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Theorem eulerpartlemelr 30750
Description: Lemma for eulerpart 30775. (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 4036 . . . 4 ((ℕ0𝑚 ℕ) ∩ 𝑅) ⊆ (ℕ0𝑚 ℕ)
21sseli 3801 . . 3 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 𝐴 ∈ (ℕ0𝑚 ℕ))
3 elmapi 8117 . . 3 (𝐴 ∈ (ℕ0𝑚 ℕ) → 𝐴:ℕ⟶ℕ0)
42, 3syl 17 . 2 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
5 inss2 4037 . . . 4 ((ℕ0𝑚 ℕ) ∩ 𝑅) ⊆ 𝑅
65sseli 3801 . . 3 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 𝐴𝑅)
7 cnveq 5504 . . . . . 6 (𝑓 = 𝐴𝑓 = 𝐴)
87imaeq1d 5682 . . . . 5 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
98eleq1d 2877 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
10 eulerpartlems.r . . . 4 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
119, 10elab2g 3555 . . 3 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴𝑅 ↔ (𝐴 “ ℕ) ∈ Fin))
126, 11mpbid 223 . 2 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴 “ ℕ) ∈ Fin)
134, 12jca 503 1 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2157  {cab 2799  cin 3775  cmpt 4930  ccnv 5317  cima 5321  wf 6100  cfv 6104  (class class class)co 6877  𝑚 cmap 8095  Fincfn 8195   · cmul 10229  cn 11308  0cn0 11562  Σcsu 14642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-fv 6112  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-1st 7401  df-2nd 7402  df-map 8097
This theorem is referenced by:  eulerpartlemsv2  30751  eulerpartlemsf  30752  eulerpartlems  30753  eulerpartlemsv3  30754  eulerpartlemgc  30755
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