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Theorem eulerpartlemelr 33845
Description: Lemma for eulerpart 33870. (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 4220 . . . 4 ((β„•0 ↑m β„•) ∩ 𝑅) βŠ† (β„•0 ↑m β„•)
21sseli 3970 . . 3 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ 𝐴 ∈ (β„•0 ↑m β„•))
3 elmapi 8839 . . 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 4221 . . . 4 ((β„•0 ↑m β„•) ∩ 𝑅) βŠ† 𝑅
65sseli 3970 . . 3 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ 𝐴 ∈ 𝑅)
7 cnveq 5863 . . . . . 6 (𝑓 = 𝐴 β†’ ◑𝑓 = ◑𝐴)
87imaeq1d 6048 . . . . 5 (𝑓 = 𝐴 β†’ (◑𝑓 β€œ β„•) = (◑𝐴 β€œ β„•))
98eleq1d 2810 . . . 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 3662 . . 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 511 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 395   = wceq 1533   ∈ wcel 2098  {cab 2701   ∩ cin 3939   ↦ cmpt 5221  β—‘ccnv 5665   β€œ cima 5669  βŸΆwf 6529  β€˜cfv 6533  (class class class)co 7401   ↑m cmap 8816  Fincfn 8935   Β· cmul 11111  β„•cn 12209  β„•0cn0 12469  Ξ£csu 15629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8818
This theorem is referenced by:  eulerpartlemsv2  33846  eulerpartlemsf  33847  eulerpartlems  33848  eulerpartlemsv3  33849  eulerpartlemgc  33850
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