Theorem 3lcm2e6woprm 12721

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 9260	. . . 4 |- 3 e. CC
2		2cn 9256	. . . 4 |- 2 e. CC
3	1, 2	mulcli 8227	. . 3 |- ( 3 x. 2 ) e. CC
4		3z 9552	. . . 4 |- 3 e. ZZ
5		2z 9551	. . . 4 |- 2 e. ZZ
6		lcmcl 12707	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> ( 3 lcm 2 ) e. NN0 )
7	6	nn0cnd 9501	. . . 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 9277	. . . . . . 7 |- 2 =/= 0
11	10	neii 2405	. . . . . 6 |- -. 2 = 0
12	11	intnan 937	. . . . 5 |- -. ( 3 = 0 /\ 2 = 0 )
13		gcdn0cl 12596	. . . . . 6 |- ( ( ( 3 e. ZZ /\ 2 e. ZZ ) /\ -. ( 3 = 0 /\ 2 = 0 ) ) -> ( 3 gcd 2 ) e. NN )
14	13	nncnd 9199	. . . . 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 9217	. . . . 5 |- ( 3 gcd 2 ) =/= 0
18	16	nnzi 9544	. . . . . 6 |- ( 3 gcd 2 ) e. ZZ
19		0z 9534	. . . . . 6 |- 0 e. ZZ
20		zapne 9598	. . . . . 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 9348	. . . . . . 7 |- 3 e. NN
25		2nn 9347	. . . . . . 7 |- 2 e. NN
26	24, 25	pm3.2i 272	. . . . . 6 |- ( 3 e. NN /\ 2 e. NN )
27		lcmgcdnn 12717	. . . . . . 7 |- ( ( 3 e. NN /\ 2 e. NN ) -> ( ( 3 lcm 2 ) x. ( 3 gcd 2 ) ) = ( 3 x. 2 ) )
28	27	eqcomd 2237	. . . . . 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 8899	. . . . 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 2237	. . 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 12607	. . . . 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 9549	. . . . . . . . 9 |- 1 e. ZZ
37		gcdid 12620	. . . . . . . . 9 |- ( 1 e. ZZ -> ( 1 gcd 1 ) = ( abs ` 1 ) )
38	36, 37	ax-mp 5	. . . . . . . 8 |- ( 1 gcd 1 ) = ( abs ` 1 )
39		abs1 11695	. . . . . . . 8 |- ( abs ` 1 ) = 1
40	38, 39	eqtr2i 2253	. . . . . . 7 |- 1 = ( 1 gcd 1 )
41		gcdadd 12619	. . . . . . . 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 9302	. . . . . . . 8 |- ( 1 + 1 ) = 2
44	43	oveq2i 6039	. . . . . . 7 |- ( 1 gcd ( 1 + 1 ) ) = ( 1 gcd 2 )
45	40, 42, 44	3eqtri 2256	. . . . . 6 |- 1 = ( 1 gcd 2 )
46		gcdcom 12607	. . . . . . 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 12619	. . . . . . 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 2256	. . . . 5 |- 1 = ( 2 gcd ( 1 + 2 ) )
51		1p2e3 9320	. . . . . 6 |- ( 1 + 2 ) = 3
52	51	oveq2i 6039	. . . . 5 |- ( 2 gcd ( 1 + 2 ) ) = ( 2 gcd 3 )
53	50, 52	eqtr2i 2253	. . . 4 |- ( 2 gcd 3 ) = 1
54	35, 53	eqtri 2252	. . 3 |- ( 3 gcd 2 ) = 1
55	54	oveq2i 6039	. 2 |- ( ( 3 x. 2 ) / ( 3 gcd 2 ) ) = ( ( 3 x. 2 ) / 1 )
56		3t2e6 9342	. . . 4 |- ( 3 x. 2 ) = 6
57	56	oveq1i 6038	. . 3 |- ( ( 3 x. 2 ) / 1 ) = ( 6 / 1 )
58		6cn 9267	. . . 4 |- 6 e. CC
59	58	div1i 8962	. . 3 |- ( 6 / 1 ) = 6
60	57, 59	eqtri 2252	. 2 |- ( ( 3 x. 2 ) / 1 ) = 6
61	33, 55, 60	3eqtri 2256	1 |- ( 3 lcm 2 ) = 6

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   =/= wne 2403   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   # cap 8803   / cdiv 8894   NNcn 9185   2c2 9236   3c3 9237   6c6 9240   ZZcz 9523   abscabs 11620   gcd cgcd 12587   lcm clcm 12695

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-sup 7226  df-inf 7227  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fz 10289  df-fzo 10423  df-fl 10576  df-mod 10631  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-dvds 12412  df-gcd 12588  df-lcm 12696

This theorem is referenced by: (None)