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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 | lcm ; |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6cn 8770 | . . . 4 | |
2 | 4cn 8766 | . . . 4 | |
3 | 1, 2 | mulcli 7739 | . . 3 |
4 | 6nn0 8966 | . . . . 5 | |
5 | 4 | nn0zi 9044 | . . . 4 |
6 | 4z 9052 | . . . 4 | |
7 | lcmcl 11680 | . . . . 5 lcm | |
8 | 7 | nn0cnd 9000 | . . . 4 lcm |
9 | 5, 6, 8 | mp2an 422 | . . 3 lcm |
10 | gcdcl 11582 | . . . . . 6 | |
11 | 10 | nn0cnd 9000 | . . . . 5 |
12 | 5, 6, 11 | mp2an 422 | . . . 4 |
13 | 5, 6 | pm3.2i 270 | . . . . . . 7 |
14 | 4ne0 8786 | . . . . . . . . 9 | |
15 | 14 | neii 2287 | . . . . . . . 8 |
16 | 15 | intnan 899 | . . . . . . 7 |
17 | gcdn0cl 11578 | . . . . . . 7 | |
18 | 13, 16, 17 | mp2an 422 | . . . . . 6 |
19 | 18 | nnne0i 8720 | . . . . 5 |
20 | 18 | nnzi 9043 | . . . . . 6 |
21 | 0z 9033 | . . . . . 6 | |
22 | zapne 9093 | . . . . . 6 # | |
23 | 20, 21, 22 | mp2an 422 | . . . . 5 # |
24 | 19, 23 | mpbir 145 | . . . 4 # |
25 | 12, 24 | pm3.2i 270 | . . 3 # |
26 | 6nn 8853 | . . . . . . . 8 | |
27 | 4nn 8851 | . . . . . . . 8 | |
28 | 26, 27 | pm3.2i 270 | . . . . . . 7 |
29 | lcmgcdnn 11690 | . . . . . . 7 lcm | |
30 | 28, 29 | mp1i 10 | . . . . . 6 lcm # lcm |
31 | 30 | eqcomd 2123 | . . . . 5 lcm # lcm |
32 | divmulap3 8405 | . . . . 5 lcm # lcm lcm | |
33 | 31, 32 | mpbird 166 | . . . 4 lcm # lcm |
34 | 33 | eqcomd 2123 | . . 3 lcm # lcm |
35 | 3, 9, 25, 34 | mp3an 1300 | . 2 lcm |
36 | 6gcd4e2 11610 | . . 3 | |
37 | 36 | oveq2i 5753 | . 2 |
38 | 2cn 8759 | . . . 4 | |
39 | 2ap0 8781 | . . . 4 # | |
40 | 1, 2, 38, 39 | divassapi 8496 | . . 3 |
41 | 4d2e2 8848 | . . . 4 | |
42 | 41 | oveq2i 5753 | . . 3 |
43 | 6t2e12 9253 | . . 3 ; | |
44 | 40, 42, 43 | 3eqtri 2142 | . 2 ; |
45 | 35, 37, 44 | 3eqtri 2142 | 1 lcm ; |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 wne 2285 class class class wbr 3899 (class class class)co 5742 cc 7586 cc0 7588 c1 7589 cmul 7593 # cap 8311 cdiv 8400 cn 8688 c2 8739 c4 8741 c6 8743 cz 9022 ;cdc 9150 cgcd 11562 lcm clcm 11668 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 ax-caucvg 7708 |
This theorem depends on definitions: df-bi 116 df-stab 801 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-isom 5102 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-sup 6839 df-inf 6840 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 df-7 8752 df-8 8753 df-9 8754 df-n0 8946 df-z 9023 df-dec 9151 df-uz 9295 df-q 9380 df-rp 9410 df-fz 9759 df-fzo 9888 df-fl 10011 df-mod 10064 df-seqfrec 10187 df-exp 10261 df-cj 10582 df-re 10583 df-im 10584 df-rsqrt 10738 df-abs 10739 df-dvds 11421 df-gcd 11563 df-lcm 11669 |
This theorem is referenced by: (None) |
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