6lcm4e12 - Intuitionistic Logic Explorer

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Theorem 6lcm4e12 12459

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 9131	. . . 4 ⊢ 6 ∈ ℂ
2		4cn 9127	. . . 4 ⊢ 4 ∈ ℂ
3	1, 2	mulcli 8090	. . 3 ⊢ (6 · 4) ∈ ℂ
4		6nn0 9329	. . . . 5 ⊢ 6 ∈ ℕ₀
5	4	nn0zi 9407	. . . 4 ⊢ 6 ∈ ℤ
6		4z 9415	. . . 4 ⊢ 4 ∈ ℤ
7		lcmcl 12444	. . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ₀)
8	7	nn0cnd 9363	. . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ)
9	5, 6, 8	mp2an 426	. . 3 ⊢ (6 lcm 4) ∈ ℂ
10		gcdcl 12337	. . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ₀)
11	10	nn0cnd 9363	. . . . 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 9147	. . . . . . . . 9 ⊢ 4 ≠ 0
15	14	neii 2379	. . . . . . . 8 ⊢ ¬ 4 = 0
16	15	intnan 931	. . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0)
17		gcdn0cl 12333	. . . . . . 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 9081	. . . . 5 ⊢ (6 gcd 4) ≠ 0
20	18	nnzi 9406	. . . . . 6 ⊢ (6 gcd 4) ∈ ℤ
21		0z 9396	. . . . . 6 ⊢ 0 ∈ ℤ
22		zapne 9460	. . . . . 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 9215	. . . . . . . 8 ⊢ 6 ∈ ℕ
27		4nn 9213	. . . . . . . 8 ⊢ 4 ∈ ℕ
28	26, 27	pm3.2i 272	. . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ)
29		lcmgcdnn 12454	. . . . . . 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 2212	. . . . 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 8763	. . . . 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 2212	. . 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 12366	. . 3 ⊢ (6 gcd 4) = 2
37	36	oveq2i 5965	. 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2)
38		2cn 9120	. . . 4 ⊢ 2 ∈ ℂ
39		2ap0 9142	. . . 4 ⊢ 2 # 0
40	1, 2, 38, 39	divassapi 8854	. . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2))
41		4d2e2 9210	. . . 4 ⊢ (4 / 2) = 2
42	41	oveq2i 5965	. . 3 ⊢ (6 · (4 / 2)) = (6 · 2)
43		6t2e12 9620	. . 3 ⊢ (6 · 2) = ;12
44	40, 42, 43	3eqtri 2231	. 2 ⊢ ((6 · 4) / 2) = ;12
45	35, 37, 44	3eqtri 2231	1 ⊢ (6 lcm 4) = ;12

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 ∧ w3a 981 = wceq 1373 ∈ wcel 2177 ≠ wne 2377 class class class wbr 4048 (class class class)co 5954 ℂcc 7936 0cc0 7938 1c1 7939 · cmul 7943 # cap 8667 / cdiv 8758 ℕcn 9049 2c2 9100 4c4 9102 6c6 9104 ℤcz 9385 cdc 9517 gcd cgcd 12324 lcm clcm 12432

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 2179 ax-14 2180 ax-ext 2188 ax-coll 4164 ax-sep 4167 ax-nul 4175 ax-pow 4223 ax-pr 4258 ax-un 4485 ax-setind 4590 ax-iinf 4641 ax-cnex 8029 ax-resscn 8030 ax-1cn 8031 ax-1re 8032 ax-icn 8033 ax-addcl 8034 ax-addrcl 8035 ax-mulcl 8036 ax-mulrcl 8037 ax-addcom 8038 ax-mulcom 8039 ax-addass 8040 ax-mulass 8041 ax-distr 8042 ax-i2m1 8043 ax-0lt1 8044 ax-1rid 8045 ax-0id 8046 ax-rnegex 8047 ax-precex 8048 ax-cnre 8049 ax-pre-ltirr 8050 ax-pre-ltwlin 8051 ax-pre-lttrn 8052 ax-pre-apti 8053 ax-pre-ltadd 8054 ax-pre-mulgt0 8055 ax-pre-mulext 8056 ax-arch 8057 ax-caucvg 8058

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 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3001 df-csb 3096 df-dif 3170 df-un 3172 df-in 3174 df-ss 3181 df-nul 3463 df-if 3574 df-pw 3620 df-sn 3641 df-pr 3642 df-op 3644 df-uni 3854 df-int 3889 df-iun 3932 df-br 4049 df-opab 4111 df-mpt 4112 df-tr 4148 df-id 4345 df-po 4348 df-iso 4349 df-iord 4418 df-on 4420 df-ilim 4421 df-suc 4423 df-iom 4644 df-xp 4686 df-rel 4687 df-cnv 4688 df-co 4689 df-dm 4690 df-rn 4691 df-res 4692 df-ima 4693 df-iota 5238 df-fun 5279 df-fn 5280 df-f 5281 df-f1 5282 df-fo 5283 df-f1o 5284 df-fv 5285 df-isom 5286 df-riota 5909 df-ov 5957 df-oprab 5958 df-mpo 5959 df-1st 6236 df-2nd 6237 df-recs 6401 df-frec 6487 df-sup 7098 df-inf 7099 df-pnf 8122 df-mnf 8123 df-xr 8124 df-ltxr 8125 df-le 8126 df-sub 8258 df-neg 8259 df-reap 8661 df-ap 8668 df-div 8759 df-inn 9050 df-2 9108 df-3 9109 df-4 9110 df-5 9111 df-6 9112 df-7 9113 df-8 9114 df-9 9115 df-n0 9309 df-z 9386 df-dec 9518 df-uz 9662 df-q 9754 df-rp 9789 df-fz 10144 df-fzo 10278 df-fl 10426 df-mod 10481 df-seqfrec 10606 df-exp 10697 df-cj 11203 df-re 11204 df-im 11205 df-rsqrt 11359 df-abs 11360 df-dvds 12149 df-gcd 12325 df-lcm 12433

This theorem is referenced by: (None)

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