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Theorem eulerpartlemelr 34339
Description: Lemma for eulerpart 34364. (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 4245 . . . 4 ((ℕ0m ℕ) ∩ 𝑅) ⊆ (ℕ0m ℕ)
21sseli 3991 . . 3 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → 𝐴 ∈ (ℕ0m ℕ))
3 elmapi 8888 . . 3 (𝐴 ∈ (ℕ0m ℕ) → 𝐴:ℕ⟶ℕ0)
42, 3syl 17 . 2 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
5 inss2 4246 . . . 4 ((ℕ0m ℕ) ∩ 𝑅) ⊆ 𝑅
65sseli 3991 . . 3 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → 𝐴𝑅)
7 cnveq 5887 . . . . . 6 (𝑓 = 𝐴𝑓 = 𝐴)
87imaeq1d 6079 . . . . 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 3683 . . 3 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝐴𝑅 ↔ (𝐴 “ ℕ) ∈ Fin))
126, 11mpbid 232 . 2 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝐴 “ ℕ) ∈ Fin)
134, 12jca 511 1 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  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  cn 12264  0cn0 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-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3060  df-rex 3069  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-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  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-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by:  eulerpartlemsv2  34340  eulerpartlemsf  34341  eulerpartlems  34342  eulerpartlemsv3  34343  eulerpartlemgc  34344
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