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Theorem lcmid 12221

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 12206	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0)
2	1	adantr 276	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 = 0) → (𝑀 lcm 0) = 0)
3		oveq2 5927	. . . . 5 ⊢ (𝑀 = 0 → (𝑀 lcm 𝑀) = (𝑀 lcm 0))
4		fveq2 5555	. . . . . 6 ⊢ (𝑀 = 0 → (abs‘𝑀) = (abs‘0))
5		abs0 11205	. . . . . 6 ⊢ (abs‘0) = 0
6	4, 5	eqtrdi 2242	. . . . 5 ⊢ (𝑀 = 0 → (abs‘𝑀) = 0)
7	3, 6	eqeq12d 2208	. . . 4 ⊢ (𝑀 = 0 → ((𝑀 lcm 𝑀) = (abs‘𝑀) ↔ (𝑀 lcm 0) = 0))
8	7	adantl 277	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 = 0) → ((𝑀 lcm 𝑀) = (abs‘𝑀) ↔ (𝑀 lcm 0) = 0))
9	2, 8	mpbird 167	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 = 0) → (𝑀 lcm 𝑀) = (abs‘𝑀))
10		df-ne 2365	. . 3 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
11		lcmcl 12213	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 lcm 𝑀) ∈ ℕ₀)
12	11	nn0cnd 9298	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 lcm 𝑀) ∈ ℂ)
13	12	anidms 397	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 𝑀) ∈ ℂ)
14	13	adantr 276	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 lcm 𝑀) ∈ ℂ)
15		zabscl 11233	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℤ)
16	15	zcnd 9443	. . . . 5 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℂ)
17	16	adantr 276	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℂ)
18		zcn 9325	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
19	18	adantr 276	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝑀 ∈ ℂ)
20		simpr 110	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝑀 ≠ 0)
21	19, 20	absne0d 11334	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ≠ 0)
22		0zd 9332	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 0 ∈ ℤ)
23		zapne 9394	. . . . . 6 ⊢ (((abs‘𝑀) ∈ ℤ ∧ 0 ∈ ℤ) → ((abs‘𝑀) # 0 ↔ (abs‘𝑀) ≠ 0))
24	15, 22, 23	syl2an2r 595	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → ((abs‘𝑀) # 0 ↔ (abs‘𝑀) ≠ 0))
25	21, 24	mpbird 167	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) # 0)
26		lcmgcd 12219	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = (abs‘(𝑀 · 𝑀)))
27	26	anidms 397	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = (abs‘(𝑀 · 𝑀)))
28		gcdid 12126	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 𝑀) = (abs‘𝑀))
29	28	oveq2d 5935	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = ((𝑀 lcm 𝑀) · (abs‘𝑀)))
30	18, 18	absmuld 11341	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (abs‘(𝑀 · 𝑀)) = ((abs‘𝑀) · (abs‘𝑀)))
31	27, 29, 30	3eqtr3d 2234	. . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (abs‘𝑀)) = ((abs‘𝑀) · (abs‘𝑀)))
32	31	adantr 276	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → ((𝑀 lcm 𝑀) · (abs‘𝑀)) = ((abs‘𝑀) · (abs‘𝑀)))
33	14, 17, 17, 25, 32	mulcanap2ad 8685	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 lcm 𝑀) = (abs‘𝑀))
34	10, 33	sylan2br 288	. 2 ⊢ ((𝑀 ∈ ℤ ∧ ¬ 𝑀 = 0) → (𝑀 lcm 𝑀) = (abs‘𝑀))
35		0z 9331	. . . 4 ⊢ 0 ∈ ℤ
36		zdceq 9395	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑀 = 0)
37	35, 36	mpan2 425	. . 3 ⊢ (𝑀 ∈ ℤ → DECID 𝑀 = 0)
38		exmiddc 837	. . 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 799	1 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 𝑀) = (abs‘𝑀))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 DECID wdc 835 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 class class class wbr 4030 ‘cfv 5255 (class class class)co 5919 ℂcc 7872 0cc0 7874 · cmul 7879 # cap 8602 ℤcz 9320 abscabs 11144 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-n0 9244 df-z 9321 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: lcmgcdeq 12224

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