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Theorem eulerpartlemelr 30728
Description: Lemma for eulerpart 30753. (Contributed by Thierry Arnoux, 8-Aug-2018.)
Hypotheses
Ref Expression
eulerpartlems.r 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
eulerpartlems.s 𝑆 = (𝑓 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))
Assertion
Ref Expression
eulerpartlemelr (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin))
Distinct variable groups:   𝑓,𝑘,𝐴   𝑅,𝑓,𝑘
Allowed substitution hints:   𝑆(𝑓,𝑘)

Proof of Theorem eulerpartlemelr
StepHypRef Expression
1 inss1 3976 . . . 4 ((ℕ0𝑚 ℕ) ∩ 𝑅) ⊆ (ℕ0𝑚 ℕ)
21sseli 3740 . . 3 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 𝐴 ∈ (ℕ0𝑚 ℕ))
3 elmapi 8045 . . 3 (𝐴 ∈ (ℕ0𝑚 ℕ) → 𝐴:ℕ⟶ℕ0)
42, 3syl 17 . 2 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
5 inss2 3977 . . . 4 ((ℕ0𝑚 ℕ) ∩ 𝑅) ⊆ 𝑅
65sseli 3740 . . 3 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 𝐴𝑅)
7 cnveq 5451 . . . . . 6 (𝑓 = 𝐴𝑓 = 𝐴)
87imaeq1d 5623 . . . . 5 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
98eleq1d 2824 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
10 eulerpartlems.r . . . 4 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
119, 10elab2g 3493 . . 3 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴𝑅 ↔ (𝐴 “ ℕ) ∈ Fin))
126, 11mpbid 222 . 2 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴 “ ℕ) ∈ Fin)
134, 12jca 555 1 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {cab 2746  cin 3714  cmpt 4881  ccnv 5265  cima 5269  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  Fincfn 8121   · cmul 10133  cn 11212  0cn0 11484  Σcsu 14615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025
This theorem is referenced by:  eulerpartlemsv2  30729  eulerpartlemsf  30730  eulerpartlems  30731  eulerpartlemsv3  30732  eulerpartlemgc  30733
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