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Mirrors > Home > MPE Home > Th. List > sgmnncl | Structured version Visualization version GIF version |
Description: Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.) |
Ref | Expression |
---|---|
sgmnncl | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0z 12584 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ) | |
2 | sgmval2 27026 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝐴)) | |
3 | 1, 2 | sylan 579 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝐴)) |
4 | fzfid 13941 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (1...𝐵) ∈ Fin) | |
5 | dvdsssfz1 16266 | . . . . . 6 ⊢ (𝐵 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} ⊆ (1...𝐵)) | |
6 | 5 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} ⊆ (1...𝐵)) |
7 | 4, 6 | ssfid 9266 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} ∈ Fin) |
8 | elrabi 3672 | . . . . . 6 ⊢ (𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} → 𝑘 ∈ ℕ) | |
9 | simpl 482 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℕ0) | |
10 | nnexpcl 14043 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ 𝐴 ∈ ℕ0) → (𝑘↑𝐴) ∈ ℕ) | |
11 | 8, 9, 10 | syl2anr 596 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) ∧ 𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵}) → (𝑘↑𝐴) ∈ ℕ) |
12 | 11 | nnzd 12586 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) ∧ 𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵}) → (𝑘↑𝐴) ∈ ℤ) |
13 | 7, 12 | fsumzcl 15685 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝐴) ∈ ℤ) |
14 | nnz 12580 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
15 | iddvds 16218 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℤ → 𝐵 ∥ 𝐵) | |
16 | 14, 15 | syl 17 | . . . . . . . 8 ⊢ (𝐵 ∈ ℕ → 𝐵 ∥ 𝐵) |
17 | breq1 5144 | . . . . . . . . 9 ⊢ (𝑝 = 𝐵 → (𝑝 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵)) | |
18 | 17 | rspcev 3606 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐵) → ∃𝑝 ∈ ℕ 𝑝 ∥ 𝐵) |
19 | 16, 18 | mpdan 684 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → ∃𝑝 ∈ ℕ 𝑝 ∥ 𝐵) |
20 | rabn0 4380 | . . . . . . 7 ⊢ ({𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} ≠ ∅ ↔ ∃𝑝 ∈ ℕ 𝑝 ∥ 𝐵) | |
21 | 19, 20 | sylibr 233 | . . . . . 6 ⊢ (𝐵 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} ≠ ∅) |
22 | 21 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} ≠ ∅) |
23 | 11 | nnrpd 13017 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) ∧ 𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵}) → (𝑘↑𝐴) ∈ ℝ+) |
24 | 7, 22, 23 | fsumrpcl 15687 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝐴) ∈ ℝ+) |
25 | 24 | rpgt0d 13022 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → 0 < Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝐴)) |
26 | elnnz 12569 | . . 3 ⊢ (Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝐴) ∈ ℕ ↔ (Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝐴) ∈ ℤ ∧ 0 < Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝐴))) | |
27 | 13, 25, 26 | sylanbrc 582 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝐴) ∈ ℕ) |
28 | 3, 27 | eqeltrd 2827 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∃wrex 3064 {crab 3426 ⊆ wss 3943 ∅c0 4317 class class class wbr 5141 (class class class)co 7404 0cc0 11109 1c1 11110 < clt 11249 ℕcn 12213 ℕ0cn0 12473 ℤcz 12559 ...cfz 13487 ↑cexp 14030 Σcsu 15636 ∥ cdvds 16202 σ csgm 26979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14031 df-fac 14237 df-bc 14266 df-hash 14294 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-dvds 16203 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19231 df-cmn 19700 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-lp 22991 df-perf 22992 df-cn 23082 df-cnp 23083 df-haus 23170 df-tx 23417 df-hmeo 23610 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-xms 24177 df-ms 24178 df-tms 24179 df-cncf 24749 df-limc 25746 df-dv 25747 df-log 26441 df-cxp 26442 df-sgm 26985 |
This theorem is referenced by: perfectlem1 27113 perfectlem2 27114 perfectALTVlem1 46942 perfectALTVlem2 46943 |
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