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Theorem pczcl 12436

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 2193	. . 3 ⊢ sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ) = sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < )
2	1	pczpre 12435	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ))
3		prmuz2 12269	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
4		eqid 2193	. . . . 5 ⊢ {𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁} = {𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}
5	4, 1	pcprecl 12427	. . . 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 283	. . 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 112	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ) ∈ ℕ₀)
8	2, 7	eqeltrd 2270	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) ∈ ℕ₀)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2164 ≠ wne 2364 {crab 2476 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 supcsup 7041 ℝcr 7871 0cc0 7872 < clt 8054 2c2 9033 ℕ₀cn0 9240 ℤcz 9317 ℤ_≥cuz 9592 ↑cexp 10609 ∥ cdvds 11930 ℙcprime 12245 pCnt cpc 12422

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-1o 6469 df-2o 6470 df-er 6587 df-en 6795 df-sup 7043 df-inf 7044 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-dvds 11931 df-gcd 12080 df-prm 12246 df-pc 12423

This theorem is referenced by: pccl 12437 pcdiv 12440 pcqmul 12441 pcqcl 12444 pcxnn0cl 12448 pcge0 12451 pcdvdsb 12458 pcdvdstr 12465 pcgcd1 12466 pc2dvds 12468 pcz 12470 pcaddlem 12477 pcadd 12478 qexpz 12490 lgsval 15120 lgsfcl2 15122 lgsdir 15151 lgsdilem2 15152 lgsdi 15153 lgsne0 15154

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