Theorem 3lcm2e6woprm 14629

Description: The least common multiple of three and two is six. In contrast to 3lcm2e6 14730, this proof does not use the property of 2 and 3 being prime, therefore it is much longer. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 27-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem 3lcm2e6woprm

Step	Hyp	Ref	Expression
1		3cn 10712	. . . 4 |- 3 e. CC
2		2cn 10708	. . . 4 |- 2 e. CC
3	1, 2	mulcli 9674	. . 3 |- ( 3 x. 2 ) e. CC
4		3z 10999	. . . 4 |- 3 e. ZZ
5		2z 10998	. . . 4 |- 2 e. ZZ
6		lcmcl 14615	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> ( 3 lcm 2 ) e. NN0 )
7	6	nn0cnd 10956	. . . 4 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> ( 3 lcm 2 ) e. CC )
8	4, 5, 7	mp2an 683	. . 3 |- ( 3 lcm 2 ) e. CC
9	4, 5	pm3.2i 461	. . . . 5 |- ( 3 e. ZZ /\ 2 e. ZZ )
10		2ne0 10730	. . . . . . 7 |- 2 =/= 0
11	10	neii 2637	. . . . . 6 |- -. 2 = 0
12	11	intnan 930	. . . . 5 |- -. ( 3 = 0 /\ 2 = 0 )
13		gcdn0cl 14525	. . . . . 6 |- ( ( ( 3 e. ZZ /\ 2 e. ZZ ) /\ -. ( 3 = 0 /\ 2 = 0 ) ) -> ( 3 gcd 2 ) e. NN )
14	13	nncnd 10653	. . . . 5 |- ( ( ( 3 e. ZZ /\ 2 e. ZZ ) /\ -. ( 3 = 0 /\ 2 = 0 ) ) -> ( 3 gcd 2 ) e. CC )
15	9, 12, 14	mp2an 683	. . . 4 |- ( 3 gcd 2 ) e. CC
16	9, 12, 13	mp2an 683	. . . . 5 |- ( 3 gcd 2 ) e. NN
17	16	nnne0i 10672	. . . 4 |- ( 3 gcd 2 ) =/= 0
18	15, 17	pm3.2i 461	. . 3 |- ( ( 3 gcd 2 ) e. CC /\ ( 3 gcd 2 ) =/= 0 )
19		3nn 10797	. . . . . . 7 |- 3 e. NN
20		2nn 10796	. . . . . . 7 |- 2 e. NN
21	19, 20	pm3.2i 461	. . . . . 6 |- ( 3 e. NN /\ 2 e. NN )
22		lcmgcdnn 14625	. . . . . . 7 |- ( ( 3 e. NN /\ 2 e. NN ) -> ( ( 3 lcm 2 ) x. ( 3 gcd 2 ) ) = ( 3 x. 2 ) )
23	22	eqcomd 2468	. . . . . 6 |- ( ( 3 e. NN /\ 2 e. NN ) -> ( 3 x. 2 ) = ( ( 3 lcm 2 ) x. ( 3 gcd 2 ) ) )
24	21, 23	mp1i 13	. . . . 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 ) ) )
25		divmul3 10303	. . . . 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 ) ) ) )
26	24, 25	mpbird 240	. . . 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 ) )
27	26	eqcomd 2468	. . 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 ) ) )
28	3, 8, 18, 27	mp3an 1373	. 2 |- ( 3 lcm 2 ) = ( ( 3 x. 2 ) / ( 3 gcd 2 ) )
29		gcdcom 14533	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> ( 3 gcd 2 ) = ( 2 gcd 3 ) )
30	4, 5, 29	mp2an 683	. . . 4 |- ( 3 gcd 2 ) = ( 2 gcd 3 )
