0sgmppw - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 0sgmppw		Structured version Visualization version GIF version

Theorem 0sgmppw 26251

Description: A prime power 𝑃↑𝐾 has 𝐾 + 1 divisors. (Contributed by Mario Carneiro, 17-May-2016.)

**Assertion**
Ref	Expression
0sgmppw	⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → (0 σ (𝑃↑𝐾)) = (𝐾 + 1))

Proof of Theorem 0sgmppw Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmnn 16307	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
2		nnexpcl 13723	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝐾 ∈ ℕ₀) → (𝑃↑𝐾) ∈ ℕ)
3	1, 2	sylan 579	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → (𝑃↑𝐾) ∈ ℕ)
4		0sgm 26198	. . . 4 ⊢ ((𝑃↑𝐾) ∈ ℕ → (0 σ (𝑃↑𝐾)) = (♯‘{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐾)}))
5	3, 4	syl 17	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → (0 σ (𝑃↑𝐾)) = (♯‘{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐾)}))
6		fzfid 13621	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → (0...𝐾) ∈ Fin)
7		eqid 2738	. . . . 5 ⊢ (𝑛 ∈ (0...𝐾) ↦ (𝑃↑𝑛)) = (𝑛 ∈ (0...𝐾) ↦ (𝑃↑𝑛))
8	7	dvdsppwf1o 26240	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → (𝑛 ∈ (0...𝐾) ↦ (𝑃↑𝑛)):(0...𝐾)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐾)})
9	6, 8	hasheqf1od 13996	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → (♯‘(0...𝐾)) = (♯‘{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃↑𝐾)}))
10	5, 9	eqtr4d 2781	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → (0 σ (𝑃↑𝐾)) = (♯‘(0...𝐾)))
11		simpr 484	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → 𝐾 ∈ ℕ₀)
12		nn0uz 12549	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
13	11, 12	eleqtrdi 2849	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → 𝐾 ∈ (ℤ_≥‘0))
14		hashfz 14070	. . 3 ⊢ (𝐾 ∈ (ℤ_≥‘0) → (♯‘(0...𝐾)) = ((𝐾 − 0) + 1))
15	13, 14	syl 17	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → (♯‘(0...𝐾)) = ((𝐾 − 0) + 1))
16		nn0cn 12173	. . . . 5 ⊢ (𝐾 ∈ ℕ₀ → 𝐾 ∈ ℂ)
17	16	adantl 481	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → 𝐾 ∈ ℂ)
18	17	subid1d 11251	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → (𝐾 − 0) = 𝐾)
19	18	oveq1d 7270	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → ((𝐾 − 0) + 1) = (𝐾 + 1))
20	10, 15, 19	3eqtrd 2782	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ₀) → (0 σ (𝑃↑𝐾)) = (𝐾 + 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 {crab 3067 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 ℂcc 10800 0cc0 10802 1c1 10803 + caddc 10805 − cmin 11135 ℕcn 11903 ℕ₀cn0 12163 ℤ_≥cuz 12511 ...cfz 13168 ↑cexp 13710 ♯chash 13972 ∥ cdvds 15891 ℙcprime 16304 σ csgm 26150

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-dvds 15892 df-gcd 16130 df-prm 16305 df-pc 16466 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-cxp 25618 df-sgm 26156

This theorem is referenced by: (None)

W3C validator