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Mirrors > Home > MPE Home > Th. List > lcmid | Structured version Visualization version GIF version |
Description: The lcm of an integer and itself is its absolute value. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
Ref | Expression |
---|---|
lcmid | ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 𝑀) = (abs‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7191 | . . 3 ⊢ (𝑀 = 0 → (𝑀 lcm 𝑀) = (𝑀 lcm 0)) | |
2 | fveq2 6687 | . . . 4 ⊢ (𝑀 = 0 → (abs‘𝑀) = (abs‘0)) | |
3 | abs0 14748 | . . . 4 ⊢ (abs‘0) = 0 | |
4 | 2, 3 | eqtrdi 2790 | . . 3 ⊢ (𝑀 = 0 → (abs‘𝑀) = 0) |
5 | 1, 4 | eqeq12d 2755 | . 2 ⊢ (𝑀 = 0 → ((𝑀 lcm 𝑀) = (abs‘𝑀) ↔ (𝑀 lcm 0) = 0)) |
6 | lcmcl 16055 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 lcm 𝑀) ∈ ℕ0) | |
7 | 6 | nn0cnd 12051 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 lcm 𝑀) ∈ ℂ) |
8 | 7 | anidms 570 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 𝑀) ∈ ℂ) |
9 | 8 | adantr 484 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 lcm 𝑀) ∈ ℂ) |
10 | zabscl 14776 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℤ) | |
11 | 10 | zcnd 12182 | . . . 4 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℂ) |
12 | 11 | adantr 484 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℂ) |
13 | zcn 12080 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
14 | 13 | adantr 484 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝑀 ∈ ℂ) |
15 | simpr 488 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝑀 ≠ 0) | |
16 | 14, 15 | absne0d 14910 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ≠ 0) |
17 | lcmgcd 16061 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = (abs‘(𝑀 · 𝑀))) | |
18 | 17 | anidms 570 | . . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = (abs‘(𝑀 · 𝑀))) |
19 | gcdid 15983 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 𝑀) = (abs‘𝑀)) | |
20 | 19 | oveq2d 7199 | . . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = ((𝑀 lcm 𝑀) · (abs‘𝑀))) |
21 | 13, 13 | absmuld 14917 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (abs‘(𝑀 · 𝑀)) = ((abs‘𝑀) · (abs‘𝑀))) |
22 | 18, 20, 21 | 3eqtr3d 2782 | . . . 4 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (abs‘𝑀)) = ((abs‘𝑀) · (abs‘𝑀))) |
23 | 22 | adantr 484 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → ((𝑀 lcm 𝑀) · (abs‘𝑀)) = ((abs‘𝑀) · (abs‘𝑀))) |
24 | 9, 12, 12, 16, 23 | mulcan2ad 11367 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 lcm 𝑀) = (abs‘𝑀)) |
25 | lcm0val 16048 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0) | |
26 | 5, 24, 25 | pm2.61ne 3020 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 𝑀) = (abs‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ‘cfv 6350 (class class class)co 7183 ℂcc 10626 0cc0 10628 · cmul 10633 ℤcz 12075 abscabs 14696 gcd cgcd 15950 lcm clcm 16042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-sup 8992 df-inf 8993 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-n0 11990 df-z 12076 df-uz 12338 df-rp 12486 df-fl 13266 df-mod 13342 df-seq 13474 df-exp 13535 df-cj 14561 df-re 14562 df-im 14563 df-sqrt 14697 df-abs 14698 df-dvds 15713 df-gcd 15951 df-lcm 16044 |
This theorem is referenced by: lcmgcdeq 16066 lcmfsn 16089 lcm1un 39674 |
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