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Mirrors > Home > MPE Home > Th. List > pcprecl | Structured version Visualization version GIF version |
Description: Closure 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 |
---|---|
pcprecl | ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pclem.2 | . . 3 ⊢ 𝑆 = sup(𝐴, ℝ, < ) | |
2 | pclem.1 | . . . . 5 ⊢ 𝐴 = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁} | |
3 | 2 | pclem 15947 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑦 ∈ ℤ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦)) |
4 | suprzcl2 12085 | . . . 4 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑦 ∈ ℤ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦) → sup(𝐴, ℝ, < ) ∈ 𝐴) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → sup(𝐴, ℝ, < ) ∈ 𝐴) |
6 | 1, 5 | syl5eqel 2863 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ 𝐴) |
7 | oveq2 6930 | . . . 4 ⊢ (𝑥 = 𝑆 → (𝑃↑𝑥) = (𝑃↑𝑆)) | |
8 | 7 | breq1d 4896 | . . 3 ⊢ (𝑥 = 𝑆 → ((𝑃↑𝑥) ∥ 𝑁 ↔ (𝑃↑𝑆) ∥ 𝑁)) |
9 | oveq2 6930 | . . . . . 6 ⊢ (𝑛 = 𝑥 → (𝑃↑𝑛) = (𝑃↑𝑥)) | |
10 | 9 | breq1d 4896 | . . . . 5 ⊢ (𝑛 = 𝑥 → ((𝑃↑𝑛) ∥ 𝑁 ↔ (𝑃↑𝑥) ∥ 𝑁)) |
11 | 10 | cbvrabv 3396 | . . . 4 ⊢ {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁} = {𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁} |
12 | 2, 11 | eqtri 2802 | . . 3 ⊢ 𝐴 = {𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁} |
13 | 8, 12 | elrab2 3576 | . 2 ⊢ (𝑆 ∈ 𝐴 ↔ (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁)) |
14 | 6, 13 | sylib 210 | 1 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∀wral 3090 ∃wrex 3091 {crab 3094 ⊆ wss 3792 ∅c0 4141 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 supcsup 8634 ℝcr 10271 0cc0 10272 < clt 10411 ≤ cle 10412 2c2 11430 ℕ0cn0 11642 ℤcz 11728 ℤ≥cuz 11992 ↑cexp 13178 ∥ cdvds 15387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-fl 12912 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-dvds 15388 |
This theorem is referenced by: pcprendvds 15949 pcprendvds2 15950 pcpre1 15951 pcpremul 15952 pceulem 15954 pczpre 15956 pczcl 15957 pczdvds 15971 |
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