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Theorem 3lcm2e6woprm 11560
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
StepHypRef Expression
1 3cn 8653 . . . 4  |-  3  e.  CC
2 2cn 8649 . . . 4  |-  2  e.  CC
31, 2mulcli 7643 . . 3  |-  ( 3  x.  2 )  e.  CC
4 3z 8935 . . . 4  |-  3  e.  ZZ
5 2z 8934 . . . 4  |-  2  e.  ZZ
6 lcmcl 11546 . . . . 5  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  ( 3 lcm  2 )  e.  NN0 )
76nn0cnd 8884 . . . 4  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  ( 3 lcm  2 )  e.  CC )
84, 5, 7mp2an 420 . . 3  |-  ( 3 lcm  2 )  e.  CC
94, 5pm3.2i 268 . . . . 5  |-  ( 3  e.  ZZ  /\  2  e.  ZZ )
10 2ne0 8670 . . . . . . 7  |-  2  =/=  0
1110neii 2269 . . . . . 6  |-  -.  2  =  0
1211intnan 882 . . . . 5  |-  -.  (
3  =  0  /\  2  =  0 )
13 gcdn0cl 11446 . . . . . 6  |-  ( ( ( 3  e.  ZZ  /\  2  e.  ZZ )  /\  -.  ( 3  =  0  /\  2  =  0 ) )  ->  ( 3  gcd  2 )  e.  NN )
1413nncnd 8592 . . . . 5  |-  ( ( ( 3  e.  ZZ  /\  2  e.  ZZ )  /\  -.  ( 3  =  0  /\  2  =  0 ) )  ->  ( 3  gcd  2 )  e.  CC )
159, 12, 14mp2an 420 . . . 4  |-  ( 3  gcd  2 )  e.  CC
169, 12, 13mp2an 420 . . . . . 6  |-  ( 3  gcd  2 )  e.  NN
1716nnne0i 8610 . . . . 5  |-  ( 3  gcd  2 )  =/=  0
1816nnzi 8927 . . . . . 6  |-  ( 3  gcd  2 )  e.  ZZ
19 0z 8917 . . . . . 6  |-  0  e.  ZZ
20 zapne 8977 . . . . . 6  |-  ( ( ( 3  gcd  2
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( 3  gcd  2 ) #  0  <->  (
3  gcd  2 )  =/=  0 ) )
2118, 19, 20mp2an 420 . . . . 5  |-  ( ( 3  gcd  2 ) #  0  <->  ( 3  gcd  2 )  =/=  0
)
2217, 21mpbir 145 . . . 4  |-  ( 3  gcd  2 ) #  0
2315, 22pm3.2i 268 . . 3  |-  ( ( 3  gcd  2 )  e.  CC  /\  (
3  gcd  2 ) #  0 )
24 3nn 8734 . . . . . . 7  |-  3  e.  NN
25 2nn 8733 . . . . . . 7  |-  2  e.  NN
2624, 25pm3.2i 268 . . . . . 6  |-  ( 3  e.  NN  /\  2  e.  NN )
27 lcmgcdnn 11556 . . . . . . 7  |-  ( ( 3  e.  NN  /\  2  e.  NN )  ->  ( ( 3 lcm  2 )  x.  ( 3  gcd  2 ) )  =  ( 3  x.  2 ) )
2827eqcomd 2105 . . . . . 6  |-  ( ( 3  e.  NN  /\  2  e.  NN )  ->  ( 3  x.  2 )  =  ( ( 3 lcm  2 )  x.  ( 3  gcd  2
) ) )
2926, 28mp1i 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 8298 . . . . 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
) ) ) )
3129, 30mpbird 166 . . . 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 ) )
3231eqcomd 2105 . . 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 ) ) )
333, 8, 23, 32mp3an 1283 . 2  |-  ( 3 lcm  2 )  =  ( ( 3  x.  2 )  /  ( 3  gcd  2 ) )
34 gcdcom 11457 . . . . 5  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  ( 3  gcd  2
)  =  ( 2  gcd  3 ) )
354, 5, 34mp2an 420 . . . 4  |-  ( 3  gcd  2 )  =  ( 2  gcd  3
