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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 11266 | . . 3 ⊢ (6 · 4) ∈ ℂ |
4 | 6nn0 12545 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
5 | 4 | nn0zi 12640 | . . . 4 ⊢ 6 ∈ ℤ |
6 | 4z 12649 | . . . 4 ⊢ 4 ∈ ℤ | |
7 | lcmcl 16635 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ0) | |
8 | 7 | nn0cnd 12587 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ) |
9 | 5, 6, 8 | mp2an 692 | . . 3 ⊢ (6 lcm 4) ∈ ℂ |
10 | gcdcl 16540 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ0) | |
11 | 10 | nn0cnd 12587 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℂ) |
12 | 5, 6, 11 | mp2an 692 | . . . 4 ⊢ (6 gcd 4) ∈ ℂ |
13 | 5, 6 | pm3.2i 470 | . . . . . 6 ⊢ (6 ∈ ℤ ∧ 4 ∈ ℤ) |
14 | 4ne0 12372 | . . . . . . . 8 ⊢ 4 ≠ 0 | |
15 | 14 | neii 2940 | . . . . . . 7 ⊢ ¬ 4 = 0 |
16 | 15 | intnan 486 | . . . . . 6 ⊢ ¬ (6 = 0 ∧ 4 = 0) |
17 | gcdn0cl 16536 | . . . . . 6 ⊢ (((6 ∈ ℤ ∧ 4 ∈ ℤ) ∧ ¬ (6 = 0 ∧ 4 = 0)) → (6 gcd 4) ∈ ℕ) | |
18 | 13, 16, 17 | mp2an 692 | . . . . 5 ⊢ (6 gcd 4) ∈ ℕ |
19 | 18 | nnne0i 12304 | . . . 4 ⊢ (6 gcd 4) ≠ 0 |
20 | 12, 19 | pm3.2i 470 | . . 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 470 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ) |
24 | lcmgcdnn 16645 | . . . . . . 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 2741 | . . . . 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 11925 | . . . . 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 2741 | . . 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 1460 | . 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4)) |
31 | 6gcd4e2 16572 | . . 3 ⊢ (6 gcd 4) = 2 | |
32 | 31 | oveq2i 7442 | . 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 7442 | . . 3 ⊢ (6 · (4 / 2)) = (6 · 2) |
38 | 6t2e12 12835 | . . 3 ⊢ (6 · 2) = ;12 | |
39 | 35, 37, 38 | 3eqtri 2767 | . 2 ⊢ ((6 · 4) / 2) = ;12 |
40 | 30, 32, 39 | 3eqtri 2767 | 1 ⊢ (6 lcm 4) = ;12 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 · cmul 11158 / cdiv 11918 ℕcn 12264 2c2 12319 4c4 12321 6c6 12323 ℤcz 12611 ;cdc 12731 gcd cgcd 16528 lcm clcm 16622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 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 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529 df-lcm 16624 |
This theorem is referenced by: lcmf2a3a4e12 16681 lcm4un 41998 |
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