6lcm4e12 - Intuitionistic Logic Explorer

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Theorem 6lcm4e12 10944

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 8439	. . . 4 ⊢ 6 ∈ ℂ
2		4cn 8435	. . . 4 ⊢ 4 ∈ ℂ
3	1, 2	mulcli 7437	. . 3 ⊢ (6 · 4) ∈ ℂ
4		6nn0 8627	. . . . 5 ⊢ 6 ∈ ℕ₀
5	4	nn0zi 8705	. . . 4 ⊢ 6 ∈ ℤ
6		4z 8713	. . . 4 ⊢ 4 ∈ ℤ
7		lcmcl 10929	. . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ₀)
8	7	nn0cnd 8661	. . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ)
9	5, 6, 8	mp2an 417	. . 3 ⊢ (6 lcm 4) ∈ ℂ
10		gcdcl 10833	. . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ₀)
11	10	nn0cnd 8661	. . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℂ)
12	5, 6, 11	mp2an 417	. . . 4 ⊢ (6 gcd 4) ∈ ℂ
13	5, 6	pm3.2i 266	. . . . . . 7 ⊢ (6 ∈ ℤ ∧ 4 ∈ ℤ)
14		4ne0 8455	. . . . . . . . 9 ⊢ 4 ≠ 0
15	14	neii 2253	. . . . . . . 8 ⊢ ¬ 4 = 0
16	15	intnan 874	. . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0)
17		gcdn0cl 10829	. . . . . . 7 ⊢ (((6 ∈ ℤ ∧ 4 ∈ ℤ) ∧ ¬ (6 = 0 ∧ 4 = 0)) → (6 gcd 4) ∈ ℕ)
18	13, 16, 17	mp2an 417	. . . . . 6 ⊢ (6 gcd 4) ∈ ℕ
19	18	nnne0i 8388	. . . . 5 ⊢ (6 gcd 4) ≠ 0
20	18	nnzi 8704	. . . . . 6 ⊢ (6 gcd 4) ∈ ℤ
21		0z 8694	. . . . . 6 ⊢ 0 ∈ ℤ
22		zapne 8754	. . . . . 6 ⊢ (((6 gcd 4) ∈ ℤ ∧ 0 ∈ ℤ) → ((6 gcd 4) # 0 ↔ (6 gcd 4) ≠ 0))
23	20, 21, 22	mp2an 417	. . . . 5 ⊢ ((6 gcd 4) # 0 ↔ (6 gcd 4) ≠ 0)
24	19, 23	mpbir 144	. . . 4 ⊢ (6 gcd 4) # 0
25	12, 24	pm3.2i 266	. . 3 ⊢ ((6 gcd 4) ∈ ℂ ∧ (6 gcd 4) # 0)
26		6nn 8515	. . . . . . . 8 ⊢ 6 ∈ ℕ
27		4nn 8513	. . . . . . . 8 ⊢ 4 ∈ ℕ
28	26, 27	pm3.2i 266	. . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ)
29		lcmgcdnn 10939	. . . . . . 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 2090	. . . . 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 8083	. . . . 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 165	. . . 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 2090	. . 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 1271	. 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4))
36		6gcd4e2 10859	. . 3 ⊢ (6 gcd 4) = 2
37	36	oveq2i 5624	. 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2)
38		2cn 8428	. . . 4 ⊢ 2 ∈ ℂ
39		2ap0 8450	. . . 4 ⊢ 2 # 0
40	1, 2, 38, 39	divassapi 8174	. . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2))
41		4d2e2 8510	. . . 4 ⊢ (4 / 2) = 2
42	41	oveq2i 5624	. . 3 ⊢ (6 · (4 / 2)) = (6 · 2)
43		6t2e12 8912	. . 3 ⊢ (6 · 2) = ;12
44	40, 42, 43	3eqtri 2109	. 2 ⊢ ((6 · 4) / 2) = ;12
45	35, 37, 44	3eqtri 2109	1 ⊢ (6 lcm 4) = ;12

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∧ wa 102 ↔ wb 103 ∧ w3a 922 = wceq 1287 ∈ wcel 1436 ≠ wne 2251 class class class wbr 3820 (class class class)co 5613 ℂcc 7292 0cc0 7294 1c1 7295 · cmul 7299 # cap 7999 / cdiv 8078 ℕcn 8357 2c2 8407 4c4 8409 6c6 8411 ℤcz 8683 cdc 8809 gcd cgcd 10813 lcm clcm 10917

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3929 ax-sep 3932 ax-nul 3940 ax-pow 3984 ax-pr 4010 ax-un 4234 ax-setind 4326 ax-iinf 4376 ax-cnex 7380 ax-resscn 7381 ax-1cn 7382 ax-1re 7383 ax-icn 7384 ax-addcl 7385 ax-addrcl 7386 ax-mulcl 7387 ax-mulrcl 7388 ax-addcom 7389 ax-mulcom 7390 ax-addass 7391 ax-mulass 7392 ax-distr 7393 ax-i2m1 7394 ax-0lt1 7395 ax-1rid 7396 ax-0id 7397 ax-rnegex 7398 ax-precex 7399 ax-cnre 7400 ax-pre-ltirr 7401 ax-pre-ltwlin 7402 ax-pre-lttrn 7403 ax-pre-apti 7404 ax-pre-ltadd 7405 ax-pre-mulgt0 7406 ax-pre-mulext 7407 ax-arch 7408 ax-caucvg 7409

This theorem depends on definitions: df-bi 115 df-dc 779 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rmo 2363 df-rab 2364 df-v 2617 df-sbc 2830 df-csb 2923 df-dif 2990 df-un 2992 df-in 2994 df-ss 3001 df-nul 3276 df-if 3380 df-pw 3417 df-sn 3437 df-pr 3438 df-op 3440 df-uni 3637 df-int 3672 df-iun 3715 df-br 3821 df-opab 3875 df-mpt 3876 df-tr 3912 df-id 4094 df-po 4097 df-iso 4098 df-iord 4167 df-on 4169 df-ilim 4170 df-suc 4172 df-iom 4379 df-xp 4417 df-rel 4418 df-cnv 4419 df-co 4420 df-dm 4421 df-rn 4422 df-res 4423 df-ima 4424 df-iota 4946 df-fun 4983 df-fn 4984 df-f 4985 df-f1 4986 df-fo 4987 df-f1o 4988 df-fv 4989 df-isom 4990 df-riota 5569 df-ov 5616 df-oprab 5617 df-mpt2 5618 df-1st 5868 df-2nd 5869 df-recs 6024 df-frec 6110 df-sup 6623 df-inf 6624 df-pnf 7468 df-mnf 7469 df-xr 7470 df-ltxr 7471 df-le 7472 df-sub 7599 df-neg 7600 df-reap 7993 df-ap 8000 df-div 8079 df-inn 8358 df-2 8416 df-3 8417 df-4 8418 df-5 8419 df-6 8420 df-7 8421 df-8 8422 df-9 8423 df-n0 8607 df-z 8684 df-dec 8810 df-uz 8952 df-q 9037 df-rp 9067 df-fz 9357 df-fzo 9482 df-fl 9605 df-mod 9658 df-iseq 9780 df-iexp 9853 df-cj 10171 df-re 10172 df-im 10173 df-rsqrt 10326 df-abs 10327 df-dvds 10672 df-gcd 10814 df-lcm 10918

This theorem is referenced by: (None)

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