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Mirrors > Home > ILE Home > Th. List > 6lcm4e12 | Unicode version |
Description: The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.) |
Ref | Expression |
---|---|
6lcm4e12 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6cn 8396 |
. . . 4
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2 | 4cn 8392 |
. . . 4
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3 | 1, 2 | mulcli 7394 |
. . 3
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4 | 6nn0 8584 |
. . . . 5
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5 | 4 | nn0zi 8666 |
. . . 4
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6 | 4z 8674 |
. . . 4
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7 | lcmcl 10832 |
. . . . 5
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8 | 7 | nn0cnd 8618 |
. . . 4
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9 | 5, 6, 8 | mp2an 417 |
. . 3
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10 | gcdcl 10736 |
. . . . . 6
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11 | 10 | nn0cnd 8618 |
. . . . 5
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12 | 5, 6, 11 | mp2an 417 |
. . . 4
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13 | 5, 6 | pm3.2i 266 |
. . . . . . 7
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14 | 4ne0 8412 |
. . . . . . . . 9
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15 | 14 | neii 2251 |
. . . . . . . 8
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16 | 15 | intnan 872 |
. . . . . . 7
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17 | gcdn0cl 10732 |
. . . . . . 7
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18 | 13, 16, 17 | mp2an 417 |
. . . . . 6
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19 | 18 | nnne0i 8345 |
. . . . 5
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20 | 18 | nnzi 8665 |
. . . . . 6
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21 | 0z 8655 |
. . . . . 6
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22 | zapne 8715 |
. . . . . 6
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23 | 20, 21, 22 | mp2an 417 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 19, 23 | mpbir 144 |
. . . 4
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25 | 12, 24 | pm3.2i 266 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 6nn 8472 |
. . . . . . . 8
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27 | 4nn 8470 |
. . . . . . . 8
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28 | 26, 27 | pm3.2i 266 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | lcmgcdnn 10842 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 28, 29 | mp1i 10 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 30 | eqcomd 2088 |
. . . . 5
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32 | divmulap3 8040 |
. . . . 5
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33 | 31, 32 | mpbird 165 |
. . . 4
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34 | 33 | eqcomd 2088 |
. . 3
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35 | 3, 9, 25, 34 | mp3an 1269 |
. 2
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36 | 6gcd4e2 10762 |
. . 3
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37 | 36 | oveq2i 5600 |
. 2
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38 | 2cn 8385 |
. . . 4
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39 | 2ap0 8407 |
. . . 4
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40 | 1, 2, 38, 39 | divassapi 8131 |
. . 3
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41 | 4d2e2 8467 |
. . . 4
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42 | 41 | oveq2i 5600 |
. . 3
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43 | 6t2e12 8873 |
. . 3
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44 | 40, 42, 43 | 3eqtri 2107 |
. 2
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45 | 35, 37, 44 | 3eqtri 2107 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 ax-cnex 7337 ax-resscn 7338 ax-1cn 7339 ax-1re 7340 ax-icn 7341 ax-addcl 7342 ax-addrcl 7343 ax-mulcl 7344 ax-mulrcl 7345 ax-addcom 7346 ax-mulcom 7347 ax-addass 7348 ax-mulass 7349 ax-distr 7350 ax-i2m1 7351 ax-0lt1 7352 ax-1rid 7353 ax-0id 7354 ax-rnegex 7355 ax-precex 7356 ax-cnre 7357 ax-pre-ltirr 7358 ax-pre-ltwlin 7359 ax-pre-lttrn 7360 ax-pre-apti 7361 ax-pre-ltadd 7362 ax-pre-mulgt0 7363 ax-pre-mulext 7364 ax-arch 7365 ax-caucvg 7366 |
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 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4083 df-po 4086 df-iso 4087 df-iord 4156 df-on 4158 df-ilim 4159 df-suc 4161 df-iom 4368 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-isom 4976 df-riota 5545 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-recs 6000 df-frec 6086 df-sup 6584 df-inf 6585 df-pnf 7425 df-mnf 7426 df-xr 7427 df-ltxr 7428 df-le 7429 df-sub 7556 df-neg 7557 df-reap 7950 df-ap 7957 df-div 8036 df-inn 8315 df-2 8373 df-3 8374 df-4 8375 df-5 8376 df-6 8377 df-7 8378 df-8 8379 df-9 8380 df-n0 8564 df-z 8645 df-dec 8771 df-uz 8913 df-q 8998 df-rp 9028 df-fz 9318 df-fzo 9442 df-fl 9564 df-mod 9617 df-iseq 9739 df-iexp 9790 df-cj 10101 df-re 10102 df-im 10103 df-rsqrt 10256 df-abs 10257 df-dvds 10575 df-gcd 10717 df-lcm 10821 |
This theorem is referenced by: (None) |
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