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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 11403 | . . . 4 ⊢ 6 ∈ ℂ | |
2 | 4cn 11395 | . . . 4 ⊢ 4 ∈ ℂ | |
3 | 1, 2 | mulcli 10334 | . . 3 ⊢ (6 · 4) ∈ ℂ |
4 | 6nn0 11599 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
5 | 4 | nn0zi 11688 | . . . . 5 ⊢ 6 ∈ ℤ |
6 | 4z 11697 | . . . . 5 ⊢ 4 ∈ ℤ | |
7 | 5, 6 | pm3.2i 463 | . . . 4 ⊢ (6 ∈ ℤ ∧ 4 ∈ ℤ) |
8 | lcmcl 15646 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ0) | |
9 | 8 | nn0cnd 11638 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ) |
10 | 7, 9 | ax-mp 5 | . . 3 ⊢ (6 lcm 4) ∈ ℂ |
11 | gcdcl 15560 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ0) | |
12 | 11 | nn0cnd 11638 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℂ) |
13 | 7, 12 | ax-mp 5 | . . . 4 ⊢ (6 gcd 4) ∈ ℂ |
14 | 4ne0 11424 | . . . . . . . . 9 ⊢ 4 ≠ 0 | |
15 | 14 | neii 2971 | . . . . . . . 8 ⊢ ¬ 4 = 0 |
16 | 15 | intnan 481 | . . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0) |
17 | 7, 16 | pm3.2i 463 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) ∧ ¬ (6 = 0 ∧ 4 = 0)) |
18 | gcdn0cl 15556 | . . . . . 6 ⊢ (((6 ∈ ℤ ∧ 4 ∈ ℤ) ∧ ¬ (6 = 0 ∧ 4 = 0)) → (6 gcd 4) ∈ ℕ) | |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ (6 gcd 4) ∈ ℕ |
20 | 19 | nnne0i 11349 | . . . 4 ⊢ (6 gcd 4) ≠ 0 |
21 | 13, 20 | pm3.2i 463 | . . 3 ⊢ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) ≠ 0) |
22 | 6nn 11401 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
23 | 4nn 11393 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
24 | 22, 23 | pm3.2i 463 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ) |
25 | lcmgcdnn 15656 | . . . . . . 7 ⊢ ((6 ∈ ℕ ∧ 4 ∈ ℕ) → ((6 lcm 4) · (6 gcd 4)) = (6 · 4)) | |
26 | 24, 25 | 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)) |
27 | 26 | eqcomd 2803 | . . . . 5 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) ≠ 0)) → (6 · 4) = ((6 lcm 4) · (6 gcd 4))) |
28 | divmul3 10980 | . . . . 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)))) | |
29 | 27, 28 | mpbird 249 | . . . 4 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) ≠ 0)) → ((6 · 4) / (6 gcd 4)) = (6 lcm 4)) |
30 | 29 | eqcomd 2803 | . . 3 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) ≠ 0)) → (6 lcm 4) = ((6 · 4) / (6 gcd 4))) |
31 | 3, 10, 21, 30 | mp3an 1586 | . 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4)) |
32 | 6gcd4e2 15587 | . . 3 ⊢ (6 gcd 4) = 2 | |
33 | 32 | oveq2i 6887 | . 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2) |
34 | 2cn 11384 | . . . 4 ⊢ 2 ∈ ℂ | |
35 | 2ne0 11420 | . . . 4 ⊢ 2 ≠ 0 | |
36 | 1, 2, 34, 35 | divassi 11071 | . . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2)) |
37 | 4d2e2 11486 | . . . 4 ⊢ (4 / 2) = 2 | |
38 | 37 | oveq2i 6887 | . . 3 ⊢ (6 · (4 / 2)) = (6 · 2) |
39 | 6t2e12 11885 | . . 3 ⊢ (6 · 2) = ;12 | |
40 | 36, 38, 39 | 3eqtri 2823 | . 2 ⊢ ((6 · 4) / 2) = ;12 |
41 | 31, 33, 40 | 3eqtri 2823 | 1 ⊢ (6 lcm 4) = ;12 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 (class class class)co 6876 ℂcc 10220 0cc0 10222 1c1 10223 · cmul 10227 / cdiv 10974 ℕcn 11310 2c2 11364 4c4 11366 6c6 11368 ℤcz 11662 ;cdc 11779 gcd cgcd 15548 lcm clcm 15633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-sup 8588 df-inf 8589 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-rp 12071 df-fl 12844 df-mod 12920 df-seq 13052 df-exp 13111 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-dvds 15317 df-gcd 15549 df-lcm 15635 |
This theorem is referenced by: lcmf2a3a4e12 15692 |
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