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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 8660 | . . . 4 ⊢ 6 ∈ ℂ | |
2 | 4cn 8656 | . . . 4 ⊢ 4 ∈ ℂ | |
3 | 1, 2 | mulcli 7643 | . . 3 ⊢ (6 · 4) ∈ ℂ |
4 | 6nn0 8850 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
5 | 4 | nn0zi 8928 | . . . 4 ⊢ 6 ∈ ℤ |
6 | 4z 8936 | . . . 4 ⊢ 4 ∈ ℤ | |
7 | lcmcl 11546 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ0) | |
8 | 7 | nn0cnd 8884 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ) |
9 | 5, 6, 8 | mp2an 420 | . . 3 ⊢ (6 lcm 4) ∈ ℂ |
10 | gcdcl 11450 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ0) | |
11 | 10 | nn0cnd 8884 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℂ) |
12 | 5, 6, 11 | mp2an 420 | . . . 4 ⊢ (6 gcd 4) ∈ ℂ |
13 | 5, 6 | pm3.2i 268 | . . . . . . 7 ⊢ (6 ∈ ℤ ∧ 4 ∈ ℤ) |
14 | 4ne0 8676 | . . . . . . . . 9 ⊢ 4 ≠ 0 | |
15 | 14 | neii 2269 | . . . . . . . 8 ⊢ ¬ 4 = 0 |
16 | 15 | intnan 882 | . . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0) |
17 | gcdn0cl 11446 | . . . . . . 7 ⊢ (((6 ∈ ℤ ∧ 4 ∈ ℤ) ∧ ¬ (6 = 0 ∧ 4 = 0)) → (6 gcd 4) ∈ ℕ) | |
18 | 13, 16, 17 | mp2an 420 | . . . . . 6 ⊢ (6 gcd 4) ∈ ℕ |
19 | 18 | nnne0i 8610 | . . . . 5 ⊢ (6 gcd 4) ≠ 0 |
20 | 18 | nnzi 8927 | . . . . . 6 ⊢ (6 gcd 4) ∈ ℤ |
21 | 0z 8917 | . . . . . 6 ⊢ 0 ∈ ℤ | |
22 | zapne 8977 | . . . . . 6 ⊢ (((6 gcd 4) ∈ ℤ ∧ 0 ∈ ℤ) → ((6 gcd 4) # 0 ↔ (6 gcd 4) ≠ 0)) | |
23 | 20, 21, 22 | mp2an 420 | . . . . 5 ⊢ ((6 gcd 4) # 0 ↔ (6 gcd 4) ≠ 0) |
24 | 19, 23 | mpbir 145 | . . . 4 ⊢ (6 gcd 4) # 0 |
25 | 12, 24 | pm3.2i 268 | . . 3 ⊢ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0) |
26 | 6nn 8737 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
27 | 4nn 8735 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
28 | 26, 27 | pm3.2i 268 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ) |
29 | lcmgcdnn 11556 | . . . . . . 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 2105 | . . . . 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 8298 | . . . . 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 166 | . . . 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 2105 | . . 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 1283 | . 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4)) |
36 | 6gcd4e2 11476 | . . 3 ⊢ (6 gcd 4) = 2 | |
37 | 36 | oveq2i 5717 | . 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2) |
38 | 2cn 8649 | . . . 4 ⊢ 2 ∈ ℂ | |
39 | 2ap0 8671 | . . . 4 ⊢ 2 # 0 | |
40 | 1, 2, 38, 39 | divassapi 8389 | . . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2)) |
41 | 4d2e2 8732 | . . . 4 ⊢ (4 / 2) = 2 | |
42 | 41 | oveq2i 5717 | . . 3 ⊢ (6 · (4 / 2)) = (6 · 2) |
43 | 6t2e12 9137 | . . 3 ⊢ (6 · 2) = ;12 | |
44 | 40, 42, 43 | 3eqtri 2124 | . 2 ⊢ ((6 · 4) / 2) = ;12 |
45 | 35, 37, 44 | 3eqtri 2124 | 1 ⊢ (6 lcm 4) = ;12 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∧ w3a 930 = wceq 1299 ∈ wcel 1448 ≠ wne 2267 class class class wbr 3875 (class class class)co 5706 ℂcc 7498 0cc0 7500 1c1 7501 · cmul 7505 # cap 8209 / cdiv 8293 ℕcn 8578 2c2 8629 4c4 8631 6c6 8633 ℤcz 8906 ;cdc 9034 gcd cgcd 11430 lcm clcm 11534 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 ax-arch 7614 ax-caucvg 7615 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-isom 5068 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-sup 6786 df-inf 6787 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-5 8640 df-6 8641 df-7 8642 df-8 8643 df-9 8644 df-n0 8830 df-z 8907 df-dec 9035 df-uz 9177 df-q 9262 df-rp 9292 df-fz 9632 df-fzo 9761 df-fl 9884 df-mod 9937 df-seqfrec 10060 df-exp 10134 df-cj 10455 df-re 10456 df-im 10457 df-rsqrt 10610 df-abs 10611 df-dvds 11289 df-gcd 11431 df-lcm 11535 |
This theorem is referenced by: (None) |
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