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Theorem eulerpartlemelr 34360
Description: Lemma for eulerpart 34385. (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 4236 . . . 4 ((ℕ0m ℕ) ∩ 𝑅) ⊆ (ℕ0m ℕ)
21sseli 3978 . . 3 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → 𝐴 ∈ (ℕ0m ℕ))
3 elmapi 8890 . . 3 (𝐴 ∈ (ℕ0m ℕ) → 𝐴:ℕ⟶ℕ0)
42, 3syl 17 . 2 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
5 inss2 4237 . . . 4 ((ℕ0m ℕ) ∩ 𝑅) ⊆ 𝑅
65sseli 3978 . . 3 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → 𝐴𝑅)
7 cnveq 5883 . . . . . 6 (𝑓 = 𝐴𝑓 = 𝐴)
87imaeq1d 6076 . . . . 5 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
98eleq1d 2825 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
10 eulerpartlems.r . . . 4 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
119, 10elab2g 3679 . . 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 1539  wcel 2107  {cab 2713  cin 3949  cmpt 5224  ccnv 5683  cima 5687  wf 6556  cfv 6560  (class class class)co 7432  m cmap 8867  Fincfn 8986   · cmul 11161  cn 12267  0cn0 12528  Σcsu 15723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869
This theorem is referenced by:  eulerpartlemsv2  34361  eulerpartlemsf  34362  eulerpartlems  34363  eulerpartlemsv3  34364  eulerpartlemgc  34365
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