6lcm4e12 - Intuitionistic Logic Explorer

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Theorem 6lcm4e12 12041

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 8960	. . . 4 ⊢ 6 ∈ ℂ
2		4cn 8956	. . . 4 ⊢ 4 ∈ ℂ
3	1, 2	mulcli 7925	. . 3 ⊢ (6 · 4) ∈ ℂ
4		6nn0 9156	. . . . 5 ⊢ 6 ∈ ℕ₀
5	4	nn0zi 9234	. . . 4 ⊢ 6 ∈ ℤ
6		4z 9242	. . . 4 ⊢ 4 ∈ ℤ
7		lcmcl 12026	. . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ₀)
8	7	nn0cnd 9190	. . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ)
9	5, 6, 8	mp2an 424	. . 3 ⊢ (6 lcm 4) ∈ ℂ
10		gcdcl 11921	. . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ₀)
11	10	nn0cnd 9190	. . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℂ)
12	5, 6, 11	mp2an 424	. . . 4 ⊢ (6 gcd 4) ∈ ℂ
13	5, 6	pm3.2i 270	. . . . . . 7 ⊢ (6 ∈ ℤ ∧ 4 ∈ ℤ)
14		4ne0 8976	. . . . . . . . 9 ⊢ 4 ≠ 0
15	14	neii 2342	. . . . . . . 8 ⊢ ¬ 4 = 0
16	15	intnan 924	. . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0)
17		gcdn0cl 11917	. . . . . . 7 ⊢ (((6 ∈ ℤ ∧ 4 ∈ ℤ) ∧ ¬ (6 = 0 ∧ 4 = 0)) → (6 gcd 4) ∈ ℕ)
18	13, 16, 17	mp2an 424	. . . . . 6 ⊢ (6 gcd 4) ∈ ℕ
19	18	nnne0i 8910	. . . . 5 ⊢ (6 gcd 4) ≠ 0
20	18	nnzi 9233	. . . . . 6 ⊢ (6 gcd 4) ∈ ℤ
21		0z 9223	. . . . . 6 ⊢ 0 ∈ ℤ
22		zapne 9286	. . . . . 6 ⊢ (((6 gcd 4) ∈ ℤ ∧ 0 ∈ ℤ) → ((6 gcd 4) # 0 ↔ (6 gcd 4) ≠ 0))
23	20, 21, 22	mp2an 424	. . . . 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 9043	. . . . . . . 8 ⊢ 6 ∈ ℕ
27		4nn 9041	. . . . . . . 8 ⊢ 4 ∈ ℕ
28	26, 27	pm3.2i 270	. . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ)
29		lcmgcdnn 12036	. . . . . . 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 2176	. . . . 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 8594	. . . . 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 2176	. . 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 1332	. 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4))
36		6gcd4e2 11950	. . 3 ⊢ (6 gcd 4) = 2
37	36	oveq2i 5864	. 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2)
38		2cn 8949	. . . 4 ⊢ 2 ∈ ℂ
39		2ap0 8971	. . . 4 ⊢ 2 # 0
40	1, 2, 38, 39	divassapi 8685	. . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2))
41		4d2e2 9038	. . . 4 ⊢ (4 / 2) = 2
42	41	oveq2i 5864	. . 3 ⊢ (6 · (4 / 2)) = (6 · 2)
43		6t2e12 9446	. . 3 ⊢ (6 · 2) = ;12
44	40, 42, 43	3eqtri 2195	. 2 ⊢ ((6 · 4) / 2) = ;12
45	35, 37, 44	3eqtri 2195	1 ⊢ (6 lcm 4) = ;12

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 class class class wbr 3989 (class class class)co 5853 ℂcc 7772 0cc0 7774 1c1 7775 · cmul 7779 # cap 8500 / cdiv 8589 ℕcn 8878 2c2 8929 4c4 8931 6c6 8933 ℤcz 9212 cdc 9343 gcd cgcd 11897 lcm clcm 12014

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-9 8944 df-n0 9136 df-z 9213 df-dec 9344 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-dvds 11750 df-gcd 11898 df-lcm 12015

This theorem is referenced by: (None)

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