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Theorem pczcl 12226

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 2165	. . 3 ⊢ sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ) = sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < )
2	1	pczpre 12225	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ))
3		prmuz2 12059	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
4		eqid 2165	. . . . 5 ⊢ {𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁} = {𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}
5	4, 1	pcprecl 12217	. . . 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 2242	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) ∈ ℕ₀)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2136 ≠ wne 2335 {crab 2447 class class class wbr 3981 ‘cfv 5187 (class class class)co 5841 supcsup 6943 ℝcr 7748 0cc0 7749 < clt 7929 2c2 8904 ℕ₀cn0 9110 ℤcz 9187 ℤ_≥cuz 9462 ↑cexp 10450 ∥ cdvds 11723 ℙcprime 12035 pCnt cpc 12212

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4096 ax-sep 4099 ax-nul 4107 ax-pow 4152 ax-pr 4186 ax-un 4410 ax-setind 4513 ax-iinf 4564 ax-cnex 7840 ax-resscn 7841 ax-1cn 7842 ax-1re 7843 ax-icn 7844 ax-addcl 7845 ax-addrcl 7846 ax-mulcl 7847 ax-mulrcl 7848 ax-addcom 7849 ax-mulcom 7850 ax-addass 7851 ax-mulass 7852 ax-distr 7853 ax-i2m1 7854 ax-0lt1 7855 ax-1rid 7856 ax-0id 7857 ax-rnegex 7858 ax-precex 7859 ax-cnre 7860 ax-pre-ltirr 7861 ax-pre-ltwlin 7862 ax-pre-lttrn 7863 ax-pre-apti 7864 ax-pre-ltadd 7865 ax-pre-mulgt0 7866 ax-pre-mulext 7867 ax-arch 7868 ax-caucvg 7869

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2296 df-ne 2336 df-nel 2431 df-ral 2448 df-rex 2449 df-reu 2450 df-rmo 2451 df-rab 2452 df-v 2727 df-sbc 2951 df-csb 3045 df-dif 3117 df-un 3119 df-in 3121 df-ss 3128 df-nul 3409 df-if 3520 df-pw 3560 df-sn 3581 df-pr 3582 df-op 3584 df-uni 3789 df-int 3824 df-iun 3867 df-br 3982 df-opab 4043 df-mpt 4044 df-tr 4080 df-id 4270 df-po 4273 df-iso 4274 df-iord 4343 df-on 4345 df-ilim 4346 df-suc 4348 df-iom 4567 df-xp 4609 df-rel 4610 df-cnv 4611 df-co 4612 df-dm 4613 df-rn 4614 df-res 4615 df-ima 4616 df-iota 5152 df-fun 5189 df-fn 5190 df-f 5191 df-f1 5192 df-fo 5193 df-f1o 5194 df-fv 5195 df-isom 5196 df-riota 5797 df-ov 5844 df-oprab 5845 df-mpo 5846 df-1st 6105 df-2nd 6106 df-recs 6269 df-frec 6355 df-1o 6380 df-2o 6381 df-er 6497 df-en 6703 df-sup 6945 df-inf 6946 df-pnf 7931 df-mnf 7932 df-xr 7933 df-ltxr 7934 df-le 7935 df-sub 8067 df-neg 8068 df-reap 8469 df-ap 8476 df-div 8565 df-inn 8854 df-2 8912 df-3 8913 df-4 8914 df-n0 9111 df-z 9188 df-uz 9463 df-q 9554 df-rp 9586 df-fz 9941 df-fzo 10074 df-fl 10201 df-mod 10254 df-seqfrec 10377 df-exp 10451 df-cj 10780 df-re 10781 df-im 10782 df-rsqrt 10936 df-abs 10937 df-dvds 11724 df-gcd 11872 df-prm 12036 df-pc 12213

This theorem is referenced by: pccl 12227 pcdiv 12230 pcqmul 12231 pcqcl 12234 pcxnn0cl 12238 pcge0 12240 pcdvdsb 12247 pcdvdstr 12254 pcgcd1 12255 pc2dvds 12257 pcz 12259 pcaddlem 12266 pcadd 12267 qexpz 12278 lgsval 13505 lgsfcl2 13507 lgsdir 13536 lgsdilem2 13537 lgsdi 13538 lgsne0 13539

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