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Theorem pczcl 12252

Description: Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

**Assertion**
Ref	Expression
pczcl	⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) ∈ ℕ₀)

Proof of Theorem pczcl Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2170	. . 3 ⊢ sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ) = sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < )
2	1	pczpre 12251	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ))
3		prmuz2 12085	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
4		eqid 2170	. . . . 5 ⊢ {𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁} = {𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}
5	4, 1	pcprecl 12243	. . . 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 281	. . 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 111	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ) ∈ ℕ₀)
8	2, 7	eqeltrd 2247	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) ∈ ℕ₀)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 ≠ wne 2340 {crab 2452 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 supcsup 6959 ℝcr 7773 0cc0 7774 < clt 7954 2c2 8929 ℕ₀cn0 9135 ℤcz 9212 ℤ_≥cuz 9487 ↑cexp 10475 ∥ cdvds 11749 ℙcprime 12061 pCnt cpc 12238

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-1o 6395 df-2o 6396 df-er 6513 df-en 6719 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-dvds 11750 df-gcd 11898 df-prm 12062 df-pc 12239

This theorem is referenced by: pccl 12253 pcdiv 12256 pcqmul 12257 pcqcl 12260 pcxnn0cl 12264 pcge0 12266 pcdvdsb 12273 pcdvdstr 12280 pcgcd1 12281 pc2dvds 12283 pcz 12285 pcaddlem 12292 pcadd 12293 qexpz 12304 lgsval 13699 lgsfcl2 13701 lgsdir 13730 lgsdilem2 13731 lgsdi 13732 lgsne0 13733

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