( ( 1 e. ZZ /\ 2 e. ZZ ) -> ( 1 gcd 2 ) = ( 2 gcd 1 ) )
47	36, 5, 46	mp2an 420	. . . . . 6 |- ( 1 gcd 2 ) = ( 2 gcd 1 )
48		gcdadd 11468	. . . . . . 7 |- ( ( 2 e. ZZ /\ 1 e. ZZ ) -> ( 2 gcd 1 ) = ( 2 gcd ( 1 + 2 ) ) )
49	5, 36, 48	mp2an 420	. . . . . 6 |- ( 2 gcd 1 ) = ( 2 gcd ( 1 + 2 ) )
50	45, 47, 49	3eqtri 2124	. . . . 5 |- 1 = ( 2 gcd ( 1 + 2 ) )
51		1p2e3 8706	. . . . . 6 |- ( 1 + 2 ) = 3
52	51	oveq2i 5717	. . . . 5 |- ( 2 gcd ( 1 + 2 ) ) = ( 2 gcd 3 )
53	50, 52	eqtr2i 2121	. . . 4 |- ( 2 gcd 3 ) = 1
54	35, 53	eqtri 2120	. . 3 |- ( 3 gcd 2 ) = 1
55	54	oveq2i 5717	. 2 |- ( ( 3 x. 2 ) / ( 3 gcd 2 ) ) = ( ( 3 x. 2 ) / 1 )
56		3t2e6 8728	. . . 4 |- ( 3 x. 2 ) = 6
57	56	oveq1i 5716	. . 3 |- ( ( 3 x. 2 ) / 1 ) = ( 6 / 1 )
58		6cn 8660	. . . 4 |- 6 e. CC
59	58	div1i 8361	. . 3 |- ( 6 / 1 ) = 6
60	57, 59	eqtri 2120	. 2 |- ( ( 3 x. 2 ) / 1 ) = 6
61	33, 55, 60	3eqtri 2124	1 |- ( 3 lcm 2 ) = 6

Syntax hints:   -. wn 3   /\ wa 103   <-> wb 104   /\ w3a 930   = wceq 1299   e. wcel 1448   =/= wne 2267   class class class wbr 3875   ` cfv 5059  (class class class)co 5706   CCcc 7498   0cc0 7500   1c1 7501   + caddc 7503   x. cmul 7505   # cap 8209   / cdiv 8293   NNcn 8578   2c2 8629   3c3 8630   6c6 8633   ZZcz 8906   abscabs 10609   gcd cgcd 11430   lcm clcm 11534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-sup 6786  df-inf 6787  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-5 8640  df-6 8641  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-fz 9632  df-fzo 9761  df-fl 9884  df-mod 9937  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-dvds 11289  df-gcd 11431  df-lcm 11535

This theorem is referenced by: (None)