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Mirrors > Home > ILE Home > Th. List > 6lcm4e12 | 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 8930 | . . . 4 ⊢ 6 ∈ ℂ | |
2 | 4cn 8926 | . . . 4 ⊢ 4 ∈ ℂ | |
3 | 1, 2 | mulcli 7895 | . . 3 ⊢ (6 · 4) ∈ ℂ |
4 | 6nn0 9126 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
5 | 4 | nn0zi 9204 | . . . 4 ⊢ 6 ∈ ℤ |
6 | 4z 9212 | . . . 4 ⊢ 4 ∈ ℤ | |
7 | lcmcl 11983 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ0) | |
8 | 7 | nn0cnd 9160 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ) |
9 | 5, 6, 8 | mp2an 423 | . . 3 ⊢ (6 lcm 4) ∈ ℂ |
10 | gcdcl 11884 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ0) | |
11 | 10 | nn0cnd 9160 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℂ) |
12 | 5, 6, 11 | mp2an 423 | . . . 4 ⊢ (6 gcd 4) ∈ ℂ |
13 | 5, 6 | pm3.2i 270 | . . . . . . 7 ⊢ (6 ∈ ℤ ∧ 4 ∈ ℤ) |
14 | 4ne0 8946 | . . . . . . . . 9 ⊢ 4 ≠ 0 | |
15 | 14 | neii 2336 | . . . . . . . 8 ⊢ ¬ 4 = 0 |
16 | 15 | intnan 919 | . . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0) |
17 | gcdn0cl 11880 | . . . . . . 7 ⊢ (((6 ∈ ℤ ∧ 4 ∈ ℤ) ∧ ¬ (6 = 0 ∧ 4 = 0)) → (6 gcd 4) ∈ ℕ) | |
18 | 13, 16, 17 | mp2an 423 | . . . . . 6 ⊢ (6 gcd 4) ∈ ℕ |
19 | 18 | nnne0i 8880 | . . . . 5 ⊢ (6 gcd 4) ≠ 0 |
20 | 18 | nnzi 9203 | . . . . . 6 ⊢ (6 gcd 4) ∈ ℤ |
21 | 0z 9193 | . . . . . 6 ⊢ 0 ∈ ℤ | |
22 | zapne 9256 | . . . . . 6 ⊢ (((6 gcd 4) ∈ ℤ ∧ 0 ∈ ℤ) → ((6 gcd 4) # 0 ↔ (6 gcd 4) ≠ 0)) | |
23 | 20, 21, 22 | mp2an 423 | . . . . 5 ⊢ ((6 gcd 4) # 0 ↔ (6 gcd 4) ≠ 0) |
24 | 19, 23 | mpbir 145 | . . . 4 ⊢ (6 gcd 4) # 0 |
25 | 12, 24 | pm3.2i 270 | . . 3 ⊢ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0) |
26 | 6nn 9013 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
27 | 4nn 9011 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
28 | 26, 27 | pm3.2i 270 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ) |
29 | lcmgcdnn 11993 | . . . . . . 7 ⊢ ((6 ∈ ℕ ∧ 4 ∈ ℕ) → ((6 lcm 4) · (6 gcd 4)) = (6 · 4)) | |
30 | 28, 29 | mp1i 10 | . . . . . 6 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0)) → ((6 lcm 4) · (6 gcd 4)) = (6 · 4)) |
31 | 30 | eqcomd 2170 | . . . . 5 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0)) → (6 · 4) = ((6 lcm 4) · (6 gcd 4))) |
32 | divmulap3 8564 | . . . . 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)))) | |
33 | 31, 32 | mpbird 166 | . . . 4 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0)) → ((6 · 4) / (6 gcd 4)) = (6 lcm 4)) |
34 | 33 | eqcomd 2170 | . . 3 ⊢ (((6 · 4) ∈ ℂ ∧ (6 lcm 4) ∈ ℂ ∧ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0)) → (6 lcm 4) = ((6 · 4) / (6 gcd 4))) |
35 | 3, 9, 25, 34 | mp3an 1326 | . 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4)) |
36 | 6gcd4e2 11913 | . . 3 ⊢ (6 gcd 4) = 2 | |
37 | 36 | oveq2i 5847 | . 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2) |
38 | 2cn 8919 | . . . 4 ⊢ 2 ∈ ℂ | |
39 | 2ap0 8941 | . . . 4 ⊢ 2 # 0 | |
40 | 1, 2, 38, 39 | divassapi 8655 | . . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2)) |
41 | 4d2e2 9008 | . . . 4 ⊢ (4 / 2) = 2 | |
42 | 41 | oveq2i 5847 | . . 3 ⊢ (6 · (4 / 2)) = (6 · 2) |
43 | 6t2e12 9416 | . . 3 ⊢ (6 · 2) = ;12 | |
44 | 40, 42, 43 | 3eqtri 2189 | . 2 ⊢ ((6 · 4) / 2) = ;12 |
45 | 35, 37, 44 | 3eqtri 2189 | 1 ⊢ (6 lcm 4) = ;12 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 class class class wbr 3976 (class class class)co 5836 ℂcc 7742 0cc0 7744 1c1 7745 · cmul 7749 # cap 8470 / cdiv 8559 ℕcn 8848 2c2 8899 4c4 8901 6c6 8903 ℤcz 9182 ;cdc 9313 gcd cgcd 11860 lcm clcm 11971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-n0 9106 df-z 9183 df-dec 9314 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-dvds 11714 df-gcd 11861 df-lcm 11972 |
This theorem is referenced by: (None) |
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