![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pcpre1 | Structured version Visualization version GIF version |
Description: Value of the prime power pre-function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 26-Apr-2016.) |
Ref | Expression |
---|---|
pclem.1 | ⊢ 𝐴 = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁} |
pclem.2 | ⊢ 𝑆 = sup(𝐴, ℝ, < ) |
Ref | Expression |
---|---|
pcpre1 | ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → 𝑆 = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 12000 | . . . . . . . . . 10 ⊢ 1 ∈ ℤ | |
2 | eleq1 2877 | . . . . . . . . . 10 ⊢ (𝑁 = 1 → (𝑁 ∈ ℤ ↔ 1 ∈ ℤ)) | |
3 | 1, 2 | mpbiri 261 | . . . . . . . . 9 ⊢ (𝑁 = 1 → 𝑁 ∈ ℤ) |
4 | ax-1ne0 10595 | . . . . . . . . . 10 ⊢ 1 ≠ 0 | |
5 | neeq1 3049 | . . . . . . . . . 10 ⊢ (𝑁 = 1 → (𝑁 ≠ 0 ↔ 1 ≠ 0)) | |
6 | 4, 5 | mpbiri 261 | . . . . . . . . 9 ⊢ (𝑁 = 1 → 𝑁 ≠ 0) |
7 | 3, 6 | jca 515 | . . . . . . . 8 ⊢ (𝑁 = 1 → (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) |
8 | pclem.1 | . . . . . . . . 9 ⊢ 𝐴 = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁} | |
9 | pclem.2 | . . . . . . . . 9 ⊢ 𝑆 = sup(𝐴, ℝ, < ) | |
10 | 8, 9 | pcprecl 16166 | . . . . . . . 8 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁)) |
11 | 7, 10 | sylan2 595 | . . . . . . 7 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁)) |
12 | 11 | simprd 499 | . . . . . 6 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → (𝑃↑𝑆) ∥ 𝑁) |
13 | simpr 488 | . . . . . 6 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → 𝑁 = 1) | |
14 | 12, 13 | breqtrd 5056 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → (𝑃↑𝑆) ∥ 1) |
15 | eluz2nn 12272 | . . . . . . . . 9 ⊢ (𝑃 ∈ (ℤ≥‘2) → 𝑃 ∈ ℕ) | |
16 | 15 | adantr 484 | . . . . . . . 8 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → 𝑃 ∈ ℕ) |
17 | 11 | simpld 498 | . . . . . . . 8 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → 𝑆 ∈ ℕ0) |
18 | 16, 17 | nnexpcld 13602 | . . . . . . 7 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → (𝑃↑𝑆) ∈ ℕ) |
19 | 18 | nnzd 12074 | . . . . . 6 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → (𝑃↑𝑆) ∈ ℤ) |
20 | 1nn 11636 | . . . . . 6 ⊢ 1 ∈ ℕ | |
21 | dvdsle 15652 | . . . . . 6 ⊢ (((𝑃↑𝑆) ∈ ℤ ∧ 1 ∈ ℕ) → ((𝑃↑𝑆) ∥ 1 → (𝑃↑𝑆) ≤ 1)) | |
22 | 19, 20, 21 | sylancl 589 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → ((𝑃↑𝑆) ∥ 1 → (𝑃↑𝑆) ≤ 1)) |
23 | 14, 22 | mpd 15 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → (𝑃↑𝑆) ≤ 1) |
24 | 16 | nncnd 11641 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → 𝑃 ∈ ℂ) |
25 | 24 | exp0d 13500 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → (𝑃↑0) = 1) |
26 | 23, 25 | breqtrrd 5058 | . . 3 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → (𝑃↑𝑆) ≤ (𝑃↑0)) |
27 | 16 | nnred 11640 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → 𝑃 ∈ ℝ) |
28 | 17 | nn0zd 12073 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → 𝑆 ∈ ℤ) |
29 | 0zd 11981 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → 0 ∈ ℤ) | |
30 | eluz2gt1 12308 | . . . . 5 ⊢ (𝑃 ∈ (ℤ≥‘2) → 1 < 𝑃) | |
31 | 30 | adantr 484 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → 1 < 𝑃) |
32 | 27, 28, 29, 31 | leexp2d 13611 | . . 3 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → (𝑆 ≤ 0 ↔ (𝑃↑𝑆) ≤ (𝑃↑0))) |
33 | 26, 32 | mpbird 260 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → 𝑆 ≤ 0) |
34 | 10 | simpld 498 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ ℕ0) |
35 | 7, 34 | sylan2 595 | . . 3 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → 𝑆 ∈ ℕ0) |
36 | nn0le0eq0 11913 | . . 3 ⊢ (𝑆 ∈ ℕ0 → (𝑆 ≤ 0 ↔ 𝑆 = 0)) | |
37 | 35, 36 | syl 17 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → (𝑆 ≤ 0 ↔ 𝑆 = 0)) |
38 | 33, 37 | mpbid 235 | 1 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 = 1) → 𝑆 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {crab 3110 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℝcr 10525 0cc0 10526 1c1 10527 < clt 10664 ≤ cle 10665 ℕcn 11625 2c2 11680 ℕ0cn0 11885 ℤcz 11969 ℤ≥cuz 12231 ↑cexp 13425 ∥ cdvds 15599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 |
This theorem is referenced by: pczpre 16174 pc1 16182 |
Copyright terms: Public domain | W3C validator |