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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
StepHypRef Expression
1 3cn 10712 . . . 4  |-  3  e.  CC
2 2cn 10708 . . . 4  |-  2  e.  CC
31, 2mulcli 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 )
76nn0cnd 10956 . . . 4  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  ( 3 lcm  2 )  e.  CC )
84, 5, 7mp2an 683 . . 3  |-  ( 3 lcm  2 )  e.  CC
94, 5pm3.2i 461 . . . . 5  |-  ( 3  e.  ZZ  /\  2  e.  ZZ )
10 2ne0 10730 . . . . . . 7  |-  2  =/=  0
1110neii 2637 . . . . . 6  |-  -.  2  =  0
1211intnan 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 )
1413nncnd 10653 . . . . 5  |-  ( ( ( 3  e.  ZZ  /\  2  e.  ZZ )  /\  -.  ( 3  =  0  /\  2  =  0 ) )  ->  ( 3  gcd  2 )  e.  CC )
159, 12, 14mp2an 683 . . . 4  |-  ( 3  gcd  2 )  e.  CC
169, 12, 13mp2an 683 . . . . 5  |-  ( 3  gcd  2 )  e.  NN
1716nnne0i 10672 . . . 4  |-  ( 3  gcd  2 )  =/=  0
1815, 17pm3.2i 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
2119, 20pm3.2i 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 ) )
2322eqcomd 2468 . . . . . 6  |-  ( ( 3  e.  NN  /\  2  e.  NN )  ->  ( 3  x.  2 )  =  ( ( 3 lcm  2 )  x.  ( 3  gcd  2
) ) )
2421, 23mp1i 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 ) ) ) )
2624, 25mpbird 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 ) )
2726eqcomd 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 ) ) )
283, 8, 18, 27mp3an 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 ) )
304, 5, 29mp2an 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
) )
3331, 32ax-mp 5 . . . . . . . 8  |-  ( 1  gcd  1 )  =  ( abs `  1
)
34 abs1 13409 . . . . . . . 8  |-  ( abs `  1 )  =  1
3533, 34eqtr2i 2485 . . . . . . 7  |-  1  =  ( 1  gcd  1 )
36 gcdadd 14543 . . . . . . . 8  |-  ( ( 1  e.  ZZ  /\  1  e.  ZZ )  ->  ( 1  gcd  1
)  =  ( 1  gcd  ( 1  +  1 ) ) )
3731, 31, 36mp2an 683 . . . . . . 7  |-  ( 1  gcd  1 )  =  ( 1  gcd  (
1  +  1 ) )
38 1p1e2 10751 . . . . . . . 8  |-  ( 1  +  1 )  =  2
3938oveq2i 6326 . . . . . . 7  |-  ( 1  gcd  ( 1  +  1 ) )  =  ( 1  gcd  2
)
4035, 37, 393eqtri 2488 . . . . . 6  |-  1  =  ( 1  gcd  2 )
41 gcdcom 14533 . . . . . . 7  |-  ( ( 1  e.  ZZ  /\  2  e.  ZZ )  ->  ( 1  gcd  2
)  =  ( 2  gcd  1 ) )
4231, 5, 41mp2an 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 ) ) )
445, 31, 43mp2an 683 . . . . . 6  |-  ( 2  gcd  1 )  =  ( 2  gcd  (
1  +  2 ) )
4540, 42, 443eqtri 2488 . . . . 5  |-  1  =  ( 2  gcd  ( 1  +  2 ) )
46 1p2e3 10763 . . . . . 6  |-  ( 1  +  2 )  =  3
4746oveq2i 6326 . . . . 5  |-  ( 2  gcd  ( 1  +  2 ) )  =  ( 2  gcd  3
)
4845, 47eqtr2i 2485 . . . 4  |-  ( 2  gcd  3 )  =  1
4930, 48eqtri 2484 . . 3  |-  ( 3  gcd  2 )  =  1
5049oveq2i 6326 . 2  |-  ( ( 3  x.  2 )  /  ( 3  gcd  2 ) )  =  ( ( 3  x.  2 )  /  1
)
51 3t2e6 10790 . . . 4  |-  ( 3  x.  2 )  =  6
5251oveq1i 6325 . . 3  |-  ( ( 3  x.  2 )  /  1 )  =  ( 6  /  1
)
53 6cn 10719 . . . 4  |-  6  e.  CC
5453div1i 10363 . . 3  |-  ( 6  /  1 )  =  6
5552, 54eqtri 2484 . 2  |-  ( ( 3  x.  2 )  /  1 )  =  6
5628, 50, 553eqtri 2488 1  |-  ( 3 lcm  2 )  =  6
Colors of variables: wff setvar class
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
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