6lcm4e12 - Intuitionistic Logic Explorer

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Theorem 6lcm4e12 12228

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 9066	. . . 4 ⊢ 6 ∈ ℂ
2		4cn 9062	. . . 4 ⊢ 4 ∈ ℂ
3	1, 2	mulcli 8026	. . 3 ⊢ (6 · 4) ∈ ℂ
4		6nn0 9264	. . . . 5 ⊢ 6 ∈ ℕ₀
5	4	nn0zi 9342	. . . 4 ⊢ 6 ∈ ℤ
6		4z 9350	. . . 4 ⊢ 4 ∈ ℤ
7		lcmcl 12213	. . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ₀)
8	7	nn0cnd 9298	. . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ)
9	5, 6, 8	mp2an 426	. . 3 ⊢ (6 lcm 4) ∈ ℂ
10		gcdcl 12106	. . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ₀)
11	10	nn0cnd 9298	. . . . 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 9082	. . . . . . . . 9 ⊢ 4 ≠ 0
15	14	neii 2366	. . . . . . . 8 ⊢ ¬ 4 = 0
16	15	intnan 930	. . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0)
17		gcdn0cl 12102	. . . . . . 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 9016	. . . . 5 ⊢ (6 gcd 4) ≠ 0
20	18	nnzi 9341	. . . . . 6 ⊢ (6 gcd 4) ∈ ℤ
21		0z 9331	. . . . . 6 ⊢ 0 ∈ ℤ
22		zapne 9394	. . . . . 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 9150	. . . . . . . 8 ⊢ 6 ∈ ℕ
27		4nn 9148	. . . . . . . 8 ⊢ 4 ∈ ℕ
28	26, 27	pm3.2i 272	. . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ)
29		lcmgcdnn 12223	. . . . . . 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 2199	. . . . 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 8698	. . . . 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 2199	. . 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 12135	. . 3 ⊢ (6 gcd 4) = 2
37	36	oveq2i 5930	. 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2)
38		2cn 9055	. . . 4 ⊢ 2 ∈ ℂ
39		2ap0 9077	. . . 4 ⊢ 2 # 0
40	1, 2, 38, 39	divassapi 8789	. . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2))
41		4d2e2 9145	. . . 4 ⊢ (4 / 2) = 2
42	41	oveq2i 5930	. . 3 ⊢ (6 · (4 / 2)) = (6 · 2)
43		6t2e12 9554	. . 3 ⊢ (6 · 2) = ;12
44	40, 42, 43	3eqtri 2218	. 2 ⊢ ((6 · 4) / 2) = ;12
45	35, 37, 44	3eqtri 2218	1 ⊢ (6 lcm 4) = ;12

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 class class class wbr 4030 (class class class)co 5919 ℂcc 7872 0cc0 7874 1c1 7875 · cmul 7879 # cap 8602 / cdiv 8693 ℕcn 8984 2c2 9035 4c4 9037 6c6 9039 ℤcz 9320 cdc 9451 gcd cgcd 12082 lcm clcm 12201

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 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

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 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-frec 6446 df-sup 7045 df-inf 7046 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-7 9048 df-8 9049 df-9 9050 df-n0 9244 df-z 9321 df-dec 9452 df-uz 9596 df-q 9688 df-rp 9723 df-fz 10078 df-fzo 10212 df-fl 10342 df-mod 10397 df-seqfrec 10522 df-exp 10613 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-dvds 11934 df-gcd 12083 df-lcm 12202

This theorem is referenced by: (None)

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