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Theorem eulerpartlemelr 30755
Description: Lemma for eulerpart 30780. (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 3981 . . . 4 ((ℕ0𝑚 ℕ) ∩ 𝑅) ⊆ (ℕ0𝑚 ℕ)
21sseli 3748 . . 3 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 𝐴 ∈ (ℕ0𝑚 ℕ))
3 elmapi 8031 . . 3 (𝐴 ∈ (ℕ0𝑚 ℕ) → 𝐴:ℕ⟶ℕ0)
42, 3syl 17 . 2 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
5 inss2 3982 . . . 4 ((ℕ0𝑚 ℕ) ∩ 𝑅) ⊆ 𝑅
65sseli 3748 . . 3 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 𝐴𝑅)
7 cnveq 5432 . . . . . 6 (𝑓 = 𝐴𝑓 = 𝐴)
87imaeq1d 5604 . . . . 5 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
98eleq1d 2835 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
10 eulerpartlems.r . . . 4 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
119, 10elab2g 3504 . . 3 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴𝑅 ↔ (𝐴 “ ℕ) ∈ Fin))
126, 11mpbid 222 . 2 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴 “ ℕ) ∈ Fin)
134, 12jca 501 1 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  cin 3722  cmpt 4863  ccnv 5248  cima 5252  wf 6025  cfv 6029  (class class class)co 6792  𝑚 cmap 8009  Fincfn 8109   · cmul 10143  cn 11222  0cn0 11495  Σcsu 14620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-fv 6037  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7315  df-2nd 7316  df-map 8011
This theorem is referenced by:  eulerpartlemsv2  30756  eulerpartlemsf  30757  eulerpartlems  30758  eulerpartlemsv3  30759  eulerpartlemgc  30760
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