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Theorem 3lcm2e6woprm 11778
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 8807 . . . 4  |-  3  e.  CC
2 2cn 8803 . . . 4  |-  2  e.  CC
31, 2mulcli 7783 . . 3  |-  ( 3  x.  2 )  e.  CC
4 3z 9095 . . . 4  |-  3  e.  ZZ
5 2z 9094 . . . 4  |-  2  e.  ZZ
6 lcmcl 11764 . . . . 5  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  ( 3 lcm  2 )  e.  NN0 )
76nn0cnd 9044 . . . 4  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  ( 3 lcm  2 )  e.  CC )
84, 5, 7mp2an 422 . . 3  |-  ( 3 lcm  2 )  e.  CC
94, 5pm3.2i 270 . . . . 5  |-  ( 3  e.  ZZ  /\  2  e.  ZZ )
10 2ne0 8824 . . . . . . 7  |-  2  =/=  0
1110neii 2310 . . . . . 6  |-  -.  2  =  0
1211intnan 914 . . . . 5  |-  -.  (
3  =  0  /\  2  =  0 )
13 gcdn0cl 11662 . . . . . 6  |-  ( ( ( 3  e.  ZZ  /\  2  e.  ZZ )  /\  -.  ( 3  =  0  /\  2  =  0 ) )  ->  ( 3  gcd  2 )  e.  NN )
1413nncnd 8746 . . . . 5  |-  ( ( ( 3  e.  ZZ  /\  2  e.  ZZ )  /\  -.  ( 3  =  0  /\  2  =  0 ) )  ->  ( 3  gcd  2 )  e.  CC )
159, 12, 14mp2an 422 . . . 4  |-  ( 3  gcd  2 )  e.  CC
169, 12, 13mp2an 422 . . . . . 6  |-  ( 3  gcd  2 )  e.  NN
1716nnne0i 8764 . . . . 5  |-  ( 3  gcd  2 )  =/=  0
1816nnzi 9087 . . . . . 6  |-  ( 3  gcd  2 )  e.  ZZ
19 0z 9077 . . . . . 6  |-  0  e.  ZZ
20 zapne 9137 . . . . . 6  |-  ( ( ( 3  gcd  2
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( 3  gcd  2 ) #  0  <->  (
3  gcd  2 )  =/=  0 ) )
2118, 19, 20mp2an 422 . . . . 5  |-  ( ( 3  gcd  2 ) #  0  <->  ( 3  gcd  2 )  =/=  0
)
2217, 21mpbir 145 . . . 4  |-  ( 3  gcd  2 ) #  0
2315, 22pm3.2i 270 . . 3  |-  ( ( 3  gcd  2 )  e.  CC  /\  (
3  gcd  2 ) #  0 )
24 3nn 8894 . . . . . . 7  |-  3  e.  NN
25 2nn 8893 . . . . . . 7  |-  2  e.  NN
2624, 25pm3.2i 270 . . . . . 6  |-  ( 3  e.  NN  /\  2  e.  NN )
27 lcmgcdnn 11774 . . . . . . 7  |-  ( ( 3  e.  NN  /\  2  e.  NN )  ->  ( ( 3 lcm  2 )  x.  ( 3  gcd  2 ) )  =  ( 3  x.  2 ) )
2827eqcomd 2145 . . . . . 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 8449 . . . . 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 2145 . . 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 1315 . 2  |-  ( 3 lcm  2 )  =  ( ( 3  x.  2 )  /  ( 3  gcd  2 ) )
34 gcdcom 11673 . . . . 5  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  ( 3  gcd  2
)  =  ( 2  gcd  3 ) )
354, 5, 34mp2an 422 . . . 4  |-  ( 3  gcd  2 )  =  ( 2  gcd  3
)
36 1z 9092 . . . . . . . . 9  |-  1  e.  ZZ
37 gcdid 11685 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  (
1  gcd  1 )  =  ( abs `  1
) )
3836, 37ax-mp 5 . . . . . . . 8  |-  ( 1  gcd  1 )  =  ( abs `  1
)
39 abs1 10856 . . . . . . . 8  |-  ( abs `  1 )  =  1
4038, 39eqtr2i 2161 . . . . . . 7  |-  1  =  ( 1  gcd  1 )
41 gcdadd 11684 . . . . . . . 8  |-  ( ( 1  e.  ZZ  /\  1  e.  ZZ )  ->  ( 1  gcd  1
)  =  ( 1  gcd  ( 1  +  1 ) ) )
4236, 36, 41mp2an 422 . . . . . . 7  |-  ( 1  gcd  1 )  =  ( 1  gcd  (
1  +  1 ) )
43 1p1e2 8849 . . . . . . . 8  |-  ( 1  +  1 )  =  2
4443oveq2i 5785 . . . . . . 7  |-  ( 1  gcd  ( 1  +  1 ) )  =  ( 1  gcd  2
)
4540, 42, 443eqtri 2164 . . . . . 6  |-  1  =  ( 1  gcd  2 )
46 gcdcom 11673 . . . . . . 7  |-  ( ( 1  e.  ZZ  /\  2  e.  ZZ )  ->  ( 1  gcd  2
)  =  ( 2  gcd  1 ) )
4736, 5, 46mp2an 422 . . . . . 6  |-  ( 1  gcd  2 )  =  ( 2  gcd  1
)
48 gcdadd 11684 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  1  e.  ZZ )  ->  ( 2  gcd  1
)  =  ( 2  gcd  ( 1  +  2 ) ) )
495, 36, 48mp2an 422 . . . . . 6  |-  ( 2  gcd  1 )  =  ( 2  gcd  (
1  +  2 ) )
5045, 47, 493eqtri 2164 . . . . 5  |-  1  =  ( 2  gcd  ( 1  +  2 ) )
51 1p2e3 8866 . . . . . 6  |-  ( 1  +  2 )  =  3
5251oveq2i 5785 . . . . 5  |-  ( 2  gcd  ( 1  +  2 ) )  =  ( 2  gcd  3
)
5350, 52eqtr2i 2161 . . . 4  |-  ( 2  gcd  3 )  =  1
5435, 53eqtri 2160 . . 3  |-  ( 3  gcd  2 )  =  1
5554oveq2i 5785 . 2  |-  ( ( 3  x.  2 )  /  ( 3  gcd  2 ) )  =  ( ( 3  x.  2 )  /  1
)
56 3t2e6 8888 . . . 4  |-  ( 3  x.  2 )  =  6
5756oveq1i 5784 . . 3  |-  ( ( 3  x.  2 )  /  1 )  =  ( 6  /  1
)
58 6cn 8814 . . . 4  |-  6  e.  CC
5958div1i 8512 . . 3  |-  ( 6  /  1 )  =  6
6057, 59eqtri 2160 . 2  |-  ( ( 3  x.  2 )  /  1 )  =  6
6133, 55, 603eqtri 2164 1  |-  ( 3 lcm  2 )  =  6
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480    =/= wne 2308   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7630   0cc0 7632   1c1 7633    + caddc 7635    x. cmul 7637   # cap 8355    / cdiv 8444   NNcn 8732   2c2 8783   3c3 8784   6c6 8787   ZZcz 9066   abscabs 10781    gcd cgcd 11646   lcm clcm 11752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-inf 6872  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-5 8794  df-6 8795  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-fz 9803  df-fzo 9932  df-fl 10055  df-mod 10108  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-dvds 11505  df-gcd 11647  df-lcm 11753
This theorem is referenced by: (None)
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