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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 12232 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
3 | 1cnd 11213 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ) | |
4 | 2, 3 | npcand 11579 | . . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
5 | 4 | oveq2d 7421 | . . . . 5 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁)) |
6 | nnm1nn0 12517 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
7 | 1, 6 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0) |
8 | nn0uz 12868 | . . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 8 | eleq2i 2819 | . . . . . . . 8 ⊢ ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (ℤ≥‘0)) |
10 | 7, 9 | sylib 217 | . . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘0)) |
11 | 1m1e0 12288 | . . . . . . . . . 10 ⊢ (1 − 1) = 0 | |
12 | 11 | fveq2i 6888 | . . . . . . . . 9 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0) |
13 | 12 | eleq2i 2819 | . . . . . . . 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 257 | . . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘(1 − 1))) |
16 | 1z 12596 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
17 | fzsuc2 13565 | . . . . . . 7 ⊢ ((1 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ≥‘(1 − 1))) → (1...((𝑁 − 1) + 1)) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) | |
18 | 16, 17 | mpan 687 | . . . . . 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 2768 | . . . 4 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
21 | 4 | sneqd 4635 | . . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁}) |
22 | 21 | uneq2d 4158 | . . . 4 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
23 | 20, 22 | eqtrd 2766 | . . 3 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
24 | 23 | fveq2d 6889 | . 2 ⊢ (𝜑 → (lcm‘(1...𝑁)) = (lcm‘((1...(𝑁 − 1)) ∪ {𝑁}))) |
25 | fzssz 13509 | . . . . 5 ⊢ (1...(𝑁 − 1)) ⊆ ℤ | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ ℤ) |
27 | fzfi 13943 | . . . . 5 ⊢ (1...(𝑁 − 1)) ∈ Fin | |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ∈ Fin) |
29 | nnz 12583 | . . . . 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 16588 | . . 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 2766 | 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 3941 ⊆ wss 3943 {csn 4623 ‘cfv 6537 (class class class)co 7405 Fincfn 8941 0cc0 11112 1c1 11113 + caddc 11115 − cmin 11448 ℕcn 12216 ℕ0cn0 12476 ℤcz 12562 ℤ≥cuz 12826 ...cfz 13490 lcm clcm 16532 lcmclcmf 16533 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-prod 15856 df-dvds 16205 df-gcd 16443 df-lcm 16534 df-lcmf 16535 |
This theorem is referenced by: lcm1un 41394 lcm2un 41395 lcm3un 41396 lcm4un 41397 lcm5un 41398 lcm6un 41399 lcm7un 41400 lcm8un 41401 lcmineqlem19 41428 lcmineqlem22 41431 |
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