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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 8121 | . . . 4 | |
2 | 4cn 8117 | . . . 4 | |
3 | 1, 2 | mulcli 7124 | . . 3 |
4 | 6nn0 8309 | . . . . 5 | |
5 | 4 | nn0zi 8373 | . . . 4 |
6 | 4z 8381 | . . . 4 | |
7 | lcmcl 10454 | . . . . 5 lcm | |
8 | 7 | nn0cnd 8343 | . . . 4 lcm |
9 | 5, 6, 8 | mp2an 416 | . . 3 lcm |
10 | gcdcl 10358 | . . . . . 6 | |
11 | 10 | nn0cnd 8343 | . . . . 5 |
12 | 5, 6, 11 | mp2an 416 | . . . 4 |
13 | 5, 6 | pm3.2i 266 | . . . . . . 7 |
14 | 4ne0 8137 | . . . . . . . . 9 | |
15 | 14 | neii 2247 | . . . . . . . 8 |
16 | 15 | intnan 871 | . . . . . . 7 |
17 | gcdn0cl 10354 | . . . . . . 7 | |
18 | 13, 16, 17 | mp2an 416 | . . . . . 6 |
19 | 18 | nnne0i 8070 | . . . . 5 |
20 | 18 | nnzi 8372 | . . . . . 6 |
21 | 0z 8362 | . . . . . 6 | |
22 | zapne 8422 | . . . . . 6 # | |
23 | 20, 21, 22 | mp2an 416 | . . . . 5 # |
24 | 19, 23 | mpbir 144 | . . . 4 # |
25 | 12, 24 | pm3.2i 266 | . . 3 # |
26 | 6nn 8197 | . . . . . . . 8 | |
27 | 4nn 8195 | . . . . . . . 8 | |
28 | 26, 27 | pm3.2i 266 | . . . . . . 7 |
29 | lcmgcdnn 10464 | . . . . . . 7 lcm | |
30 | 28, 29 | mp1i 10 | . . . . . 6 lcm # lcm |
31 | 30 | eqcomd 2086 | . . . . 5 lcm # lcm |
32 | divmulap3 7765 | . . . . 5 lcm # lcm lcm | |
33 | 31, 32 | mpbird 165 | . . . 4 lcm # lcm |
34 | 33 | eqcomd 2086 | . . 3 lcm # lcm |
35 | 3, 9, 25, 34 | mp3an 1268 | . 2 lcm |
36 | 6gcd4e2 10384 | . . 3 | |
37 | 36 | oveq2i 5543 | . 2 |
38 | 2cn 8110 | . . . 4 | |
39 | 2ap0 8132 | . . . 4 # | |
40 | 1, 2, 38, 39 | divassapi 7856 | . . 3 |
41 | 4d2e2 8192 | . . . 4 | |
42 | 41 | oveq2i 5543 | . . 3 |
43 | 6t2e12 8580 | . . 3 ; | |
44 | 40, 42, 43 | 3eqtri 2105 | . 2 ; |
45 | 35, 37, 44 | 3eqtri 2105 | 1 lcm ; |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 wne 2245 class class class wbr 3785 (class class class)co 5532 cc 6979 cc0 6981 c1 6982 cmul 6986 # cap 7681 cdiv 7760 cn 8039 c2 8089 c4 8091 c6 8093 cz 8351 ;cdc 8477 cgcd 10338 lcm clcm 10442 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 ax-caucvg 7096 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-isom 4931 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-sup 6397 df-inf 6398 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-5 8101 df-6 8102 df-7 8103 df-8 8104 df-9 8105 df-n0 8289 df-z 8352 df-dec 8478 df-uz 8620 df-q 8705 df-rp 8735 df-fz 9030 df-fzo 9153 df-fl 9274 df-mod 9325 df-iseq 9432 df-iexp 9476 df-cj 9729 df-re 9730 df-im 9731 df-rsqrt 9884 df-abs 9885 df-dvds 10196 df-gcd 10339 df-lcm 10443 |
This theorem is referenced by: (None) |
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