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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 12258 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
3 | 1cnd 11239 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ) | |
4 | 2, 3 | npcand 11605 | . . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
5 | 4 | oveq2d 7432 | . . . . 5 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁)) |
6 | nnm1nn0 12543 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
7 | 1, 6 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0) |
8 | nn0uz 12894 | . . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 8 | eleq2i 2817 | . . . . . . . 8 ⊢ ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (ℤ≥‘0)) |
10 | 7, 9 | sylib 217 | . . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘0)) |
11 | 1m1e0 12314 | . . . . . . . . . 10 ⊢ (1 − 1) = 0 | |
12 | 11 | fveq2i 6895 | . . . . . . . . 9 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0) |
13 | 12 | eleq2i 2817 | . . . . . . . 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 12622 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
17 | fzsuc2 13591 | . . . . . . 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 2767 | . . . 4 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
21 | 4 | sneqd 4636 | . . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁}) |
22 | 21 | uneq2d 4156 | . . . 4 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
23 | 20, 22 | eqtrd 2765 | . . 3 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
24 | 23 | fveq2d 6896 | . 2 ⊢ (𝜑 → (lcm‘(1...𝑁)) = (lcm‘((1...(𝑁 − 1)) ∪ {𝑁}))) |
25 | fzssz 13535 | . . . . 5 ⊢ (1...(𝑁 − 1)) ⊆ ℤ | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ ℤ) |
27 | fzfi 13969 | . . . . 5 ⊢ (1...(𝑁 − 1)) ∈ Fin | |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ∈ Fin) |
29 | nnz 12609 | . . . . 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 16614 | . . 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 2765 | 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 3937 ⊆ wss 3939 {csn 4624 ‘cfv 6543 (class class class)co 7416 Fincfn 8962 0cc0 11138 1c1 11139 + caddc 11141 − cmin 11474 ℕcn 12242 ℕ0cn0 12502 ℤcz 12588 ℤ≥cuz 12852 ...cfz 13516 lcm clcm 16558 lcmclcmf 16559 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-prod 15882 df-dvds 16231 df-gcd 16469 df-lcm 16560 df-lcmf 16561 |
This theorem is referenced by: lcm1un 41540 lcm2un 41541 lcm3un 41542 lcm4un 41543 lcm5un 41544 lcm6un 41545 lcm7un 41546 lcm8un 41547 lcmineqlem19 41574 lcmineqlem22 41577 |
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