Theorem eulerpartlemelr 34324

Description: Lemma for eulerpart 34349. (Contributed by Thierry Arnoux, 8-Aug-2018.)

Hypotheses

Ref	Expression
eulerpartlems.r	⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
eulerpartlems.s	⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))

Assertion

Ref	Expression
eulerpartlemelr	⊢ (𝐴 ∈ ((ℕ0 ↑m 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

Step	Hyp	Ref	Expression
1		inss1 4190	. . . 4 ⊢ ((ℕ0 ↑m ℕ) ∩ 𝑅) ⊆ (ℕ0 ↑m ℕ)
2	1	sseli 3933	. . 3 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 𝐴 ∈ (ℕ0 ↑m ℕ))
3		elmapi 8783	. . 3 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴:ℕ⟶ℕ0)
4	2, 3	syl 17	. 2 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
5		inss2 4191	. . . 4 ⊢ ((ℕ0 ↑m ℕ) ∩ 𝑅) ⊆ 𝑅
6	5	sseli 3933	. . 3 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 𝐴 ∈ 𝑅)
7		cnveq 5820	. . . . . 6 ⊢ (𝑓 = 𝐴 → ◡𝑓 = ◡𝐴)
8	7	imaeq1d 6014	. . . . 5 ⊢ (𝑓 = 𝐴 → (◡𝑓 “ ℕ) = (◡𝐴 “ ℕ))
9	8	eleq1d 2813	. . . 4 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐴 “ ℕ) ∈ Fin))
10		eulerpartlems.r	. . . 4 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
11	9, 10	elab2g 3638	. . 3 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝐴 ∈ 𝑅 ↔ (◡𝐴 “ ℕ) ∈ Fin))
12	6, 11	mpbid 232	. 2 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (◡𝐴 “ ℕ) ∈ Fin)
13	4, 12	jca 511	1 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707   ∩ cin 3904   ↦ cmpt 5176  ^◡ccnv 5622   “ cima 5626  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353   ↑_m cmap 8760  Fincfn 8879   · cmul 11033  ℕcn 12146  ℕ₀cn0 12402  Σcsu 15611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-map 8762

This theorem is referenced by:  eulerpartlemsv2  34325  eulerpartlemsf  34326  eulerpartlems  34327  eulerpartlemsv3  34328  eulerpartlemgc  34329