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Theorem eulerpartlemelr 32769
Description: Lemma for eulerpart 32794. (Contributed by Thierry Arnoux, 8-Aug-2018.)
Hypotheses
Ref Expression
eulerpartlems.r 𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}
eulerpartlems.s 𝑆 = (𝑓 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ↦ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜))
Assertion
Ref Expression
eulerpartlemelr (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (𝐴:β„•βŸΆβ„•0 ∧ (◑𝐴 β€œ β„•) ∈ Fin))
Distinct variable groups:   𝑓,π‘˜,𝐴   𝑅,𝑓,π‘˜
Allowed substitution hints:   𝑆(𝑓,π‘˜)

Proof of Theorem eulerpartlemelr
StepHypRef Expression
1 inss1 4186 . . . 4 ((β„•0 ↑m β„•) ∩ 𝑅) βŠ† (β„•0 ↑m β„•)
21sseli 3938 . . 3 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ 𝐴 ∈ (β„•0 ↑m β„•))
3 elmapi 8745 . . 3 (𝐴 ∈ (β„•0 ↑m β„•) β†’ 𝐴:β„•βŸΆβ„•0)
42, 3syl 17 . 2 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ 𝐴:β„•βŸΆβ„•0)
5 inss2 4187 . . . 4 ((β„•0 ↑m β„•) ∩ 𝑅) βŠ† 𝑅
65sseli 3938 . . 3 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ 𝐴 ∈ 𝑅)
7 cnveq 5827 . . . . . 6 (𝑓 = 𝐴 β†’ ◑𝑓 = ◑𝐴)
87imaeq1d 6010 . . . . 5 (𝑓 = 𝐴 β†’ (◑𝑓 β€œ β„•) = (◑𝐴 β€œ β„•))
98eleq1d 2822 . . . 4 (𝑓 = 𝐴 β†’ ((◑𝑓 β€œ β„•) ∈ Fin ↔ (◑𝐴 β€œ β„•) ∈ Fin))
10 eulerpartlems.r . . . 4 𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}
119, 10elab2g 3630 . . 3 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (𝐴 ∈ 𝑅 ↔ (◑𝐴 β€œ β„•) ∈ Fin))
126, 11mpbid 231 . 2 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (◑𝐴 β€œ β„•) ∈ Fin)
134, 12jca 512 1 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (𝐴:β„•βŸΆβ„•0 ∧ (◑𝐴 β€œ β„•) ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2714   ∩ cin 3907   ↦ cmpt 5186  β—‘ccnv 5630   β€œ cima 5634  βŸΆwf 6489  β€˜cfv 6493  (class class class)co 7351   ↑m cmap 8723  Fincfn 8841   Β· cmul 11014  β„•cn 12111  β„•0cn0 12371  Ξ£csu 15530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-map 8725
This theorem is referenced by:  eulerpartlemsv2  32770  eulerpartlemsf  32771  eulerpartlems  32772  eulerpartlemsv3  32773  eulerpartlemgc  32774
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