pczcl - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > pczcl		GIF version

Theorem pczcl 12706

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 2206	. . 3 ⊢ sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ) = sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < )
2	1	pczpre 12705	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = sup({𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}, ℝ, < ))
3		prmuz2 12538	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
4		eqid 2206	. . . . 5 ⊢ {𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁} = {𝑥 ∈ ℕ₀ ∣ (𝑃↑𝑥) ∥ 𝑁}
5	4, 1	pcprecl 12697	. . . 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 2283	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) ∈ ℕ₀)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2177 ≠ wne 2377 {crab 2489 class class class wbr 4054 ‘cfv 5285 (class class class)co 5962 supcsup 7105 ℝcr 7954 0cc0 7955 < clt 8137 2c2 9117 ℕ₀cn0 9325 ℤcz 9402 ℤ_≥cuz 9678 ↑cexp 10715 ∥ cdvds 12183 ℙcprime 12514 pCnt cpc 12692

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4170 ax-sep 4173 ax-nul 4181 ax-pow 4229 ax-pr 4264 ax-un 4493 ax-setind 4598 ax-iinf 4649 ax-cnex 8046 ax-resscn 8047 ax-1cn 8048 ax-1re 8049 ax-icn 8050 ax-addcl 8051 ax-addrcl 8052 ax-mulcl 8053 ax-mulrcl 8054 ax-addcom 8055 ax-mulcom 8056 ax-addass 8057 ax-mulass 8058 ax-distr 8059 ax-i2m1 8060 ax-0lt1 8061 ax-1rid 8062 ax-0id 8063 ax-rnegex 8064 ax-precex 8065 ax-cnre 8066 ax-pre-ltirr 8067 ax-pre-ltwlin 8068 ax-pre-lttrn 8069 ax-pre-apti 8070 ax-pre-ltadd 8071 ax-pre-mulgt0 8072 ax-pre-mulext 8073 ax-arch 8074 ax-caucvg 8075

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-if 3576 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3860 df-int 3895 df-iun 3938 df-br 4055 df-opab 4117 df-mpt 4118 df-tr 4154 df-id 4353 df-po 4356 df-iso 4357 df-iord 4426 df-on 4428 df-ilim 4429 df-suc 4431 df-iom 4652 df-xp 4694 df-rel 4695 df-cnv 4696 df-co 4697 df-dm 4698 df-rn 4699 df-res 4700 df-ima 4701 df-iota 5246 df-fun 5287 df-fn 5288 df-f 5289 df-f1 5290 df-fo 5291 df-f1o 5292 df-fv 5293 df-isom 5294 df-riota 5917 df-ov 5965 df-oprab 5966 df-mpo 5967 df-1st 6244 df-2nd 6245 df-recs 6409 df-frec 6495 df-1o 6520 df-2o 6521 df-er 6638 df-en 6846 df-sup 7107 df-inf 7108 df-pnf 8139 df-mnf 8140 df-xr 8141 df-ltxr 8142 df-le 8143 df-sub 8275 df-neg 8276 df-reap 8678 df-ap 8685 df-div 8776 df-inn 9067 df-2 9125 df-3 9126 df-4 9127 df-n0 9326 df-z 9403 df-uz 9679 df-q 9771 df-rp 9806 df-fz 10161 df-fzo 10295 df-fl 10445 df-mod 10500 df-seqfrec 10625 df-exp 10716 df-cj 11238 df-re 11239 df-im 11240 df-rsqrt 11394 df-abs 11395 df-dvds 12184 df-gcd 12360 df-prm 12515 df-pc 12693

This theorem is referenced by: pccl 12707 pcdiv 12710 pcqmul 12711 pcqcl 12714 pcxnn0cl 12718 pcge0 12721 pcdvdsb 12728 pcdvdstr 12735 pcgcd1 12736 pc2dvds 12738 pcz 12740 pcaddlem 12747 pcadd 12748 qexpz 12760 lgsval 15566 lgsfcl2 15568 lgsdir 15597 lgsdilem2 15598 lgsdi 15599 lgsne0 15600

W3C validator