![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlemsf | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 30988. (Contributed by Thierry Arnoux, 8-Aug-2018.) |
Ref | Expression |
---|---|
eulerpartlems.r | ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} |
eulerpartlems.s | ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) |
Ref | Expression |
---|---|
eulerpartlemsf | ⊢ 𝑆:((ℕ0 ↑𝑚 ℕ) ∩ 𝑅)⟶ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eulerpartlems.s | . 2 ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) | |
2 | simpl 476 | . . . . . . 7 ⊢ ((𝑔 = 𝑓 ∧ 𝑘 ∈ ℕ) → 𝑔 = 𝑓) | |
3 | 2 | fveq1d 6434 | . . . . . 6 ⊢ ((𝑔 = 𝑓 ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) = (𝑓‘𝑘)) |
4 | 3 | oveq1d 6919 | . . . . 5 ⊢ ((𝑔 = 𝑓 ∧ 𝑘 ∈ ℕ) → ((𝑔‘𝑘) · 𝑘) = ((𝑓‘𝑘) · 𝑘)) |
5 | 4 | sumeq2dv 14809 | . . . 4 ⊢ (𝑔 = 𝑓 → Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) |
6 | 5 | eleq1d 2890 | . . 3 ⊢ (𝑔 = 𝑓 → (Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) ∈ ℕ0 ↔ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) ∈ ℕ0)) |
7 | eulerpartlems.r | . . . . . 6 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
8 | 7, 1 | eulerpartlemsv2 30964 | . . . . 5 ⊢ (𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝑆‘𝑔) = Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘)) |
9 | 7, 1 | eulerpartlemsv1 30962 | . . . . 5 ⊢ (𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝑆‘𝑔) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘)) |
10 | 8, 9 | eqtr3d 2862 | . . . 4 ⊢ (𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘)) |
11 | 7, 1 | eulerpartlemelr 30963 | . . . . . 6 ⊢ (𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin)) |
12 | 11 | simprd 491 | . . . . 5 ⊢ (𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (◡𝑔 “ ℕ) ∈ Fin) |
13 | 11 | simpld 490 | . . . . . . . 8 ⊢ (𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → 𝑔:ℕ⟶ℕ0) |
14 | 13 | adantr 474 | . . . . . . 7 ⊢ ((𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑔:ℕ⟶ℕ0) |
15 | cnvimass 5725 | . . . . . . . . 9 ⊢ (◡𝑔 “ ℕ) ⊆ dom 𝑔 | |
16 | 15, 13 | fssdm 6293 | . . . . . . . 8 ⊢ (𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (◡𝑔 “ ℕ) ⊆ ℕ) |
17 | 16 | sselda 3826 | . . . . . . 7 ⊢ ((𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ) |
18 | 14, 17 | ffvelrnd 6608 | . . . . . 6 ⊢ ((𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → (𝑔‘𝑘) ∈ ℕ0) |
19 | 17 | nnnn0d 11677 | . . . . . 6 ⊢ ((𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ0) |
20 | 18, 19 | nn0mulcld 11682 | . . . . 5 ⊢ ((𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℕ0) |
21 | 12, 20 | fsumnn0cl 14843 | . . . 4 ⊢ (𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) ∈ ℕ0) |
22 | 10, 21 | eqeltrrd 2906 | . . 3 ⊢ (𝑔 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) ∈ ℕ0) |
23 | 6, 22 | vtoclga 3488 | . 2 ⊢ (𝑓 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) ∈ ℕ0) |
24 | 1, 23 | fmpti 6630 | 1 ⊢ 𝑆:((ℕ0 ↑𝑚 ℕ) ∩ 𝑅)⟶ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1658 ∈ wcel 2166 {cab 2810 ∩ cin 3796 ↦ cmpt 4951 ◡ccnv 5340 “ cima 5344 ⟶wf 6118 ‘cfv 6122 (class class class)co 6904 ↑𝑚 cmap 8121 Fincfn 8221 · cmul 10256 ℕcn 11349 ℕ0cn0 11617 Σcsu 14792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-map 8123 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-sup 8616 df-oi 8683 df-card 9077 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-z 11704 df-uz 11968 df-rp 12112 df-fz 12619 df-fzo 12760 df-seq 13095 df-exp 13154 df-hash 13410 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-clim 14595 df-sum 14793 |
This theorem is referenced by: eulerpartlems 30966 eulerpartlemsv3 30967 eulerpartlemgc 30968 |
Copyright terms: Public domain | W3C validator |