Theorem 3lcm2e6woprm 12088

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 8996	. . . 4 |- 3 e. CC
2		2cn 8992	. . . 4 |- 2 e. CC
3	1, 2	mulcli 7964	. . 3 |- ( 3 x. 2 ) e. CC
4		3z 9284	. . . 4 |- 3 e. ZZ
5		2z 9283	. . . 4 |- 2 e. ZZ
6		lcmcl 12074	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> ( 3 lcm 2 ) e. NN0 )
7	6	nn0cnd 9233	. . . 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 9013	. . . . . . 7 |- 2 =/= 0
11	10	neii 2349	. . . . . 6 |- -. 2 = 0
12	11	intnan 929	. . . . 5 |- -. ( 3 = 0 /\ 2 = 0 )
13		gcdn0cl 11965	. . . . . 6 |- ( ( ( 3 e. ZZ /\ 2 e. ZZ ) /\ -. ( 3 = 0 /\ 2 = 0 ) ) -> ( 3 gcd 2 ) e. NN )
14	13	nncnd 8935	. . . . 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 8953	. . . . 5 |- ( 3 gcd 2 ) =/= 0
18	16	nnzi 9276	. . . . . 6 |- ( 3 gcd 2 ) e. ZZ
19		0z 9266	. . . . . 6 |- 0 e. ZZ
20		zapne 9329	. . . . . 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 9083	. . . . . . 7 |- 3 e. NN
25		2nn 9082	. . . . . . 7 |- 2 e. NN
26	24, 25	pm3.2i 272	. . . . . 6 |- ( 3 e. NN /\ 2 e. NN )
27		lcmgcdnn 12084	. . . . . . 7 |- ( ( 3 e. NN /\ 2 e. NN ) -> ( ( 3 lcm 2 ) x. ( 3 gcd 2 ) ) = ( 3 x. 2 ) )
28	27	eqcomd 2183	. . . . . 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 8636	. . . . 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 2183	. . 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 1337	. 2 |- ( 3 lcm 2 ) = ( ( 3 x. 2 ) / ( 3 gcd 2 ) )
34		gcdcom 11976	. . . . 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 9281	. . . . . . . . 9 |- 1 e. ZZ
37		gcdid 11989	. . . . . . . . 9 |- ( 1 e. ZZ -> ( 1 gcd 1 ) = ( abs ` 1 ) )
38	36, 37	ax-mp 5	. . . . . . . 8 |- ( 1 gcd 1 ) = ( abs ` 1 )
39		abs1 11083	. . . . . . . 8 |- ( abs ` 1 ) = 1
40	38, 39	eqtr2i 2199	. . . . . . 7 |- 1 = ( 1 gcd 1 )
41		gcdadd 11988	. . . . . . . 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 9038	. . . . . . . 8 |- ( 1 + 1 ) = 2
44	43	oveq2i 5888	. . . . . . 7 |- ( 1 gcd ( 1 + 1 ) ) = ( 1 gcd 2 )
45	40, 42, 44	3eqtri 2202	. . . . . 6 |- 1 = ( 1 gcd 2 )
46		gcdcom 11976	. . . . . . 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 11988	. . . . . . 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 2202	. . . . 5 |- 1 = ( 2 gcd ( 1 + 2 ) )
51		1p2e3 9055	. . . . . 6 |- ( 1 + 2 ) = 3
52	51	oveq2i 5888	. . . . 5 |- ( 2 gcd ( 1 + 2 ) ) = ( 2 gcd 3 )
53	50, 52	eqtr2i 2199	. . . 4 |- ( 2 gcd 3 ) = 1
54	35, 53	eqtri 2198	. . 3 |- ( 3 gcd 2 ) = 1
55	54	oveq2i 5888	. 2 |- ( ( 3 x. 2 ) / ( 3 gcd 2 ) ) = ( ( 3 x. 2 ) / 1 )
56		3t2e6 9077	. . . 4 |- ( 3 x. 2 ) = 6
57	56	oveq1i 5887	. . 3 |- ( ( 3 x. 2 ) / 1 ) = ( 6 / 1 )
58		6cn 9003	. . . 4 |- 6 e. CC
59	58	div1i 8699	. . 3 |- ( 6 / 1 ) = 6
60	57, 59	eqtri 2198	. 2 |- ( ( 3 x. 2 ) / 1 ) = 6
61	33, 55, 60	3eqtri 2202	1 |- ( 3 lcm 2 ) = 6

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   CCcc 7811   0cc0 7813   1c1 7814   + caddc 7816   x. cmul 7818   # cap 8540   / cdiv 8631   NNcn 8921   2c2 8972   3c3 8973   6c6 8976   ZZcz 9255   abscabs 11008   gcd cgcd 11945   lcm clcm 12062

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946  df-lcm 12063

This theorem is referenced by: (None)