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Mirrors > Home > MPE Home > Th. List > pczcl | Structured version Visualization version GIF version |
Description: Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Ref | Expression |
---|---|
pczcl | ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ sup({𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ) = sup({𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ) | |
2 | 1 | pczpre 16558 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = sup({𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < )) |
3 | prmuz2 16411 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2)) | |
4 | eqid 2738 | . . . . 5 ⊢ {𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁} = {𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁} | |
5 | 4, 1 | pcprecl 16550 | . . . 4 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (sup({𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ) ∈ ℕ0 ∧ (𝑃↑sup({𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < )) ∥ 𝑁)) |
6 | 3, 5 | sylan 580 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (sup({𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ) ∈ ℕ0 ∧ (𝑃↑sup({𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < )) ∥ 𝑁)) |
7 | 6 | simpld 495 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → sup({𝑥 ∈ ℕ0 ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ) ∈ ℕ0) |
8 | 2, 7 | eqeltrd 2839 | 1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ≠ wne 2943 {crab 3068 class class class wbr 5073 ‘cfv 6426 (class class class)co 7267 supcsup 9186 ℝcr 10880 0cc0 10881 < clt 11019 2c2 12038 ℕ0cn0 12243 ℤcz 12329 ℤ≥cuz 12592 ↑cexp 13792 ∥ cdvds 15973 ℙcprime 16386 pCnt cpc 16547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-2o 8285 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-sup 9188 df-inf 9189 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-n0 12244 df-z 12330 df-uz 12593 df-q 12699 df-rp 12741 df-fl 13522 df-mod 13600 df-seq 13732 df-exp 13793 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-dvds 15974 df-gcd 16212 df-prm 16387 df-pc 16548 |
This theorem is referenced by: pccl 16560 pcqmul 16564 pcqcl 16567 pcxnn0cl 16571 pcge0 16573 pcdvdsb 16580 pcdvdstr 16587 pcgcd1 16588 pc2dvds 16590 pcz 16592 pcaddlem 16599 pcadd 16600 qexpz 16612 lgsfcl2 26461 lgsdir 26490 lgsdi 26492 lgsne0 26493 |
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