Theorem 3lcm2e6woprm 12069

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 8983	. . . 4 |- 3 e. CC
2		2cn 8979	. . . 4 |- 2 e. CC
3	1, 2	mulcli 7953	. . 3 |- ( 3 x. 2 ) e. CC
4		3z 9271	. . . 4 |- 3 e. ZZ
5		2z 9270	. . . 4 |- 2 e. ZZ
6		lcmcl 12055	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> ( 3 lcm 2 ) e. NN0 )
7	6	nn0cnd 9220	. . . 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 9000	. . . . . . 7 |- 2 =/= 0
11	10	neii 2349	. . . . . 6 |- -. 2 = 0
12	11	intnan 929	. . . . 5 |- -. ( 3 = 0 /\ 2 = 0 )
13		gcdn0cl 11946	. . . . . 6 |- ( ( ( 3 e. ZZ /\ 2 e. ZZ ) /\ -. ( 3 = 0 /\ 2 = 0 ) ) -> ( 3 gcd 2 ) e. NN )
14	13	nncnd 8922	. . . . 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 8940	. . . . 5 |- ( 3 gcd 2 ) =/= 0
18	16	nnzi 9263	. . . . . 6 |- ( 3 gcd 2 ) e. ZZ
19		0z 9253	. . . . . 6 |- 0 e. ZZ
20		zapne 9316	. . . . . 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 9070	. . . . . . 7 |- 3 e. NN
25		2nn 9069	. . . . . . 7 |- 2 e. NN
26	24, 25	pm3.2i 272	. . . . . 6 |- ( 3 e. NN /\ 2 e. NN )
27		lcmgcdnn 12065	. . . . . . 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 8623	. . . . 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 11957	. . . . 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 9268	. . . . . . . . 9 |- 1 e. ZZ
37		gcdid 11970	. . . . . . . . 9 |- ( 1 e. ZZ -> ( 1 gcd 1 ) = ( abs ` 1 ) )
38	36, 37	ax-mp 5	. . . . . . . 8 |- ( 1 gcd 1 ) = ( abs ` 1 )
39		abs1 11065	. . . . . . . 8 |- ( abs ` 1 ) = 1
40	38, 39	eqtr2i 2199	. . . . . . 7 |- 1 = ( 1 gcd 1 )
41		gcdadd 11969	. . . . . . . 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 9025	. . . . . . . 8 |- ( 1 + 1 ) = 2
44	43	oveq2i 5880	. . . . . . 7 |- ( 1 gcd ( 1 + 1 ) ) = ( 1 gcd 2 )
45	40, 42, 44	3eqtri 2202	. . . . . 6 |- 1 = ( 1 gcd 2 )
46		gcdcom 11957	. . . . . . 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 11969	. . . . . . 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 9042	. . . . . 6 |- ( 1 + 2 ) = 3
52	51	oveq2i 5880	. . . . 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 5880	. 2 |- ( ( 3 x. 2 ) / ( 3 gcd 2 ) ) = ( ( 3 x. 2 ) / 1 )
56		3t2e6 9064	. . . 4 |- ( 3 x. 2 ) = 6
57	56	oveq1i 5879	. . 3 |- ( ( 3 x. 2 ) / 1 ) = ( 6 / 1 )
58		6cn 8990	. . . 4 |- 6 e. CC
59	58	div1i 8686	. . 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 4000   ` cfv 5212  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   # cap 8528   / cdiv 8618   NNcn 8908   2c2 8959   3c3 8960   6c6 8963   ZZcz 9242   abscabs 10990   gcd cgcd 11926   lcm clcm 12043

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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927  df-lcm 12044

This theorem is referenced by: (None)