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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 9032 | . . . 4 ⊢ 6 ∈ ℂ | |
2 | 4cn 9028 | . . . 4 ⊢ 4 ∈ ℂ | |
3 | 1, 2 | mulcli 7993 | . . 3 ⊢ (6 · 4) ∈ ℂ |
4 | 6nn0 9228 | . . . . 5 ⊢ 6 ∈ ℕ0 | |
5 | 4 | nn0zi 9306 | . . . 4 ⊢ 6 ∈ ℤ |
6 | 4z 9314 | . . . 4 ⊢ 4 ∈ ℤ | |
7 | lcmcl 12107 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ0) | |
8 | 7 | nn0cnd 9262 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ) |
9 | 5, 6, 8 | mp2an 426 | . . 3 ⊢ (6 lcm 4) ∈ ℂ |
10 | gcdcl 12002 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ0) | |
11 | 10 | nn0cnd 9262 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℂ) |
12 | 5, 6, 11 | mp2an 426 | . . . 4 ⊢ (6 gcd 4) ∈ ℂ |
13 | 5, 6 | pm3.2i 272 | . . . . . . 7 ⊢ (6 ∈ ℤ ∧ 4 ∈ ℤ) |
14 | 4ne0 9048 | . . . . . . . . 9 ⊢ 4 ≠ 0 | |
15 | 14 | neii 2362 | . . . . . . . 8 ⊢ ¬ 4 = 0 |
16 | 15 | intnan 930 | . . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0) |
17 | gcdn0cl 11998 | . . . . . . 7 ⊢ (((6 ∈ ℤ ∧ 4 ∈ ℤ) ∧ ¬ (6 = 0 ∧ 4 = 0)) → (6 gcd 4) ∈ ℕ) | |
18 | 13, 16, 17 | mp2an 426 | . . . . . 6 ⊢ (6 gcd 4) ∈ ℕ |
19 | 18 | nnne0i 8982 | . . . . 5 ⊢ (6 gcd 4) ≠ 0 |
20 | 18 | nnzi 9305 | . . . . . 6 ⊢ (6 gcd 4) ∈ ℤ |
21 | 0z 9295 | . . . . . 6 ⊢ 0 ∈ ℤ | |
22 | zapne 9358 | . . . . . 6 ⊢ (((6 gcd 4) ∈ ℤ ∧ 0 ∈ ℤ) → ((6 gcd 4) # 0 ↔ (6 gcd 4) ≠ 0)) | |
23 | 20, 21, 22 | mp2an 426 | . . . . 5 ⊢ ((6 gcd 4) # 0 ↔ (6 gcd 4) ≠ 0) |
24 | 19, 23 | mpbir 146 | . . . 4 ⊢ (6 gcd 4) # 0 |
25 | 12, 24 | pm3.2i 272 | . . 3 ⊢ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0) |
26 | 6nn 9115 | . . . . . . . 8 ⊢ 6 ∈ ℕ | |
27 | 4nn 9113 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
28 | 26, 27 | pm3.2i 272 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ) |
29 | lcmgcdnn 12117 | . . . . . . 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 2195 | . . . . 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 8665 | . . . . 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 167 | . . . 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 2195 | . . 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 1348 | . 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4)) |
36 | 6gcd4e2 12031 | . . 3 ⊢ (6 gcd 4) = 2 | |
37 | 36 | oveq2i 5908 | . 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2) |
38 | 2cn 9021 | . . . 4 ⊢ 2 ∈ ℂ | |
39 | 2ap0 9043 | . . . 4 ⊢ 2 # 0 | |
40 | 1, 2, 38, 39 | divassapi 8756 | . . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2)) |
41 | 4d2e2 9110 | . . . 4 ⊢ (4 / 2) = 2 | |
42 | 41 | oveq2i 5908 | . . 3 ⊢ (6 · (4 / 2)) = (6 · 2) |
43 | 6t2e12 9518 | . . 3 ⊢ (6 · 2) = ;12 | |
44 | 40, 42, 43 | 3eqtri 2214 | . 2 ⊢ ((6 · 4) / 2) = ;12 |
45 | 35, 37, 44 | 3eqtri 2214 | 1 ⊢ (6 lcm 4) = ;12 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 class class class wbr 4018 (class class class)co 5897 ℂcc 7840 0cc0 7842 1c1 7843 · cmul 7847 # cap 8569 / cdiv 8660 ℕcn 8950 2c2 9001 4c4 9003 6c6 9005 ℤcz 9284 ;cdc 9415 gcd cgcd 11978 lcm clcm 12095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 ax-arch 7961 ax-caucvg 7962 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-frec 6417 df-sup 7014 df-inf 7015 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-5 9012 df-6 9013 df-7 9014 df-8 9015 df-9 9016 df-n0 9208 df-z 9285 df-dec 9416 df-uz 9560 df-q 9652 df-rp 9686 df-fz 10041 df-fzo 10175 df-fl 10303 df-mod 10356 df-seqfrec 10479 df-exp 10554 df-cj 10886 df-re 10887 df-im 10888 df-rsqrt 11042 df-abs 11043 df-dvds 11830 df-gcd 11979 df-lcm 12096 |
This theorem is referenced by: (None) |
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