![]() |
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 34364. (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 482 | . . . . . . 7 ⊢ ((𝑔 = 𝑓 ∧ 𝑘 ∈ ℕ) → 𝑔 = 𝑓) | |
3 | 2 | fveq1d 6909 | . . . . . 6 ⊢ ((𝑔 = 𝑓 ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) = (𝑓‘𝑘)) |
4 | 3 | oveq1d 7446 | . . . . 5 ⊢ ((𝑔 = 𝑓 ∧ 𝑘 ∈ ℕ) → ((𝑔‘𝑘) · 𝑘) = ((𝑓‘𝑘) · 𝑘)) |
5 | 4 | sumeq2dv 15735 | . . . 4 ⊢ (𝑔 = 𝑓 → Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) |
6 | 5 | eleq1d 2824 | . . 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 34340 | . . . . 5 ⊢ (𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝑔) = Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘)) |
9 | 7, 1 | eulerpartlemsv1 34338 | . . . . 5 ⊢ (𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝑔) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘)) |
10 | 8, 9 | eqtr3d 2777 | . . . 4 ⊢ (𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘)) |
11 | 7, 1 | eulerpartlemelr 34339 | . . . . . 6 ⊢ (𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin)) |
12 | 11 | simprd 495 | . . . . 5 ⊢ (𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (◡𝑔 “ ℕ) ∈ Fin) |
13 | 11 | simpld 494 | . . . . . . . 8 ⊢ (𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 𝑔:ℕ⟶ℕ0) |
14 | 13 | adantr 480 | . . . . . . 7 ⊢ ((𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑔:ℕ⟶ℕ0) |
15 | cnvimass 6102 | . . . . . . . . 9 ⊢ (◡𝑔 “ ℕ) ⊆ dom 𝑔 | |
16 | 15, 13 | fssdm 6756 | . . . . . . . 8 ⊢ (𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (◡𝑔 “ ℕ) ⊆ ℕ) |
17 | 16 | sselda 3995 | . . . . . . 7 ⊢ ((𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ) |
18 | 14, 17 | ffvelcdmd 7105 | . . . . . 6 ⊢ ((𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → (𝑔‘𝑘) ∈ ℕ0) |
19 | 17 | nnnn0d 12585 | . . . . . 6 ⊢ ((𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ0) |
20 | 18, 19 | nn0mulcld 12590 | . . . . 5 ⊢ ((𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℕ0) |
21 | 12, 20 | fsumnn0cl 15769 | . . . 4 ⊢ (𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) ∈ ℕ0) |
22 | 10, 21 | eqeltrrd 2840 | . . 3 ⊢ (𝑔 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) ∈ ℕ0) |
23 | 6, 22 | vtoclga 3577 | . 2 ⊢ (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) ∈ ℕ0) |
24 | 1, 23 | fmpti 7132 | 1 ⊢ 𝑆:((ℕ0 ↑m ℕ) ∩ 𝑅)⟶ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 ∩ cin 3962 ↦ cmpt 5231 ◡ccnv 5688 “ cima 5692 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Fincfn 8984 · cmul 11158 ℕcn 12264 ℕ0cn0 12524 Σcsu 15719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 |
This theorem is referenced by: eulerpartlems 34342 eulerpartlemsv3 34343 eulerpartlemgc 34344 |
Copyright terms: Public domain | W3C validator |