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Mirrors > Home > ILE Home > Th. List > 6lcm4e12 | GIF version |
Description: The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.) |
Ref | Expression |
---|---|
6lcm4e12 | ⊢ (6 lcm 4) = ;12 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6cn 8999 | . . . 4 ⊢ 6 ∈ ℂ | |
2 | 4cn 8995 | . . . 4 ⊢ 4 ∈ ℂ | |
3 | 1, 2 | mulcli 7961 | . . 3 ⊢ (6 · 4) ∈ ℂ |
4 | 6nn0 9195 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
5 | 4 | nn0zi 9273 | . . . 4 ⊢ 6 ∈ ℤ |
6 | 4z 9281 | . . . 4 ⊢ 4 ∈ ℤ | |
7 | lcmcl 12066 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ0) | |
8 | 7 | nn0cnd 9229 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ) |
9 | 5, 6, 8 | mp2an 426 | . . 3 ⊢ (6 lcm 4) ∈ ℂ |
10 | gcdcl 11961 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ0) | |
11 | 10 | nn0cnd 9229 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℂ) |
12 | 5, 6, 11 | mp2an 426 | . . . 4 ⊢ (6 gcd 4) ∈ ℂ |
13 | 5, 6 | pm3.2i 272 | . . . . . . 7 ⊢ (6 ∈ ℤ ∧ 4 ∈ ℤ) |
14 | 4ne0 9015 | . . . . . . . . 9 ⊢ 4 ≠ 0 | |
15 | 14 | neii 2349 | . . . . . . . 8 ⊢ ¬ 4 = 0 |
16 | 15 | intnan 929 | . . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0) |
17 | gcdn0cl 11957 | . . . . . . 7 ⊢ (((6 ∈ ℤ ∧ 4 ∈ ℤ) ∧ ¬ (6 = 0 ∧ 4 = 0)) → (6 gcd 4) ∈ ℕ) | |
18 | 13, 16, 17 | mp2an 426 | . . . . . 6 ⊢ (6 gcd 4) ∈ ℕ |
19 | 18 | nnne0i 8949 | . . . . 5 ⊢ (6 gcd 4) ≠ 0 |
20 | 18 | nnzi 9272 | . . . . . 6 ⊢ (6 gcd 4) ∈ ℤ |
21 | 0z 9262 | . . . . . 6 ⊢ 0 ∈ ℤ | |
22 | zapne 9325 | . . . . . 6 ⊢ (((6 gcd 4) ∈ ℤ ∧ 0 ∈ ℤ) → ((6 gcd 4) # 0 ↔ (6 gcd 4) ≠ 0)) | |
23 | 20, 21, 22 | mp2an 426 | . . . . 5 ⊢ ((6 gcd 4) # 0 ↔ (6 gcd 4) ≠ 0) |
24 | 19, 23 | mpbir 146 | . . . 4 ⊢ (6 gcd 4) # 0 |
25 | 12, 24 | pm3.2i 272 | . . 3 ⊢ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0) |
26 | 6nn 9082 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
27 | 4nn 9080 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
28 | 26, 27 | pm3.2i 272 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ) |
29 | lcmgcdnn 12076 | . . . . . . 7 ⊢ ((6 ∈ ℕ ∧ 4 ∈ ℕ) → ((6 lcm 4) · (6 gcd 4)) = (6 · 4)) | |
30 | 28, 29 | mp1i 10 | . . . . . 6 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0)) → ((6 lcm 4) · (6 gcd 4)) = (6 · 4)) |
31 | 30 | eqcomd 2183 | . . . . 5 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0)) → (6 · 4) = ((6 lcm 4) · (6 gcd 4))) |
32 | divmulap3 8632 | . . . . 5 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0)) → (((6 · 4) / (6 gcd 4)) = (6 lcm 4) ↔ (6 · 4) = ((6 lcm 4) · (6 gcd 4)))) | |
33 | 31, 32 | mpbird 167 | . . . 4 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0)) → ((6 · 4) / (6 gcd 4)) = (6 lcm 4)) |
34 | 33 | eqcomd 2183 | . . 3 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0)) → (6 lcm 4) = ((6 · 4) / (6 gcd 4))) |
35 | 3, 9, 25, 34 | mp3an 1337 | . 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4)) |
36 | 6gcd4e2 11990 | . . 3 ⊢ (6 gcd 4) = 2 | |
37 | 36 | oveq2i 5885 | . 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2) |
38 | 2cn 8988 | . . . 4 ⊢ 2 ∈ ℂ | |
39 | 2ap0 9010 | . . . 4 ⊢ 2 # 0 | |
40 | 1, 2, 38, 39 | divassapi 8723 | . . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2)) |
41 | 4d2e2 9077 | . . . 4 ⊢ (4 / 2) = 2 | |
42 | 41 | oveq2i 5885 | . . 3 ⊢ (6 · (4 / 2)) = (6 · 2) |
43 | 6t2e12 9485 | . . 3 ⊢ (6 · 2) = ;12 | |
44 | 40, 42, 43 | 3eqtri 2202 | . 2 ⊢ ((6 · 4) / 2) = ;12 |
45 | 35, 37, 44 | 3eqtri 2202 | 1 ⊢ (6 lcm 4) = ;12 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 class class class wbr 4003 (class class class)co 5874 ℂcc 7808 0cc0 7810 1c1 7811 · cmul 7815 # cap 8536 / cdiv 8627 ℕcn 8917 2c2 8968 4c4 8970 6c6 8972 ℤcz 9251 ;cdc 9382 gcd cgcd 11937 lcm clcm 12054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-sup 6982 df-inf 6983 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-5 8979 df-6 8980 df-7 8981 df-8 8982 df-9 8983 df-n0 9175 df-z 9252 df-dec 9383 df-uz 9527 df-q 9618 df-rp 9652 df-fz 10007 df-fzo 10140 df-fl 10267 df-mod 10320 df-seqfrec 10443 df-exp 10517 df-cj 10846 df-re 10847 df-im 10848 df-rsqrt 11002 df-abs 11003 df-dvds 11790 df-gcd 11938 df-lcm 12055 |
This theorem is referenced by: (None) |
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