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Mirrors > Home > MPE Home > Th. List > pcprendvds2 | Structured version Visualization version GIF version |
Description: Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Ref | Expression |
---|---|
pclem.1 | ⊢ 𝐴 = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁} |
pclem.2 | ⊢ 𝑆 = sup(𝐴, ℝ, < ) |
Ref | Expression |
---|---|
pcprendvds2 | ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ¬ 𝑃 ∥ (𝑁 / (𝑃↑𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pclem.1 | . . 3 ⊢ 𝐴 = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁} | |
2 | pclem.2 | . . 3 ⊢ 𝑆 = sup(𝐴, ℝ, < ) | |
3 | 1, 2 | pcprendvds 16873 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ¬ (𝑃↑(𝑆 + 1)) ∥ 𝑁) |
4 | eluz2nn 12921 | . . . . . 6 ⊢ (𝑃 ∈ (ℤ≥‘2) → 𝑃 ∈ ℕ) | |
5 | 4 | adantr 480 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℕ) |
6 | 5 | nnzd 12637 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℤ) |
7 | 1, 2 | pcprecl 16872 | . . . . . 6 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁)) |
8 | 7 | simprd 495 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑𝑆) ∥ 𝑁) |
9 | 7 | simpld 494 | . . . . . . . 8 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ ℕ0) |
10 | 5, 9 | nnexpcld 14280 | . . . . . . 7 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑𝑆) ∈ ℕ) |
11 | 10 | nnzd 12637 | . . . . . 6 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑𝑆) ∈ ℤ) |
12 | 10 | nnne0d 12313 | . . . . . 6 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑𝑆) ≠ 0) |
13 | simprl 771 | . . . . . 6 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℤ) | |
14 | dvdsval2 16289 | . . . . . 6 ⊢ (((𝑃↑𝑆) ∈ ℤ ∧ (𝑃↑𝑆) ≠ 0 ∧ 𝑁 ∈ ℤ) → ((𝑃↑𝑆) ∥ 𝑁 ↔ (𝑁 / (𝑃↑𝑆)) ∈ ℤ)) | |
15 | 11, 12, 13, 14 | syl3anc 1370 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ((𝑃↑𝑆) ∥ 𝑁 ↔ (𝑁 / (𝑃↑𝑆)) ∈ ℤ)) |
16 | 8, 15 | mpbid 232 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑁 / (𝑃↑𝑆)) ∈ ℤ) |
17 | dvdscmul 16316 | . . . 4 ⊢ ((𝑃 ∈ ℤ ∧ (𝑁 / (𝑃↑𝑆)) ∈ ℤ ∧ (𝑃↑𝑆) ∈ ℤ) → (𝑃 ∥ (𝑁 / (𝑃↑𝑆)) → ((𝑃↑𝑆) · 𝑃) ∥ ((𝑃↑𝑆) · (𝑁 / (𝑃↑𝑆))))) | |
18 | 6, 16, 11, 17 | syl3anc 1370 | . . 3 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 ∥ (𝑁 / (𝑃↑𝑆)) → ((𝑃↑𝑆) · 𝑃) ∥ ((𝑃↑𝑆) · (𝑁 / (𝑃↑𝑆))))) |
19 | 5 | nncnd 12279 | . . . . . 6 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℂ) |
20 | 19, 9 | expp1d 14183 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑(𝑆 + 1)) = ((𝑃↑𝑆) · 𝑃)) |
21 | 20 | eqcomd 2740 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ((𝑃↑𝑆) · 𝑃) = (𝑃↑(𝑆 + 1))) |
22 | zcn 12615 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
23 | 22 | ad2antrl 728 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℂ) |
24 | 10 | nncnd 12279 | . . . . 5 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑𝑆) ∈ ℂ) |
25 | 23, 24, 12 | divcan2d 12042 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ((𝑃↑𝑆) · (𝑁 / (𝑃↑𝑆))) = 𝑁) |
26 | 21, 25 | breq12d 5160 | . . 3 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (((𝑃↑𝑆) · 𝑃) ∥ ((𝑃↑𝑆) · (𝑁 / (𝑃↑𝑆))) ↔ (𝑃↑(𝑆 + 1)) ∥ 𝑁)) |
27 | 18, 26 | sylibd 239 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 ∥ (𝑁 / (𝑃↑𝑆)) → (𝑃↑(𝑆 + 1)) ∥ 𝑁)) |
28 | 3, 27 | mtod 198 | 1 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ¬ 𝑃 ∥ (𝑁 / (𝑃↑𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 {crab 3432 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 supcsup 9477 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 / cdiv 11917 ℕcn 12263 2c2 12318 ℕ0cn0 12523 ℤcz 12610 ℤ≥cuz 12875 ↑cexp 14098 ∥ cdvds 16286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fl 13828 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 |
This theorem is referenced by: pcpremul 16876 pczndvds2 16900 |
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