6lcm4e12 - Intuitionistic Logic Explorer

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Theorem 6lcm4e12 10676

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 8240	. . . 4 ⊢ 6 ∈ ℂ
2		4cn 8236	. . . 4 ⊢ 4 ∈ ℂ
3	1, 2	mulcli 7238	. . 3 ⊢ (6 · 4) ∈ ℂ
4		6nn0 8428	. . . . 5 ⊢ 6 ∈ ℕ₀
5	4	nn0zi 8506	. . . 4 ⊢ 6 ∈ ℤ
6		4z 8514	. . . 4 ⊢ 4 ∈ ℤ
7		lcmcl 10661	. . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ₀)
8	7	nn0cnd 8462	. . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ)
9	5, 6, 8	mp2an 417	. . 3 ⊢ (6 lcm 4) ∈ ℂ
10		gcdcl 10565	. . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ₀)
11	10	nn0cnd 8462	. . . . 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 8256	. . . . . . . . 9 ⊢ 4 ≠ 0
15	14	neii 2251	. . . . . . . 8 ⊢ ¬ 4 = 0
16	15	intnan 872	. . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0)
17		gcdn0cl 10561	. . . . . . 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 8189	. . . . 5 ⊢ (6 gcd 4) ≠ 0
20	18	nnzi 8505	. . . . . 6 ⊢ (6 gcd 4) ∈ ℤ
21		0z 8495	. . . . . 6 ⊢ 0 ∈ ℤ
22		zapne 8555	. . . . . 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 8316	. . . . . . . 8 ⊢ 6 ∈ ℕ
27		4nn 8314	. . . . . . . 8 ⊢ 4 ∈ ℕ
28	26, 27	pm3.2i 266	. . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ)
29		lcmgcdnn 10671	. . . . . . 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 2088	. . . . 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 7884	. . . . 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 2088	. . 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 1269	. 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4))
36		6gcd4e2 10591	. . 3 ⊢ (6 gcd 4) = 2
37	36	oveq2i 5574	. 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2)
38		2cn 8229	. . . 4 ⊢ 2 ∈ ℂ
39		2ap0 8251	. . . 4 ⊢ 2 # 0
40	1, 2, 38, 39	divassapi 7975	. . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2))
41		4d2e2 8311	. . . 4 ⊢ (4 / 2) = 2
42	41	oveq2i 5574	. . 3 ⊢ (6 · (4 / 2)) = (6 · 2)
43		6t2e12 8713	. . 3 ⊢ (6 · 2) = ;12
44	40, 42, 43	3eqtri 2107	. 2 ⊢ ((6 · 4) / 2) = ;12
45	35, 37, 44	3eqtri 2107	1 ⊢ (6 lcm 4) = ;12

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∧ wa 102 ↔ wb 103 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 ≠ wne 2249 class class class wbr 3805 (class class class)co 5563 ℂcc 7093 0cc0 7095 1c1 7096 · cmul 7100 # cap 7800 / cdiv 7879 ℕcn 8158 2c2 8208 4c4 8210 6c6 8212 ℤcz 8484 cdc 8610 gcd cgcd 10545 lcm clcm 10649

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 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3913 ax-sep 3916 ax-nul 3924 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-iinf 4357 ax-cnex 7181 ax-resscn 7182 ax-1cn 7183 ax-1re 7184 ax-icn 7185 ax-addcl 7186 ax-addrcl 7187 ax-mulcl 7188 ax-mulrcl 7189 ax-addcom 7190 ax-mulcom 7191 ax-addass 7192 ax-mulass 7193 ax-distr 7194 ax-i2m1 7195 ax-0lt1 7196 ax-1rid 7197 ax-0id 7198 ax-rnegex 7199 ax-precex 7200 ax-cnre 7201 ax-pre-ltirr 7202 ax-pre-ltwlin 7203 ax-pre-lttrn 7204 ax-pre-apti 7205 ax-pre-ltadd 7206 ax-pre-mulgt0 7207 ax-pre-mulext 7208 ax-arch 7209 ax-caucvg 7210

This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2612 df-sbc 2825 df-csb 2918 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-nul 3268 df-if 3369 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-tr 3896 df-id 4076 df-po 4079 df-iso 4080 df-iord 4149 df-on 4151 df-ilim 4152 df-suc 4154 df-iom 4360 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-f1 4957 df-fo 4958 df-f1o 4959 df-fv 4960 df-isom 4961 df-riota 5519 df-ov 5566 df-oprab 5567 df-mpt2 5568 df-1st 5818 df-2nd 5819 df-recs 5974 df-frec 6060 df-sup 6491 df-inf 6492 df-pnf 7269 df-mnf 7270 df-xr 7271 df-ltxr 7272 df-le 7273 df-sub 7400 df-neg 7401 df-reap 7794 df-ap 7801 df-div 7880 df-inn 8159 df-2 8217 df-3 8218 df-4 8219 df-5 8220 df-6 8221 df-7 8222 df-8 8223 df-9 8224 df-n0 8408 df-z 8485 df-dec 8611 df-uz 8753 df-q 8838 df-rp 8868 df-fz 9158 df-fzo 9282 df-fl 9404 df-mod 9457 df-iseq 9574 df-iexp 9625 df-cj 9930 df-re 9931 df-im 9932 df-rsqrt 10085 df-abs 10086 df-dvds 10404 df-gcd 10546 df-lcm 10650

This theorem is referenced by: (None)

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