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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcmfunnnd | Structured version Visualization version GIF version |
Description: Useful equation to calculate the least common multiple of 1 to n. (Contributed by metakunt, 29-Apr-2024.) |
Ref | Expression |
---|---|
lcmfunnnd.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
lcmfunnnd | ⊢ (𝜑 → (lcm‘(1...𝑁)) = ((lcm‘(1...(𝑁 − 1))) lcm 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmfunnnd.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
2 | 1 | nncnd 12264 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
3 | 1cnd 11245 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ) | |
4 | 2, 3 | npcand 11611 | . . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
5 | 4 | oveq2d 7440 | . . . . 5 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁)) |
6 | nnm1nn0 12549 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
7 | 1, 6 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0) |
8 | nn0uz 12900 | . . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 8 | eleq2i 2820 | . . . . . . . 8 ⊢ ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (ℤ≥‘0)) |
10 | 7, 9 | sylib 217 | . . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘0)) |
11 | 1m1e0 12320 | . . . . . . . . . 10 ⊢ (1 − 1) = 0 | |
12 | 11 | fveq2i 6903 | . . . . . . . . 9 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0) |
13 | 12 | eleq2i 2820 | . . . . . . . 8 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(1 − 1)) ↔ (𝑁 − 1) ∈ (ℤ≥‘0)) |
14 | 13 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ((𝑁 − 1) ∈ (ℤ≥‘(1 − 1)) ↔ (𝑁 − 1) ∈ (ℤ≥‘0))) |
15 | 10, 14 | mpbird 256 | . . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘(1 − 1))) |
16 | 1z 12628 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
17 | fzsuc2 13597 | . . . . . . 7 ⊢ ((1 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ≥‘(1 − 1))) → (1...((𝑁 − 1) + 1)) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) | |
18 | 16, 17 | mpan 688 | . . . . . 6 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(1 − 1)) → (1...((𝑁 − 1) + 1)) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
19 | 15, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
20 | 5, 19 | eqtr3d 2769 | . . . 4 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
21 | 4 | sneqd 4642 | . . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁}) |
22 | 21 | uneq2d 4162 | . . . 4 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
23 | 20, 22 | eqtrd 2767 | . . 3 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
24 | 23 | fveq2d 6904 | . 2 ⊢ (𝜑 → (lcm‘(1...𝑁)) = (lcm‘((1...(𝑁 − 1)) ∪ {𝑁}))) |
25 | fzssz 13541 | . . . . 5 ⊢ (1...(𝑁 − 1)) ⊆ ℤ | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ ℤ) |
27 | fzfi 13975 | . . . . 5 ⊢ (1...(𝑁 − 1)) ∈ Fin | |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ∈ Fin) |
29 | nnz 12615 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
30 | 1, 29 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
31 | 26, 28, 30 | 3jca 1125 | . . 3 ⊢ (𝜑 → ((1...(𝑁 − 1)) ⊆ ℤ ∧ (1...(𝑁 − 1)) ∈ Fin ∧ 𝑁 ∈ ℤ)) |
32 | lcmfunsn 16620 | . . 3 ⊢ (((1...(𝑁 − 1)) ⊆ ℤ ∧ (1...(𝑁 − 1)) ∈ Fin ∧ 𝑁 ∈ ℤ) → (lcm‘((1...(𝑁 − 1)) ∪ {𝑁})) = ((lcm‘(1...(𝑁 − 1))) lcm 𝑁)) | |
33 | 31, 32 | syl 17 | . 2 ⊢ (𝜑 → (lcm‘((1...(𝑁 − 1)) ∪ {𝑁})) = ((lcm‘(1...(𝑁 − 1))) lcm 𝑁)) |
34 | 24, 33 | eqtrd 2767 | 1 ⊢ (𝜑 → (lcm‘(1...𝑁)) = ((lcm‘(1...(𝑁 − 1))) lcm 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∪ cun 3945 ⊆ wss 3947 {csn 4630 ‘cfv 6551 (class class class)co 7424 Fincfn 8968 0cc0 11144 1c1 11145 + caddc 11147 − cmin 11480 ℕcn 12248 ℕ0cn0 12508 ℤcz 12594 ℤ≥cuz 12858 ...cfz 13522 lcm clcm 16564 lcmclcmf 16565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-inf 9472 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-fz 13523 df-fzo 13666 df-fl 13795 df-mod 13873 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 df-prod 15888 df-dvds 16237 df-gcd 16475 df-lcm 16566 df-lcmf 16567 |
This theorem is referenced by: lcm1un 41488 lcm2un 41489 lcm3un 41490 lcm4un 41491 lcm5un 41492 lcm6un 41493 lcm7un 41494 lcm8un 41495 lcmineqlem19 41522 lcmineqlem22 41525 |
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