ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3lcm2e6woprm Unicode version

Theorem 3lcm2e6woprm 10935
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 8425 . . . 4  |-  3  e.  CC
2 2cn 8421 . . . 4  |-  2  e.  CC
31, 2mulcli 7430 . . 3  |-  ( 3  x.  2 )  e.  CC
4 3z 8705 . . . 4  |-  3  e.  ZZ
5 2z 8704 . . . 4  |-  2  e.  ZZ
6 lcmcl 10921 . . . . 5  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  ( 3 lcm  2 )  e.  NN0 )
76nn0cnd 8654 . . . 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 8442 . . . . . . 7  |-  2  =/=  0
1110neii 2253 . . . . . 6  |-  -.  2  =  0
1211intnan 874 . . . . 5  |-  -.  (
3  =  0  /\  2  =  0 )
13 gcdn0cl 10821 . . . . . 6  |-  ( ( ( 3  e.  ZZ  /\  2  e.  ZZ )  /\  -.  ( 3  =  0  /\  2  =  0 ) )  ->  ( 3  gcd  2 )  e.  NN )
1413nncnd 8364 . . . . 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 8381 . . . . 5  |-  ( 3  gcd  2 )  =/=  0
1816nnzi 8697 . . . . . 6  |-  ( 3  gcd  2 )  e.  ZZ
19 0z 8687 . . . . . 6  |-  0  e.  ZZ
20 zapne 8747 . . . . . 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 8505 . . . . . . 7  |-  3  e.  NN
25 2nn 8504 . . . . . . 7  |-  2  e.  NN
2624, 25pm3.2i 266 . . . . . 6  |-  ( 3  e.  NN  /\  2  e.  NN )
27 lcmgcdnn 10931 . . . . . . 7  |-  ( ( 3  e.  NN  /\  2  e.  NN )  ->  ( ( 3 lcm  2 )  x.  ( 3  gcd  2 ) )  =  ( 3  x.  2 ) )
2827eqcomd 2090 . . . . . 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 8076 . . . . 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 2090 . . 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 1271 . 2  |-  ( 3 lcm  2 )  =  ( ( 3  x.  2 )  /  ( 3  gcd  2 ) )
34 gcdcom 10832 . . . . 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 8702 . . . . . . . . 9  |-  1  e.  ZZ
37 gcdid 10844 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  (
1  gcd  1 )  =  ( abs `  1
) )
3836, 37ax-mp 7 . . . . . . . 8  |-  ( 1  gcd  1 )  =  ( abs `  1
)
39 abs1 10393 . . . . . . . 8  |-  ( abs `  1 )  =  1
4038, 39eqtr2i 2106 . . . . . . 7  |-  1  =  ( 1  gcd  1 )
41 gcdadd 10843 . . . . . . . 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 8466 . . . . . . . 8  |-  ( 1  +  1 )  =  2
4443oveq2i 5618 . . . . . . 7  |-  ( 1  gcd  ( 1  +  1 ) )  =  ( 1  gcd  2
)
4540, 42, 443eqtri 2109 . . . . . 6  |-  1  =  ( 1  gcd  2 )
46 gcdcom 10832 . . . . . . 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 10843 . . . . . . 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 2109 . . . . 5  |-  1  =  ( 2  gcd  ( 1  +  2 ) )
51 1p2e3 8477 . . . . . 6  |-  ( 1  +  2 )  =  3
5251oveq2i 5618 . . . . 5  |-  ( 2  gcd  ( 1  +  2 ) )  =  ( 2  gcd  3
)
5350, 52eqtr2i 2106 . . . 4  |-  ( 2  gcd  3 )  =  1
5435, 53eqtri 2105 . . 3  |-  ( 3  gcd  2 )  =  1
5554oveq2i 5618 . 2  |-  ( ( 3  x.  2 )  /  ( 3  gcd  2 ) )  =  ( ( 3  x.  2 )  /  1
)
56 3t2e6 8499 . . . 4  |-  ( 3  x.  2 )  =  6
5756oveq1i 5617 . . 3  |-  ( ( 3  x.  2 )  /  1 )  =  ( 6  /  1
)
58 6cn 8432 . . . 4  |-  6  e.  CC
5958div1i 8139 . . 3  |-  ( 6  /  1 )  =  6
6057, 59eqtri 2105 . 2  |-  ( ( 3  x.  2 )  /  1 )  =  6
6133, 55, 603eqtri 2109 1  |-  ( 3 lcm  2 )  =  6
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    <-> wb 103    /\ w3a 922    = wceq 1287    e. wcel 1436    =/= wne 2251   class class class wbr 3820   ` cfv 4978  (class class class)co 5607   CCcc 7285   0cc0 7287   1c1 7288    + caddc 7290    x. cmul 7292   # cap 7992    / cdiv 8071   NNcn 8350   2c2 8400   3c3 8401   6c6 8404   ZZcz 8676   abscabs 10318    gcd cgcd 10805   lcm clcm 10909
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3928  ax-sep 3931  ax-nul 3939  ax-pow 3983  ax-pr 4009  ax-un 4233  ax-setind 4325  ax-iinf 4375  ax-cnex 7373  ax-resscn 7374  ax-1cn 7375  ax-1re 7376  ax-icn 7377  ax-addcl 7378  ax-addrcl 7379  ax-mulcl 7380  ax-mulrcl 7381  ax-addcom 7382  ax-mulcom 7383  ax-addass 7384  ax-mulass 7385  ax-distr 7386  ax-i2m1 7387  ax-0lt1 7388  ax-1rid 7389  ax-0id 7390  ax-rnegex 7391  ax-precex 7392  ax-cnre 7393  ax-pre-ltirr 7394  ax-pre-ltwlin 7395  ax-pre-lttrn 7396  ax-pre-apti 7397  ax-pre-ltadd 7398  ax-pre-mulgt0 7399  ax-pre-mulext 7400  ax-arch 7401  ax-caucvg 7402
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3911  df-id 4093  df-po 4096  df-iso 4097  df-iord 4166  df-on 4168  df-ilim 4169  df-suc 4171  df-iom 4378  df-xp 4416  df-rel 4417  df-cnv 4418  df-co 4419  df-dm 4420  df-rn 4421  df-res 4422  df-ima 4423  df-iota 4943  df-fun 4980  df-fn 4981  df-f 4982  df-f1 4983  df-fo 4984  df-f1o 4985  df-fv 4986  df-isom 4987  df-riota 5563  df-ov 5610  df-oprab 5611  df-mpt2 5612  df-1st 5862  df-2nd 5863  df-recs 6018  df-frec 6104  df-sup 6616  df-inf 6617  df-pnf 7461  df-mnf 7462  df-xr 7463  df-ltxr 7464  df-le 7465  df-sub 7592  df-neg 7593  df-reap 7986  df-ap 7993  df-div 8072  df-inn 8351  df-2 8409  df-3 8410  df-4 8411  df-5 8412  df-6 8413  df-n0 8600  df-z 8677  df-uz 8945  df-q 9030  df-rp 9060  df-fz 9350  df-fzo 9475  df-fl 9598  df-mod 9651  df-iseq 9773  df-iexp 9846  df-cj 10164  df-re 10165  df-im 10166  df-rsqrt 10319  df-abs 10320  df-dvds 10664  df-gcd 10806  df-lcm 10910
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator