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Theorem pczcl 12816

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 2229	. . 3 ⊢ sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ) = sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < )
2	1	pczpre 12815	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ))
3		prmuz2 12648	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
4		eqid 2229	. . . . 5 ⊢ {𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁} = {𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}
5	4, 1	pcprecl 12807	. . . 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 2306	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) ∈ ℕ₀)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2200 ≠ wne 2400 {crab 2512 class class class wbr 4082 ‘cfv 5317 (class class class)co 6000 supcsup 7145 ℝcr 7994 0cc0 7995 < clt 8177 2c2 9157 ℕ₀cn0 9365 ℤcz 9442 ℤ_≥cuz 9718 ↑cexp 10755 ∥ cdvds 12293 ℙcprime 12624 pCnt cpc 12802

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114 ax-caucvg 8115

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-isom 5326 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-frec 6535 df-1o 6560 df-2o 6561 df-er 6678 df-en 6886 df-sup 7147 df-inf 7148 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-2 9165 df-3 9166 df-4 9167 df-n0 9366 df-z 9443 df-uz 9719 df-q 9811 df-rp 9846 df-fz 10201 df-fzo 10335 df-fl 10485 df-mod 10540 df-seqfrec 10665 df-exp 10756 df-cj 11348 df-re 11349 df-im 11350 df-rsqrt 11504 df-abs 11505 df-dvds 12294 df-gcd 12470 df-prm 12625 df-pc 12803

This theorem is referenced by: pccl 12817 pcdiv 12820 pcqmul 12821 pcqcl 12824 pcxnn0cl 12828 pcge0 12831 pcdvdsb 12838 pcdvdstr 12845 pcgcd1 12846 pc2dvds 12848 pcz 12850 pcaddlem 12857 pcadd 12858 qexpz 12870 lgsval 15677 lgsfcl2 15679 lgsdir 15708 lgsdilem2 15709 lgsdi 15710 lgsne0 15711

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