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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 16177 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑦 ∈ ℤ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑦)) |
4 | suprzcl2 12341 | . . . 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 | eqeltrid 2919 | . 2 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈ 𝐴) |
7 | oveq2 7166 | . . . 4 ⊢ (𝑥 = 𝑆 → (𝑃↑𝑥) = (𝑃↑𝑆)) | |
8 | 7 | breq1d 5078 | . . 3 ⊢ (𝑥 = 𝑆 → ((𝑃↑𝑥) ∥ 𝑁 ↔ (𝑃↑𝑆) ∥ 𝑁)) |
9 | oveq2 7166 | . . . . . 6 ⊢ (𝑛 = 𝑥 → (𝑃↑𝑛) = (𝑃↑𝑥)) | |
10 | 9 | breq1d 5078 | . . . . 5 ⊢ (𝑛 = 𝑥 → ((𝑃↑𝑛) ∥ 𝑁 ↔ (𝑃↑𝑥) ∥ 𝑁)) |
11 | 10 | cbvrabv 3493 | . . . 4 ⊢ {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁} = {𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁} |
12 | 2, 11 | eqtri 2846 | . . 3 ⊢ 𝐴 = {𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁} |
13 | 8, 12 | elrab2 3685 | . 2 ⊢ (𝑆 ∈ 𝐴 ↔ (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁)) |
14 | 6, 13 | sylib 220 | 1 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 {crab 3144 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 supcsup 8906 ℝcr 10538 0cc0 10539 < clt 10677 ≤ cle 10678 2c2 11695 ℕ0cn0 11900 ℤcz 11984 ℤ≥cuz 12246 ↑cexp 13432 ∥ cdvds 15609 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fl 13165 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 |
This theorem is referenced by: pcprendvds 16179 pcprendvds2 16180 pcpre1 16181 pcpremul 16182 pceulem 16184 pczpre 16186 pczcl 16187 pczdvds 16201 |
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