6lcm4e12 - Intuitionistic Logic Explorer

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Theorem 6lcm4e12 12652

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 9218	. . . 4 ⊢ 6 ∈ ℂ
2		4cn 9214	. . . 4 ⊢ 4 ∈ ℂ
3	1, 2	mulcli 8177	. . 3 ⊢ (6 · 4) ∈ ℂ
4		6nn0 9416	. . . . 5 ⊢ 6 ∈ ℕ₀
5	4	nn0zi 9494	. . . 4 ⊢ 6 ∈ ℤ
6		4z 9502	. . . 4 ⊢ 4 ∈ ℤ
7		lcmcl 12637	. . . . 5 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℕ₀)
8	7	nn0cnd 9450	. . . 4 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 lcm 4) ∈ ℂ)
9	5, 6, 8	mp2an 426	. . 3 ⊢ (6 lcm 4) ∈ ℂ
10		gcdcl 12530	. . . . . 6 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) ∈ ℕ₀)
11	10	nn0cnd 9450	. . . . 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 9234	. . . . . . . . 9 ⊢ 4 ≠ 0
15	14	neii 2402	. . . . . . . 8 ⊢ ¬ 4 = 0
16	15	intnan 934	. . . . . . 7 ⊢ ¬ (6 = 0 ∧ 4 = 0)
17		gcdn0cl 12526	. . . . . . 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 9168	. . . . 5 ⊢ (6 gcd 4) ≠ 0
20	18	nnzi 9493	. . . . . 6 ⊢ (6 gcd 4) ∈ ℤ
21		0z 9483	. . . . . 6 ⊢ 0 ∈ ℤ
22		zapne 9547	. . . . . 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 9302	. . . . . . . 8 ⊢ 6 ∈ ℕ
27		4nn 9300	. . . . . . . 8 ⊢ 4 ∈ ℕ
28	26, 27	pm3.2i 272	. . . . . . 7 ⊢ (6 ∈ ℕ ∧ 4 ∈ ℕ)
29		lcmgcdnn 12647	. . . . . . 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 2235	. . . . 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 8850	. . . . 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 2235	. . 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 1371	. 2 ⊢ (6 lcm 4) = ((6 · 4) / (6 gcd 4))
36		6gcd4e2 12559	. . 3 ⊢ (6 gcd 4) = 2
37	36	oveq2i 6024	. 2 ⊢ ((6 · 4) / (6 gcd 4)) = ((6 · 4) / 2)
38		2cn 9207	. . . 4 ⊢ 2 ∈ ℂ
39		2ap0 9229	. . . 4 ⊢ 2 # 0
40	1, 2, 38, 39	divassapi 8941	. . 3 ⊢ ((6 · 4) / 2) = (6 · (4 / 2))
41		4d2e2 9297	. . . 4 ⊢ (4 / 2) = 2
42	41	oveq2i 6024	. . 3 ⊢ (6 · (4 / 2)) = (6 · 2)
43		6t2e12 9707	. . 3 ⊢ (6 · 2) = ;12
44	40, 42, 43	3eqtri 2254	. 2 ⊢ ((6 · 4) / 2) = ;12
45	35, 37, 44	3eqtri 2254	1 ⊢ (6 lcm 4) = ;12

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 class class class wbr 4086 (class class class)co 6013 ℂcc 8023 0cc0 8025 1c1 8026 · cmul 8030 # cap 8754 / cdiv 8845 ℕcn 9136 2c2 9187 4c4 9189 6c6 9191 ℤcz 9472 cdc 9604 gcd cgcd 12517 lcm clcm 12625

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-pre-mulext 8143 ax-arch 8144 ax-caucvg 8145

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-frec 6552 df-sup 7177 df-inf 7178 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 df-div 8846 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-5 9198 df-6 9199 df-7 9200 df-8 9201 df-9 9202 df-n0 9396 df-z 9473 df-dec 9605 df-uz 9749 df-q 9847 df-rp 9882 df-fz 10237 df-fzo 10371 df-fl 10523 df-mod 10578 df-seqfrec 10703 df-exp 10794 df-cj 11396 df-re 11397 df-im 11398 df-rsqrt 11552 df-abs 11553 df-dvds 12342 df-gcd 12518 df-lcm 12626

This theorem is referenced by: (None)

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