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Mirrors > Home > MPE Home > Th. List > prmgaplcmlem1 | Structured version Visualization version GIF version |
Description: Lemma for prmgaplcm 16396: 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 | breq1 5069 | . . 3 ⊢ (𝑥 = 𝐼 → (𝑥 ∥ (lcm‘(1...𝑁)) ↔ 𝐼 ∥ (lcm‘(1...𝑁)))) | |
2 | fzssz 12910 | . . . . 5 ⊢ (1...𝑁) ⊆ ℤ | |
3 | fzfi 13341 | . . . . 5 ⊢ (1...𝑁) ∈ Fin | |
4 | 2, 3 | pm3.2i 473 | . . . 4 ⊢ ((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) |
5 | dvdslcmf 15975 | . . . 4 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁))) | |
6 | 4, 5 | mp1i 13 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁))) |
7 | 2eluzge1 12295 | . . . . 5 ⊢ 2 ∈ (ℤ≥‘1) | |
8 | fzss1 12947 | . . . . 5 ⊢ (2 ∈ (ℤ≥‘1) → (2...𝑁) ⊆ (1...𝑁)) | |
9 | 7, 8 | mp1i 13 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2...𝑁) ⊆ (1...𝑁)) |
10 | 9 | sselda 3967 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∈ (1...𝑁)) |
11 | 1, 6, 10 | rspcdva 3625 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ (lcm‘(1...𝑁))) |
12 | elfzelz 12909 | . . . 4 ⊢ (𝐼 ∈ (2...𝑁) → 𝐼 ∈ ℤ) | |
13 | iddvds 15623 | . . . 4 ⊢ (𝐼 ∈ ℤ → 𝐼 ∥ 𝐼) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝐼 ∈ (2...𝑁) → 𝐼 ∥ 𝐼) |
15 | 14 | adantl 484 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ 𝐼) |
16 | 12 | adantl 484 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∈ ℤ) |
17 | lcmfcl 15972 | . . . . 5 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → (lcm‘(1...𝑁)) ∈ ℕ0) | |
18 | 17 | nn0zd 12086 | . . . 4 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → (lcm‘(1...𝑁)) ∈ ℤ) |
19 | 4, 18 | mp1i 13 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → (lcm‘(1...𝑁)) ∈ ℤ) |
20 | dvds2add 15643 | . . 3 ⊢ ((𝐼 ∈ ℤ ∧ (lcm‘(1...𝑁)) ∈ ℤ ∧ 𝐼 ∈ ℤ) → ((𝐼 ∥ (lcm‘(1...𝑁)) ∧ 𝐼 ∥ 𝐼) → 𝐼 ∥ ((lcm‘(1...𝑁)) + 𝐼))) | |
21 | 16, 19, 16, 20 | syl3anc 1367 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → ((𝐼 ∥ (lcm‘(1...𝑁)) ∧ 𝐼 ∥ 𝐼) → 𝐼 ∥ ((lcm‘(1...𝑁)) + 𝐼))) |
22 | 11, 15, 21 | mp2and 697 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((lcm‘(1...𝑁)) + 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 1c1 10538 + caddc 10540 ℕcn 11638 2c2 11693 ℤcz 11982 ℤ≥cuz 12244 ...cfz 12893 ∥ cdvds 15607 lcmclcmf 15933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-prod 15260 df-dvds 15608 df-lcmf 15935 |
This theorem is referenced by: prmgaplcmlem2 16388 |
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