31		1z 10996	. . . . . . . . 9 |- 1 e. ZZ
32		gcdid 14544	. . . . . . . . 9 |- ( 1 e. ZZ -> ( 1 gcd 1 ) = ( abs ` 1 ) )
33	31, 32	ax-mp 5	. . . . . . . 8 |- ( 1 gcd 1 ) = ( abs ` 1 )
34		abs1 13409	. . . . . . . 8 |- ( abs ` 1 ) = 1
35	33, 34	eqtr2i 2485	. . . . . . 7 |- 1 = ( 1 gcd 1 )
36		gcdadd 14543	. . . . . . . 8 |- ( ( 1 e. ZZ /\ 1 e. ZZ ) -> ( 1 gcd 1 ) = ( 1 gcd ( 1 + 1 ) ) )
37	31, 31, 36	mp2an 683	. . . . . . 7 |- ( 1 gcd 1 ) = ( 1 gcd ( 1 + 1 ) )
38		1p1e2 10751	. . . . . . . 8 |- ( 1 + 1 ) = 2
39	38	oveq2i 6326	. . . . . . 7 |- ( 1 gcd ( 1 + 1 ) ) = ( 1 gcd 2 )
40	35, 37, 39	3eqtri 2488	. . . . . 6 |- 1 = ( 1 gcd 2 )
41		gcdcom 14533	. . . . . . 7 |- ( ( 1 e. ZZ /\ 2 e. ZZ ) -> ( 1 gcd 2 ) = ( 2 gcd 1 ) )
42	31, 5, 41	mp2an 683	. . . . . 6 |- ( 1 gcd 2 ) = ( 2 gcd 1 )
43		gcdadd 14543	. . . . . . 7 |- ( ( 2 e. ZZ /\ 1 e. ZZ ) -> ( 2 gcd 1 ) = ( 2 gcd ( 1 + 2 ) ) )
44	5, 31, 43	mp2an 683	. . . . . 6 |- ( 2 gcd 1 ) = ( 2 gcd ( 1 + 2 ) )
45	40, 42, 44	3eqtri 2488	. . . . 5 |- 1 = ( 2 gcd ( 1 + 2 ) )
46		1p2e3 10763	. . . . . 6 |- ( 1 + 2 ) = 3
47	46	oveq2i 6326	. . . . 5 |- ( 2 gcd ( 1 + 2 ) ) = ( 2 gcd 3 )
48	45, 47	eqtr2i 2485	. . . 4 |- ( 2 gcd 3 ) = 1
49	30, 48	eqtri 2484	. . 3 |- ( 3 gcd 2 ) = 1
50	49	oveq2i 6326	. 2 |- ( ( 3 x. 2 ) / ( 3 gcd 2 ) ) = ( ( 3 x. 2 ) / 1 )
51		3t2e6 10790	. . . 4 |- ( 3 x. 2 ) = 6
52	51	oveq1i 6325	. . 3 |- ( ( 3 x. 2 ) / 1 ) = ( 6 / 1 )
53		6cn 10719	. . . 4 |- 6 e. CC
54	53	div1i 10363	. . 3 |- ( 6 / 1 ) = 6
55	52, 54	eqtri 2484	. 2 |- ( ( 3 x. 2 ) / 1 ) = 6
56	28, 50, 55	3eqtri 2488	1 |- ( 3 lcm 2 ) = 6

Syntax hints:   -. wn 3   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   =/= wne 2633   ` cfv 5601  (class class class)co 6315   CCcc 9563   0cc0 9565   1c1 9566   + caddc 9568   x. cmul 9570   / cdiv 10297   NNcn 10637   2c2 10687   3c3 10688   6c6 10691   ZZcz 10966   abscabs 13346   gcd cgcd 14517   lcm clcm 14596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-sup 7982  df-inf 7983  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-n0 10899  df-z 10967  df-uz 11189  df-rp 11332  df-fl 12060  df-mod 12129  df-seq 12246  df-exp 12305  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-dvds 14355  df-gcd 14518  df-lcm 14600

This theorem is referenced by:  lcmf2a3a4e12  14669