6lcm4e12 - Intuitionistic Logic Explorer

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Theorem 6lcm4e12 12570

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 9155	. . . 4 ⊢ 6 ∈ ℂ
2		4cn 9151	. . . 4 ⊢ 4 ∈ ℂ
3	1, 2	mulcli 8114	. . 3 ⊢ (6 · 4) ∈ ℂ
4		6nn0 9353	. . . . 5 ⊢ 6 ∈ ℕ₀
5	4	nn0zi 9431	. . . 4 ⊢ 6 ∈ ℤ
6		4z 9439	. . . 4 ⊢ 4 ∈ ℤ
7		lcmcl 12555	. . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ₀)
8	7	nn0cnd 9387	. . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ)
9	5, 6, 8	mp2an 426	. . 3 ⊢ (6 lcm 4) ∈ ℂ
10		gcdcl 12448	. . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ₀)
11	10	nn0cnd 9387	. . . . 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 9171	. . . . . . . . 9 ⊢ 4 ≠ 0
15	14	neii 2380	. . . . . . . 8 ⊢ ¬ 4 = 0
16	15	intnan 931	. . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0)
17		gcdn0cl 12444	. . . . . . 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 9105	. . . . 5 ⊢ (6 gcd 4) ≠ 0
20	18	nnzi 9430	. . . . . 6 ⊢ (6 gcd 4) ∈ ℤ
21		0z 9420	. . . . . 6 ⊢ 0 ∈ ℤ
22		zapne 9484	. . . . . 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 9239	. . . . . . . 8 ⊢ 6 ∈ ℕ
27		4nn 9237	. . . . . . . 8 ⊢ 4 ∈ ℕ
28	26, 27	pm3.2i 272	. . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ)
29		lcmgcdnn 12565	. . . . . . 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 2213	. . . . 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 8787	. . . . 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 2213	. . 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 1350	. 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4))
36		6gcd4e2 12477	. . 3 ⊢ (6 gcd 4) = 2
37	36	oveq2i 5980	. 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2)
38		2cn 9144	. . . 4 ⊢ 2 ∈ ℂ
39		2ap0 9166	. . . 4 ⊢ 2 # 0
40	1, 2, 38, 39	divassapi 8878	. . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2))
41		4d2e2 9234	. . . 4 ⊢ (4 / 2) = 2
42	41	oveq2i 5980	. . 3 ⊢ (6 · (4 / 2)) = (6 · 2)
43		6t2e12 9644	. . 3 ⊢ (6 · 2) = ;12
44	40, 42, 43	3eqtri 2232	. 2 ⊢ ((6 · 4) / 2) = ;12
45	35, 37, 44	3eqtri 2232	1 ⊢ (6 lcm 4) = ;12

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 ∧ w3a 981 = wceq 1373 ∈ wcel 2178 ≠ wne 2378 class class class wbr 4060 (class class class)co 5969 ℂcc 7960 0cc0 7962 1c1 7963 · cmul 7967 # cap 8691 / cdiv 8782 ℕcn 9073 2c2 9124 4c4 9126 6c6 9128 ℤcz 9409 cdc 9541 gcd cgcd 12435 lcm clcm 12543

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4176 ax-sep 4179 ax-nul 4187 ax-pow 4235 ax-pr 4270 ax-un 4499 ax-setind 4604 ax-iinf 4655 ax-cnex 8053 ax-resscn 8054 ax-1cn 8055 ax-1re 8056 ax-icn 8057 ax-addcl 8058 ax-addrcl 8059 ax-mulcl 8060 ax-mulrcl 8061 ax-addcom 8062 ax-mulcom 8063 ax-addass 8064 ax-mulass 8065 ax-distr 8066 ax-i2m1 8067 ax-0lt1 8068 ax-1rid 8069 ax-0id 8070 ax-rnegex 8071 ax-precex 8072 ax-cnre 8073 ax-pre-ltirr 8074 ax-pre-ltwlin 8075 ax-pre-lttrn 8076 ax-pre-apti 8077 ax-pre-ltadd 8078 ax-pre-mulgt0 8079 ax-pre-mulext 8080 ax-arch 8081 ax-caucvg 8082

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2779 df-sbc 3007 df-csb 3103 df-dif 3177 df-un 3179 df-in 3181 df-ss 3188 df-nul 3470 df-if 3581 df-pw 3629 df-sn 3650 df-pr 3651 df-op 3653 df-uni 3866 df-int 3901 df-iun 3944 df-br 4061 df-opab 4123 df-mpt 4124 df-tr 4160 df-id 4359 df-po 4362 df-iso 4363 df-iord 4432 df-on 4434 df-ilim 4435 df-suc 4437 df-iom 4658 df-xp 4700 df-rel 4701 df-cnv 4702 df-co 4703 df-dm 4704 df-rn 4705 df-res 4706 df-ima 4707 df-iota 5252 df-fun 5293 df-fn 5294 df-f 5295 df-f1 5296 df-fo 5297 df-f1o 5298 df-fv 5299 df-isom 5300 df-riota 5924 df-ov 5972 df-oprab 5973 df-mpo 5974 df-1st 6251 df-2nd 6252 df-recs 6416 df-frec 6502 df-sup 7114 df-inf 7115 df-pnf 8146 df-mnf 8147 df-xr 8148 df-ltxr 8149 df-le 8150 df-sub 8282 df-neg 8283 df-reap 8685 df-ap 8692 df-div 8783 df-inn 9074 df-2 9132 df-3 9133 df-4 9134 df-5 9135 df-6 9136 df-7 9137 df-8 9138 df-9 9139 df-n0 9333 df-z 9410 df-dec 9542 df-uz 9686 df-q 9778 df-rp 9813 df-fz 10168 df-fzo 10302 df-fl 10452 df-mod 10507 df-seqfrec 10632 df-exp 10723 df-cj 11314 df-re 11315 df-im 11316 df-rsqrt 11470 df-abs 11471 df-dvds 12260 df-gcd 12436 df-lcm 12544

This theorem is referenced by: (None)

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