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Theorem lcmid 11554

Description: The lcm of an integer and itself is its absolute value. (Contributed by Steve Rodriguez, 20-Jan-2020.)

**Assertion**
Ref	Expression
lcmid	⊢ (𝑀 ∈ ℤ → (𝑀 lcm 𝑀) = (abs‘𝑀))

**Proof of Theorem lcmid**
Step	Hyp	Ref	Expression
1		lcm0val 11539	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0)
2	1	adantr 272	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 = 0) → (𝑀 lcm 0) = 0)
3		oveq2 5714	. . . . 5 ⊢ (𝑀 = 0 → (𝑀 lcm 𝑀) = (𝑀 lcm 0))
4		fveq2 5353	. . . . . 6 ⊢ (𝑀 = 0 → (abs‘𝑀) = (abs‘0))
5		abs0 10670	. . . . . 6 ⊢ (abs‘0) = 0
6	4, 5	syl6eq 2148	. . . . 5 ⊢ (𝑀 = 0 → (abs‘𝑀) = 0)
7	3, 6	eqeq12d 2114	. . . 4 ⊢ (𝑀 = 0 → ((𝑀 lcm 𝑀) = (abs‘𝑀) ↔ (𝑀 lcm 0) = 0))
8	7	adantl 273	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 = 0) → ((𝑀 lcm 𝑀) = (abs‘𝑀) ↔ (𝑀 lcm 0) = 0))
9	2, 8	mpbird 166	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 = 0) → (𝑀 lcm 𝑀) = (abs‘𝑀))
10		df-ne 2268	. . 3 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
11		lcmcl 11546	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 lcm 𝑀) ∈ ℕ₀)
12	11	nn0cnd 8884	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 lcm 𝑀) ∈ ℂ)
13	12	anidms 392	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 𝑀) ∈ ℂ)
14	13	adantr 272	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 lcm 𝑀) ∈ ℂ)
15		zabscl 10698	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℤ)
16	15	zcnd 9026	. . . . 5 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℂ)
17	16	adantr 272	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℂ)
18		zcn 8911	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
19	18	adantr 272	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝑀 ∈ ℂ)
20		simpr 109	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝑀 ≠ 0)
21	19, 20	absne0d 10799	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ≠ 0)
22		0zd 8918	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 0 ∈ ℤ)
23		zapne 8977	. . . . . 6 ⊢ (((abs‘𝑀) ∈ ℤ ∧ 0 ∈ ℤ) → ((abs‘𝑀) # 0 ↔ (abs‘𝑀) ≠ 0))
24	15, 22, 23	syl2an2r 565	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → ((abs‘𝑀) # 0 ↔ (abs‘𝑀) ≠ 0))
25	21, 24	mpbird 166	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) # 0)
26		lcmgcd 11552	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = (abs‘(𝑀 · 𝑀)))
27	26	anidms 392	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = (abs‘(𝑀 · 𝑀)))
28		gcdid 11469	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 𝑀) = (abs‘𝑀))
29	28	oveq2d 5722	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = ((𝑀 lcm 𝑀) · (abs‘𝑀)))
30	18, 18	absmuld 10806	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (abs‘(𝑀 · 𝑀)) = ((abs‘𝑀) · (abs‘𝑀)))
31	27, 29, 30	3eqtr3d 2140	. . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (abs‘𝑀)) = ((abs‘𝑀) · (abs‘𝑀)))
32	31	adantr 272	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → ((𝑀 lcm 𝑀) · (abs‘𝑀)) = ((abs‘𝑀) · (abs‘𝑀)))
33	14, 17, 17, 25, 32	mulcanap2ad 8286	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 lcm 𝑀) = (abs‘𝑀))
34	10, 33	sylan2br 284	. 2 ⊢ ((𝑀 ∈ ℤ ∧ ¬ 𝑀 = 0) → (𝑀 lcm 𝑀) = (abs‘𝑀))
35		0z 8917	. . . 4 ⊢ 0 ∈ ℤ
36		zdceq 8978	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑀 = 0)
37	35, 36	mpan2 419	. . 3 ⊢ (𝑀 ∈ ℤ → DECID 𝑀 = 0)
38		exmiddc 788	. . 3 ⊢ (DECID 𝑀 = 0 → (𝑀 = 0 ∨ ¬ 𝑀 = 0))
39	37, 38	syl 14	. 2 ⊢ (𝑀 ∈ ℤ → (𝑀 = 0 ∨ ¬ 𝑀 = 0))
40	9, 34, 39	mpjaodan 753	1 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 𝑀) = (abs‘𝑀))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 670 DECID wdc 786 = wceq 1299 ∈ wcel 1448 ≠ wne 2267 class class class wbr 3875 ‘cfv 5059 (class class class)co 5706 ℂcc 7498 0cc0 7500 · cmul 7505 # cap 8209 ℤcz 8906 abscabs 10609 gcd cgcd 11430 lcm clcm 11534

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 ax-arch 7614 ax-caucvg 7615

This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-isom 5068 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-sup 6786 df-inf 6787 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-n0 8830 df-z 8907 df-uz 9177 df-q 9262 df-rp 9292 df-fz 9632 df-fzo 9761 df-fl 9884 df-mod 9937 df-seqfrec 10060 df-exp 10134 df-cj 10455 df-re 10456 df-im 10457 df-rsqrt 10610 df-abs 10611 df-dvds 11289 df-gcd 11431 df-lcm 11535

This theorem is referenced by: lcmgcdeq 11557

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