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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 12307 | . . . 4 ⊢ 6 ∈ ℂ | |
2 | 4cn 12301 | . . . 4 ⊢ 4 ∈ ℂ | |
3 | 1, 2 | mulcli 11225 | . . 3 ⊢ (6 · 4) ∈ ℂ |
4 | 6nn0 12497 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
5 | 4 | nn0zi 12591 | . . . 4 ⊢ 6 ∈ ℤ |
6 | 4z 12600 | . . . 4 ⊢ 4 ∈ ℤ | |
7 | lcmcl 16545 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ0) | |
8 | 7 | nn0cnd 12538 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ) |
9 | 5, 6, 8 | mp2an 689 | . . 3 ⊢ (6 lcm 4) ∈ ℂ |
10 | gcdcl 16454 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ0) | |
11 | 10 | nn0cnd 12538 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℂ) |
12 | 5, 6, 11 | mp2an 689 | . . . 4 ⊢ (6 gcd 4) ∈ ℂ |
13 | 5, 6 | pm3.2i 470 | . . . . . 6 ⊢ (6 ∈ ℤ ∧ 4 ∈ ℤ) |
14 | 4ne0 12324 | . . . . . . . 8 ⊢ 4 ≠ 0 | |
15 | 14 | neii 2936 | . . . . . . 7 ⊢ ¬ 4 = 0 |
16 | 15 | intnan 486 | . . . . . 6 ⊢ ¬ (6 = 0 ∧ 4 = 0) |
17 | gcdn0cl 16450 | . . . . . 6 ⊢ (((6 ∈ ℤ ∧ 4 ∈ ℤ) ∧ ¬ (6 = 0 ∧ 4 = 0)) → (6 gcd 4) ∈ ℕ) | |
18 | 13, 16, 17 | mp2an 689 | . . . . 5 ⊢ (6 gcd 4) ∈ ℕ |
19 | 18 | nnne0i 12256 | . . . 4 ⊢ (6 gcd 4) ≠ 0 |
20 | 12, 19 | pm3.2i 470 | . . 3 ⊢ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) ≠ 0) |
21 | 6nn 12305 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
22 | 4nn 12299 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
23 | 21, 22 | pm3.2i 470 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ) |
24 | lcmgcdnn 16555 | . . . . . . 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 2732 | . . . . 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 11881 | . . . . 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 2732 | . . 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 1457 | . 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4)) |
31 | 6gcd4e2 16487 | . . 3 ⊢ (6 gcd 4) = 2 | |
32 | 31 | oveq2i 7416 | . 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2) |
33 | 2cn 12291 | . . . 4 ⊢ 2 ∈ ℂ | |
34 | 2ne0 12320 | . . . 4 ⊢ 2 ≠ 0 | |
35 | 1, 2, 33, 34 | divassi 11974 | . . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2)) |
36 | 4d2e2 12386 | . . . 4 ⊢ (4 / 2) = 2 | |
37 | 36 | oveq2i 7416 | . . 3 ⊢ (6 · (4 / 2)) = (6 · 2) |
38 | 6t2e12 12785 | . . 3 ⊢ (6 · 2) = ;12 | |
39 | 35, 37, 38 | 3eqtri 2758 | . 2 ⊢ ((6 · 4) / 2) = ;12 |
40 | 30, 32, 39 | 3eqtri 2758 | 1 ⊢ (6 lcm 4) = ;12 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 (class class class)co 7405 ℂcc 11110 0cc0 11112 1c1 11113 · cmul 11117 / cdiv 11875 ℕcn 12216 2c2 12271 4c4 12273 6c6 12275 ℤcz 12562 ;cdc 12681 gcd cgcd 16442 lcm clcm 16532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12981 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16205 df-gcd 16443 df-lcm 16534 |
This theorem is referenced by: lcmf2a3a4e12 16591 lcm4un 41397 |
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