Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eulerpartlemelr Structured version   Visualization version   GIF version
Theorem eulerpartlemelr 33345
Description: Lemma for eulerpart 33370. (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 4228 . . . 4 ((β„•0 ↑m β„•) ∩ 𝑅) βŠ† (β„•0 ↑m β„•)
21sseli 3978 . . 3 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ 𝐴 ∈ (β„•0 ↑m β„•))
3 elmapi 8840 . . 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 4229 . . . 4 ((β„•0 ↑m β„•) ∩ 𝑅) βŠ† 𝑅
65sseli 3978 . . 3 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ 𝐴 ∈ 𝑅)
7 cnveq 5872 . . . . . 6 (𝑓 = 𝐴 β†’ ◑𝑓 = ◑𝐴)
87imaeq1d 6057 . . . . 5 (𝑓 = 𝐴 β†’ (◑𝑓 β€œ β„•) = (◑𝐴 β€œ β„•))
98eleq1d 2819 . . . 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 3670 . . 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 513 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 397   = wceq 1542   ∈ wcel 2107  {cab 2710   ∩ cin 3947   ↦ cmpt 5231  β—‘ccnv 5675   β€œ cima 5679  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406   ↑m cmap 8817  Fincfn 8936   Β· cmul 11112  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 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-map 8819
This theorem is referenced by:  eulerpartlemsv2  33346  eulerpartlemsf  33347  eulerpartlems  33348  eulerpartlemsv3  33349  eulerpartlemgc  33350
  Copyright terms: Public domain W3C validator