![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 6lcm4e12 | Structured version Visualization version 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 12251 | . . . 4 ⊢ 6 ∈ ℂ | |
2 | 4cn 12245 | . . . 4 ⊢ 4 ∈ ℂ | |
3 | 1, 2 | mulcli 11169 | . . 3 ⊢ (6 · 4) ∈ ℂ |
4 | 6nn0 12441 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
5 | 4 | nn0zi 12535 | . . . 4 ⊢ 6 ∈ ℤ |
6 | 4z 12544 | . . . 4 ⊢ 4 ∈ ℤ | |
7 | lcmcl 16484 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ0) | |
8 | 7 | nn0cnd 12482 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ) |
9 | 5, 6, 8 | mp2an 691 | . . 3 ⊢ (6 lcm 4) ∈ ℂ |
10 | gcdcl 16393 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ0) | |
11 | 10 | nn0cnd 12482 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℂ) |
12 | 5, 6, 11 | mp2an 691 | . . . 4 ⊢ (6 gcd 4) ∈ ℂ |
13 | 5, 6 | pm3.2i 472 | . . . . . 6 ⊢ (6 ∈ ℤ ∧ 4 ∈ ℤ) |
14 | 4ne0 12268 | . . . . . . . 8 ⊢ 4 ≠ 0 | |
15 | 14 | neii 2946 | . . . . . . 7 ⊢ ¬ 4 = 0 |
16 | 15 | intnan 488 | . . . . . 6 ⊢ ¬ (6 = 0 ∧ 4 = 0) |
17 | gcdn0cl 16389 | . . . . . 6 ⊢ (((6 ∈ ℤ ∧ 4 ∈ ℤ) ∧ ¬ (6 = 0 ∧ 4 = 0)) → (6 gcd 4) ∈ ℕ) | |
18 | 13, 16, 17 | mp2an 691 | . . . . 5 ⊢ (6 gcd 4) ∈ ℕ |
19 | 18 | nnne0i 12200 | . . . 4 ⊢ (6 gcd 4) ≠ 0 |
20 | 12, 19 | pm3.2i 472 | . . 3 ⊢ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) ≠ 0) |
21 | 6nn 12249 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
22 | 4nn 12243 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
23 | 21, 22 | pm3.2i 472 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ) |
24 | lcmgcdnn 16494 | . . . . . . 7 ⊢ ((6 ∈ ℕ ∧ 4 ∈ ℕ) → ((6 lcm 4) · (6 gcd 4)) = (6 · 4)) | |
25 | 23, 24 | mp1i 13 | . . . . . 6 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) ≠ 0)) → ((6 lcm 4) · (6 gcd 4)) = (6 · 4)) |
26 | 25 | eqcomd 2743 | . . . . 5 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) ≠ 0)) → (6 · 4) = ((6 lcm 4) · (6 gcd 4))) |
27 | divmul3 11825 | . . . . 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)))) | |
28 | 26, 27 | mpbird 257 | . . . 4 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) ≠ 0)) → ((6 · 4) / (6 gcd 4)) = (6 lcm 4)) |
29 | 28 | eqcomd 2743 | . . 3 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) ≠ 0)) → (6 lcm 4) = ((6 · 4) / (6 gcd 4))) |
30 | 3, 9, 20, 29 | mp3an 1462 | . 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4)) |
31 | 6gcd4e2 16426 | . . 3 ⊢ (6 gcd 4) = 2 | |
32 | 31 | oveq2i 7373 | . 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2) |
33 | 2cn 12235 | . . . 4 ⊢ 2 ∈ ℂ | |
34 | 2ne0 12264 | . . . 4 ⊢ 2 ≠ 0 | |
35 | 1, 2, 33, 34 | divassi 11918 | . . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2)) |
36 | 4d2e2 12330 | . . . 4 ⊢ (4 / 2) = 2 | |
37 | 36 | oveq2i 7373 | . . 3 ⊢ (6 · (4 / 2)) = (6 · 2) |
38 | 6t2e12 12729 | . . 3 ⊢ (6 · 2) = ;12 | |
39 | 35, 37, 38 | 3eqtri 2769 | . 2 ⊢ ((6 · 4) / 2) = ;12 |
40 | 30, 32, 39 | 3eqtri 2769 | 1 ⊢ (6 lcm 4) = ;12 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 (class class class)co 7362 ℂcc 11056 0cc0 11058 1c1 11059 · cmul 11063 / cdiv 11819 ℕcn 12160 2c2 12215 4c4 12217 6c6 12219 ℤcz 12506 ;cdc 12625 gcd cgcd 16381 lcm clcm 16471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-rp 12923 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-dvds 16144 df-gcd 16382 df-lcm 16473 |
This theorem is referenced by: lcmf2a3a4e12 16530 lcm4un 40502 |
Copyright terms: Public domain | W3C validator |