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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 12355 | . . . 4 ⊢ 6 ∈ ℂ | |
2 | 4cn 12349 | . . . 4 ⊢ 4 ∈ ℂ | |
3 | 1, 2 | mulcli 11271 | . . 3 ⊢ (6 · 4) ∈ ℂ |
4 | 6nn0 12545 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
5 | 4 | nn0zi 12639 | . . . 4 ⊢ 6 ∈ ℤ |
6 | 4z 12648 | . . . 4 ⊢ 4 ∈ ℤ | |
7 | lcmcl 16602 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ0) | |
8 | 7 | nn0cnd 12586 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ) |
9 | 5, 6, 8 | mp2an 690 | . . 3 ⊢ (6 lcm 4) ∈ ℂ |
10 | gcdcl 16506 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ0) | |
11 | 10 | nn0cnd 12586 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℂ) |
12 | 5, 6, 11 | mp2an 690 | . . . 4 ⊢ (6 gcd 4) ∈ ℂ |
13 | 5, 6 | pm3.2i 469 | . . . . . 6 ⊢ (6 ∈ ℤ ∧ 4 ∈ ℤ) |
14 | 4ne0 12372 | . . . . . . . 8 ⊢ 4 ≠ 0 | |
15 | 14 | neii 2932 | . . . . . . 7 ⊢ ¬ 4 = 0 |
16 | 15 | intnan 485 | . . . . . 6 ⊢ ¬ (6 = 0 ∧ 4 = 0) |
17 | gcdn0cl 16502 | . . . . . 6 ⊢ (((6 ∈ ℤ ∧ 4 ∈ ℤ) ∧ ¬ (6 = 0 ∧ 4 = 0)) → (6 gcd 4) ∈ ℕ) | |
18 | 13, 16, 17 | mp2an 690 | . . . . 5 ⊢ (6 gcd 4) ∈ ℕ |
19 | 18 | nnne0i 12304 | . . . 4 ⊢ (6 gcd 4) ≠ 0 |
20 | 12, 19 | pm3.2i 469 | . . 3 ⊢ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) ≠ 0) |
21 | 6nn 12353 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
22 | 4nn 12347 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
23 | 21, 22 | pm3.2i 469 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ) |
24 | lcmgcdnn 16612 | . . . . . . 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 11928 | . . . . 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 256 | . . . 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 1458 | . 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4)) |
31 | 6gcd4e2 16539 | . . 3 ⊢ (6 gcd 4) = 2 | |
32 | 31 | oveq2i 7435 | . 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2) |
33 | 2cn 12339 | . . . 4 ⊢ 2 ∈ ℂ | |
34 | 2ne0 12368 | . . . 4 ⊢ 2 ≠ 0 | |
35 | 1, 2, 33, 34 | divassi 12021 | . . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2)) |
36 | 4d2e2 12434 | . . . 4 ⊢ (4 / 2) = 2 | |
37 | 36 | oveq2i 7435 | . . 3 ⊢ (6 · (4 / 2)) = (6 · 2) |
38 | 6t2e12 12833 | . . 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 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 (class class class)co 7424 ℂcc 11156 0cc0 11158 1c1 11159 · cmul 11163 / cdiv 11921 ℕcn 12264 2c2 12319 4c4 12321 6c6 12323 ℤcz 12610 ;cdc 12729 gcd cgcd 16494 lcm clcm 16589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-inf 9486 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-rp 13029 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-dvds 16257 df-gcd 16495 df-lcm 16591 |
This theorem is referenced by: lcmf2a3a4e12 16648 lcm4un 41715 |
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