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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 8949 | . . . 4 | |
2 | 4cn 8945 | . . . 4 | |
3 | 1, 2 | mulcli 7914 | . . 3 |
4 | 6nn0 9145 | . . . . 5 | |
5 | 4 | nn0zi 9223 | . . . 4 |
6 | 4z 9231 | . . . 4 | |
7 | lcmcl 12015 | . . . . 5 lcm | |
8 | 7 | nn0cnd 9179 | . . . 4 lcm |
9 | 5, 6, 8 | mp2an 424 | . . 3 lcm |
10 | gcdcl 11910 | . . . . . 6 | |
11 | 10 | nn0cnd 9179 | . . . . 5 |
12 | 5, 6, 11 | mp2an 424 | . . . 4 |
13 | 5, 6 | pm3.2i 270 | . . . . . . 7 |
14 | 4ne0 8965 | . . . . . . . . 9 | |
15 | 14 | neii 2342 | . . . . . . . 8 |
16 | 15 | intnan 924 | . . . . . . 7 |
17 | gcdn0cl 11906 | . . . . . . 7 | |
18 | 13, 16, 17 | mp2an 424 | . . . . . 6 |
19 | 18 | nnne0i 8899 | . . . . 5 |
20 | 18 | nnzi 9222 | . . . . . 6 |
21 | 0z 9212 | . . . . . 6 | |
22 | zapne 9275 | . . . . . 6 # | |
23 | 20, 21, 22 | mp2an 424 | . . . . 5 # |
24 | 19, 23 | mpbir 145 | . . . 4 # |
25 | 12, 24 | pm3.2i 270 | . . 3 # |
26 | 6nn 9032 | . . . . . . . 8 | |
27 | 4nn 9030 | . . . . . . . 8 | |
28 | 26, 27 | pm3.2i 270 | . . . . . . 7 |
29 | lcmgcdnn 12025 | . . . . . . 7 lcm | |
30 | 28, 29 | mp1i 10 | . . . . . 6 lcm # lcm |
31 | 30 | eqcomd 2176 | . . . . 5 lcm # lcm |
32 | divmulap3 8583 | . . . . 5 lcm # lcm lcm | |
33 | 31, 32 | mpbird 166 | . . . 4 lcm # lcm |
34 | 33 | eqcomd 2176 | . . 3 lcm # lcm |
35 | 3, 9, 25, 34 | mp3an 1332 | . 2 lcm |
36 | 6gcd4e2 11939 | . . 3 | |
37 | 36 | oveq2i 5862 | . 2 |
38 | 2cn 8938 | . . . 4 | |
39 | 2ap0 8960 | . . . 4 # | |
40 | 1, 2, 38, 39 | divassapi 8674 | . . 3 |
41 | 4d2e2 9027 | . . . 4 | |
42 | 41 | oveq2i 5862 | . . 3 |
43 | 6t2e12 9435 | . . 3 ; | |
44 | 40, 42, 43 | 3eqtri 2195 | . 2 ; |
45 | 35, 37, 44 | 3eqtri 2195 | 1 lcm ; |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 wne 2340 class class class wbr 3987 (class class class)co 5851 cc 7761 cc0 7763 c1 7764 cmul 7768 # cap 8489 cdiv 8578 cn 8867 c2 8918 c4 8920 c6 8922 cz 9201 ;cdc 9332 cgcd 11886 lcm clcm 12003 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-mulrcl 7862 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-precex 7873 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 ax-pre-mulgt0 7880 ax-pre-mulext 7881 ax-arch 7882 ax-caucvg 7883 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-sup 6958 df-inf 6959 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-reap 8483 df-ap 8490 df-div 8579 df-inn 8868 df-2 8926 df-3 8927 df-4 8928 df-5 8929 df-6 8930 df-7 8931 df-8 8932 df-9 8933 df-n0 9125 df-z 9202 df-dec 9333 df-uz 9477 df-q 9568 df-rp 9600 df-fz 9955 df-fzo 10088 df-fl 10215 df-mod 10268 df-seqfrec 10391 df-exp 10465 df-cj 10795 df-re 10796 df-im 10797 df-rsqrt 10951 df-abs 10952 df-dvds 11739 df-gcd 11887 df-lcm 12004 |
This theorem is referenced by: (None) |
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