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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 12280 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
3 | 1cnd 11254 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ) | |
4 | 2, 3 | npcand 11622 | . . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
5 | 4 | oveq2d 7447 | . . . . 5 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁)) |
6 | nnm1nn0 12565 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
7 | 1, 6 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0) |
8 | nn0uz 12918 | . . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 8 | eleq2i 2831 | . . . . . . . 8 ⊢ ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (ℤ≥‘0)) |
10 | 7, 9 | sylib 218 | . . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘0)) |
11 | 1m1e0 12336 | . . . . . . . . . 10 ⊢ (1 − 1) = 0 | |
12 | 11 | fveq2i 6910 | . . . . . . . . 9 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0) |
13 | 12 | eleq2i 2831 | . . . . . . . 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 12645 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
17 | fzsuc2 13619 | . . . . . . 7 ⊢ ((1 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ≥‘(1 − 1))) → (1...((𝑁 − 1) + 1)) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) | |
18 | 16, 17 | mpan 690 | . . . . . 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 2777 | . . . 4 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
21 | 4 | sneqd 4643 | . . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁}) |
22 | 21 | uneq2d 4178 | . . . 4 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
23 | 20, 22 | eqtrd 2775 | . . 3 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
24 | 23 | fveq2d 6911 | . 2 ⊢ (𝜑 → (lcm‘(1...𝑁)) = (lcm‘((1...(𝑁 − 1)) ∪ {𝑁}))) |
25 | fzssz 13563 | . . . . 5 ⊢ (1...(𝑁 − 1)) ⊆ ℤ | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ ℤ) |
27 | fzfi 14010 | . . . . 5 ⊢ (1...(𝑁 − 1)) ∈ Fin | |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ∈ Fin) |
29 | nnz 12632 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
30 | 1, 29 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
31 | 26, 28, 30 | 3jca 1127 | . . 3 ⊢ (𝜑 → ((1...(𝑁 − 1)) ⊆ ℤ ∧ (1...(𝑁 − 1)) ∈ Fin ∧ 𝑁 ∈ ℤ)) |
32 | lcmfunsn 16678 | . . 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 2775 | 1 ⊢ (𝜑 → (lcm‘(1...𝑁)) = ((lcm‘(1...(𝑁 − 1))) lcm 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ⊆ wss 3963 {csn 4631 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 0cc0 11153 1c1 11154 + caddc 11156 − cmin 11490 ℕcn 12264 ℕ0cn0 12524 ℤcz 12611 ℤ≥cuz 12876 ...cfz 13544 lcm clcm 16622 lcmclcmf 16623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-prod 15937 df-dvds 16288 df-gcd 16529 df-lcm 16624 df-lcmf 16625 |
This theorem is referenced by: lcm1un 41995 lcm2un 41996 lcm3un 41997 lcm4un 41998 lcm5un 41999 lcm6un 42000 lcm7un 42001 lcm8un 42002 lcmineqlem19 42029 lcmineqlem22 42032 |
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