)
36 1z 8932 . . . . . . . . 9  |-  1  e.  ZZ
37 gcdid 11469 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  (
1  gcd  1 )  =  ( abs `  1
) )
3836, 37ax-mp 7 . . . . . . . 8  |-  ( 1  gcd  1 )  =  ( abs `  1
)
39 abs1 10684 . . . . . . . 8  |-  ( abs `  1 )  =  1
4038, 39eqtr2i 2121 . . . . . . 7  |-  1  =  ( 1  gcd  1 )
41 gcdadd 11468 . . . . . . . 8  |-  ( ( 1  e.  ZZ  /\  1  e.  ZZ )  ->  ( 1  gcd  1
)  =  ( 1  gcd  ( 1  +  1 ) ) )
4236, 36, 41mp2an 420 . . . . . . 7  |-  ( 1  gcd  1 )  =  ( 1  gcd  (
1  +  1 ) )
43 1p1e2 8695 . . . . . . . 8  |-  ( 1  +  1 )  =  2
4443oveq2i 5717 . . . . . . 7  |-  ( 1  gcd  ( 1  +  1 ) )  =  ( 1  gcd  2
)
4540, 42, 443eqtri 2124 . . . . . 6  |-  1  =  ( 1  gcd  2 )
46 gcdcom 11457 . . . . . . 7  |-  ( ( 1  e.  ZZ  /\  2  e.  ZZ )  ->  ( 1  gcd  2
)  =  ( 2  gcd  1 ) )
4736, 5, 46mp2an 420 . . . . . 6  |-  ( 1  gcd  2 )  =  ( 2  gcd  1
)
48 gcdadd 11468 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  1  e.  ZZ )  ->  ( 2  gcd  1
)  =  ( 2  gcd  ( 1  +  2 ) ) )
495, 36, 48mp2an 420 . . . . . 6  |-  ( 2  gcd  1 )  =  ( 2  gcd  (
1  +  2 ) )
5045, 47, 493eqtri 2124 . . . . 5  |-  1  =  ( 2  gcd  ( 1  +  2 ) )
51 1p2e3 8706 . . . . . 6  |-  ( 1  +  2 )  =  3
5251oveq2i 5717 . . . . 5  |-  ( 2  gcd  ( 1  +  2 ) )  =  ( 2  gcd  3
)
5350, 52eqtr2i 2121 . . . 4  |-  ( 2  gcd  3 )  =  1
5435, 53eqtri 2120 . . 3  |-  ( 3  gcd  2 )  =  1
5554oveq2i 5717 . 2  |-  ( ( 3  x.  2 )  /  ( 3  gcd  2 ) )  =  ( ( 3  x.  2 )  /  1
)
56 3t2e6 8728 . . . 4  |-  ( 3  x.  2 )  =  6
5756oveq1i 5716 . . 3  |-  ( ( 3  x.  2 )  /  1 )  =  ( 6  /  1
)
58 6cn 8660 . . . 4  |-  6  e.  CC
5958div1i 8361 . . 3  |-  ( 6  /  1 )  =  6
6057, 59eqtri 2120 . 2  |-  ( ( 3  x.  2 )  /  1 )  =  6
6133, 55, 603eqtri 2124 1  |-  ( 3 lcm  2 )  =  6
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    /\ w3a 930    = wceq 1299    e. wcel 1448    =/= wne 2267   class class class wbr 3875   ` cfv 5059  (class class class)co 5706   CCcc 7498   0cc0 7500   1c1 7501    + caddc 7503    x. cmul 7505   # cap 8209    / cdiv 8293   NNcn 8578   2c2 8629   3c3 8630   6c6 8633   ZZcz 8906   abscabs 10609    gcd cgcd 11430   lcm clcm 11534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-sup 6786  df-inf 6787  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-5 8640  df-6 8641  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-fz 9632  df-fzo 9761  df-fl 9884  df-mod 9937  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-dvds 11289  df-gcd 11431  df-lcm 11535
This theorem is referenced by: (None)
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