6lcm4e12 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > 6lcm4e12		GIF version

Theorem 6lcm4e12 12019

Description: The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.)

**Assertion**
Ref	Expression
6lcm4e12	⊢ (6 lcm 4) = ;12

**Proof of Theorem 6lcm4e12**
Step	Hyp	Ref	Expression
1		6cn 8939	. . . 4 ⊢ 6 ∈ ℂ
2		4cn 8935	. . . 4 ⊢ 4 ∈ ℂ
3	1, 2	mulcli 7904	. . 3 ⊢ (6 · 4) ∈ ℂ
4		6nn0 9135	. . . . 5 ⊢ 6 ∈ ℕ₀
5	4	nn0zi 9213	. . . 4 ⊢ 6 ∈ ℤ
6		4z 9221	. . . 4 ⊢ 4 ∈ ℤ
7		lcmcl 12004	. . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ₀)
8	7	nn0cnd 9169	. . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ)
9	5, 6, 8	mp2an 423	. . 3 ⊢ (6 lcm 4) ∈ ℂ
10		gcdcl 11899	. . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ₀)
11	10	nn0cnd 9169	. . . . 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 8955	. . . . . . . . 9 ⊢ 4 ≠ 0
15	14	neii 2338	. . . . . . . 8 ⊢ ¬ 4 = 0
16	15	intnan 919	. . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0)
17		gcdn0cl 11895	. . . . . . 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 8889	. . . . 5 ⊢ (6 gcd 4) ≠ 0
20	18	nnzi 9212	. . . . . 6 ⊢ (6 gcd 4) ∈ ℤ
21		0z 9202	. . . . . 6 ⊢ 0 ∈ ℤ
22		zapne 9265	. . . . . 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 9022	. . . . . . . 8 ⊢ 6 ∈ ℕ
27		4nn 9020	. . . . . . . 8 ⊢ 4 ∈ ℕ
28	26, 27	pm3.2i 270	. . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ)
29		lcmgcdnn 12014	. . . . . . 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 2171	. . . . 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 8573	. . . . 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 2171	. . 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 1327	. 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4))
36		6gcd4e2 11928	. . 3 ⊢ (6 gcd 4) = 2
37	36	oveq2i 5853	. 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2)
38		2cn 8928	. . . 4 ⊢ 2 ∈ ℂ
39		2ap0 8950	. . . 4 ⊢ 2 # 0
40	1, 2, 38, 39	divassapi 8664	. . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2))
41		4d2e2 9017	. . . 4 ⊢ (4 / 2) = 2
42	41	oveq2i 5853	. . 3 ⊢ (6 · (4 / 2)) = (6 · 2)
43		6t2e12 9425	. . 3 ⊢ (6 · 2) = ;12
44	40, 42, 43	3eqtri 2190	. 2 ⊢ ((6 · 4) / 2) = ;12
45	35, 37, 44	3eqtri 2190	1 ⊢ (6 lcm 4) = ;12

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 ≠ wne 2336 class class class wbr 3982 (class class class)co 5842 ℂcc 7751 0cc0 7753 1c1 7754 · cmul 7758 # cap 8479 / cdiv 8568 ℕcn 8857 2c2 8908 4c4 8910 6c6 8912 ℤcz 9191 cdc 9322 gcd cgcd 11875 lcm clcm 11992

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-5 8919 df-6 8920 df-7 8921 df-8 8922 df-9 8923 df-n0 9115 df-z 9192 df-dec 9323 df-uz 9467 df-q 9558 df-rp 9590 df-fz 9945 df-fzo 10078 df-fl 10205 df-mod 10258 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-dvds 11728 df-gcd 11876 df-lcm 11993

This theorem is referenced by: (None)

W3C validator