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Mirrors > Home > MPE Home > Th. List > prmgaplcmlem1 | Structured version Visualization version GIF version |
Description: Lemma for prmgaplcm 16867: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.) |
Ref | Expression |
---|---|
prmgaplcmlem1 | ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((lcm‘(1...𝑁)) + 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzelz 13370 | . . 3 ⊢ (𝐼 ∈ (2...𝑁) → 𝐼 ∈ ℤ) | |
2 | 1 | adantl 483 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∈ ℤ) |
3 | fzssz 13372 | . . . 4 ⊢ (1...𝑁) ⊆ ℤ | |
4 | fzfi 13806 | . . . 4 ⊢ (1...𝑁) ∈ Fin | |
5 | 3, 4 | pm3.2i 472 | . . 3 ⊢ ((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) |
6 | lcmfcl 16439 | . . . 4 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → (lcm‘(1...𝑁)) ∈ ℕ0) | |
7 | 6 | nn0zd 12538 | . . 3 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → (lcm‘(1...𝑁)) ∈ ℤ) |
8 | 5, 7 | mp1i 13 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → (lcm‘(1...𝑁)) ∈ ℤ) |
9 | breq1 5107 | . . 3 ⊢ (𝑥 = 𝐼 → (𝑥 ∥ (lcm‘(1...𝑁)) ↔ 𝐼 ∥ (lcm‘(1...𝑁)))) | |
10 | dvdslcmf 16442 | . . . 4 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁))) | |
11 | 5, 10 | mp1i 13 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁))) |
12 | 2eluzge1 12748 | . . . . 5 ⊢ 2 ∈ (ℤ≥‘1) | |
13 | fzss1 13409 | . . . . 5 ⊢ (2 ∈ (ℤ≥‘1) → (2...𝑁) ⊆ (1...𝑁)) | |
14 | 12, 13 | mp1i 13 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2...𝑁) ⊆ (1...𝑁)) |
15 | 14 | sselda 3943 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∈ (1...𝑁)) |
16 | 9, 11, 15 | rspcdva 3581 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ (lcm‘(1...𝑁))) |
17 | iddvds 16087 | . . . 4 ⊢ (𝐼 ∈ ℤ → 𝐼 ∥ 𝐼) | |
18 | 1, 17 | syl 17 | . . 3 ⊢ (𝐼 ∈ (2...𝑁) → 𝐼 ∥ 𝐼) |
19 | 18 | adantl 483 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ 𝐼) |
20 | 2, 8, 2, 16, 19 | dvds2addd 16109 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((lcm‘(1...𝑁)) + 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ∀wral 3063 ⊆ wss 3909 class class class wbr 5104 ‘cfv 6492 (class class class)co 7350 Fincfn 8817 1c1 10986 + caddc 10988 ℕcn 12087 2c2 12142 ℤcz 12433 ℤ≥cuz 12696 ...cfz 13353 ∥ cdvds 16071 lcmclcmf 16400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-inf2 9511 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-sup 9312 df-inf 9313 df-oi 9380 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-n0 12348 df-z 12434 df-uz 12697 df-rp 12845 df-fz 13354 df-fzo 13497 df-seq 13836 df-exp 13897 df-hash 14159 df-cj 14918 df-re 14919 df-im 14920 df-sqrt 15054 df-abs 15055 df-clim 15305 df-prod 15724 df-dvds 16072 df-lcmf 16402 |
This theorem is referenced by: prmgaplcmlem2 16859 |
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