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Theorem 3lcm2e6woprm 10675
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 8233 . . . 4  |-  3  e.  CC
2 2cn 8229 . . . 4  |-  2  e.  CC
31, 2mulcli 7238 . . 3  |-  ( 3  x.  2 )  e.  CC
4 3z 8513 . . . 4  |-  3  e.  ZZ
5 2z 8512 . . . 4  |-  2  e.  ZZ
6 lcmcl 10661 . . . . 5  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  ( 3 lcm  2 )  e.  NN0 )
76nn0cnd 8462 . . . 4  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  ( 3 lcm  2 )  e.  CC )
84, 5, 7mp2an 417 . . 3  |-  ( 3 lcm  2 )  e.  CC
94, 5pm3.2i 266 . . . . 5  |-  ( 3  e.  ZZ  /\  2  e.  ZZ )
10 2ne0 8250 . . . . . . 7  |-  2  =/=  0
1110neii 2251 . . . . . 6  |-  -.  2  =  0
1211intnan 872 . . . . 5  |-  -.  (
3  =  0  /\  2  =  0 )
13 gcdn0cl 10561 . . . . . 6  |-  ( ( ( 3  e.  ZZ  /\  2  e.  ZZ )  /\  -.  ( 3  =  0  /\  2  =  0 ) )  ->  ( 3  gcd  2 )  e.  NN )
1413nncnd 8172 . . . . 5  |-  ( ( ( 3  e.  ZZ  /\  2  e.  ZZ )  /\  -.  ( 3  =  0  /\  2  =  0 ) )  ->  ( 3  gcd  2 )  e.  CC )
159, 12, 14mp2an 417 . . . 4  |-  ( 3  gcd  2 )  e.  CC
169, 12, 13mp2an 417 . . . . . 6  |-  ( 3  gcd  2 )  e.  NN
1716nnne0i 8189 . . . . 5  |-  ( 3  gcd  2 )  =/=  0
1816nnzi 8505 . . . . . 6  |-  ( 3  gcd  2 )  e.  ZZ
19 0z 8495 . . . . . 6  |-  0  e.  ZZ
20 zapne 8555 . . . . . 6  |-  ( ( ( 3  gcd  2
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( 3  gcd  2 ) #  0  <->  (
3  gcd  2 )  =/=  0 ) )
2118, 19, 20mp2an 417 . . . . 5  |-  ( ( 3  gcd  2 ) #  0  <->  ( 3  gcd  2 )  =/=  0
)
2217, 21mpbir 144 . . . 4  |-  ( 3  gcd  2 ) #  0
2315, 22pm3.2i 266 . . 3  |-  ( ( 3  gcd  2 )  e.  CC  /\  (
3  gcd  2 ) #  0 )
24 3nn 8313 . . . . . . 7  |-  3  e.  NN
25 2nn 8312 . . . . . . 7  |-  2  e.  NN
2624, 25pm3.2i 266 . . . . . 6  |-  ( 3  e.  NN  /\  2  e.  NN )
27 lcmgcdnn 10671 . . . . . . 7  |-  ( ( 3  e.  NN  /\  2  e.  NN )  ->  ( ( 3 lcm  2 )  x.  ( 3  gcd  2 ) )  =  ( 3  x.  2 ) )
2827eqcomd 2088 . . . . . 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 7884 . . . . 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 165 . . . 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 2088 . . 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 1269 . 2  |-  ( 3 lcm  2 )  =  ( ( 3  x.  2 )  /  ( 3  gcd  2 ) )
34 gcdcom 10572 . . . . 5  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  ( 3  gcd  2
)  =  ( 2  gcd  3 ) )
354, 5, 34mp2an 417 . . . 4  |-  ( 3  gcd  2 )  =  ( 2  gcd  3
)
36 1z 8510 . . . . . . . . 9  |-  1  e.  ZZ
37 gcdid 10584 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  (
1  gcd  1 )  =  ( abs `  1
) )
3836, 37ax-mp 7 . . . . . . . 8  |-  ( 1  gcd  1 )  =  ( abs `  1
)
39 abs1 10159 . . . . . . . 8  |-  ( abs `  1 )  =  1
4038, 39eqtr2i 2104 . . . . . . 7  |-  1  =  ( 1  gcd  1 )
41 gcdadd 10583 . . . . . . . 8  |-  ( ( 1  e.  ZZ  /\  1  e.  ZZ )  ->  ( 1  gcd  1
)  =  ( 1  gcd  ( 1  +  1 ) ) )
4236, 36, 41mp2an 417 . . . . . . 7  |-  ( 1  gcd  1 )  =  ( 1  gcd  (
1  +  1 ) )
43 1p1e2 8274 . . . . . . . 8  |-  ( 1  +  1 )  =  2
4443oveq2i 5574 . . . . . . 7  |-  ( 1  gcd  ( 1  +  1 ) )  =  ( 1  gcd  2
)
4540, 42, 443eqtri 2107 . . . . . 6  |-  1  =  ( 1  gcd  2 )
46 gcdcom 10572 . . . . . . 7  |-  ( ( 1  e.  ZZ  /\  2  e.  ZZ )  ->  ( 1  gcd  2
)  =  ( 2  gcd  1 ) )
4736, 5, 46mp2an 417 . . . . . 6  |-  ( 1  gcd  2 )  =  ( 2  gcd  1
)
48 gcdadd 10583 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  1  e.  ZZ )  ->  ( 2  gcd  1
)  =  ( 2  gcd  ( 1  +  2 ) ) )
495, 36, 48mp2an 417 . . . . . 6  |-  ( 2  gcd  1 )  =  ( 2  gcd  (
1  +  2 ) )
5045, 47, 493eqtri 2107 . . . . 5  |-  1  =  ( 2  gcd  ( 1  +  2 ) )
51 1p2e3 8285 . . . . . 6  |-  ( 1  +  2 )  =  3
5251oveq2i 5574 . . . . 5  |-  ( 2  gcd  ( 1  +  2 ) )  =  ( 2  gcd  3
)
5350, 52eqtr2i 2104 . . . 4  |-  ( 2  gcd  3 )  =  1
5435, 53eqtri 2103 . . 3  |-  ( 3  gcd  2 )  =  1
5554oveq2i 5574 . 2  |-  ( ( 3  x.  2 )  /  ( 3  gcd  2 ) )  =  ( ( 3  x.  2 )  /  1
)
56 3t2e6 8307 . . . 4  |-  ( 3  x.  2 )  =  6
5756oveq1i 5573 . . 3  |-  ( ( 3  x.  2 )  /  1 )  =  ( 6  /  1
)
58 6cn 8240 . . . 4  |-  6  e.  CC
5958div1i 7947 . . 3  |-  ( 6  /  1 )  =  6
6057, 59eqtri 2103 . 2  |-  ( ( 3  x.  2 )  /  1 )  =  6
6133, 55, 603eqtri 2107 1  |-  ( 3 lcm  2 )  =  6
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434    =/= wne 2249   class class class wbr 3805   ` cfv 4952  (class class class)co 5563   CCcc 7093   0cc0 7095   1c1 7096    + caddc 7098    x. cmul 7100   # cap 7800    / cdiv 7879   NNcn 8158   2c2 8208   3c3 8209   6c6 8212   ZZcz 8484   abscabs 10084    gcd cgcd 10545   lcm clcm 10649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-isom 4961  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-sup 6491  df-inf 6492  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-5 8220  df-6 8221  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-gcd 10546  df-lcm 10650
This theorem is referenced by: (None)
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