Theorem eulerpartlemelr 34360

Description: Lemma for eulerpart 34385. (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 4185	. . . 4 ⊢ ((ℕ0 ↑m ℕ) ∩ 𝑅) ⊆ (ℕ0 ↑m ℕ)
2	1	sseli 3928	. . 3 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 𝐴 ∈ (ℕ0 ↑m ℕ))
3		elmapi 8768	. . 3 ⊢ (𝐴 ∈ (ℕ0 ↑m ℕ) → 𝐴:ℕ⟶ℕ0)
4	2, 3	syl 17	. 2 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
5		inss2 4186	. . . 4 ⊢ ((ℕ0 ↑m ℕ) ∩ 𝑅) ⊆ 𝑅
6	5	sseli 3928	. . 3 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 𝐴 ∈ 𝑅)
7		cnveq 5811	. . . . . 6 ⊢ (𝑓 = 𝐴 → ◡𝑓 = ◡𝐴)
8	7	imaeq1d 6005	. . . . 5 ⊢ (𝑓 = 𝐴 → (◡𝑓 “ ℕ) = (◡𝐴 “ ℕ))
9	8	eleq1d 2814	. . . 4 ⊢ (𝑓 = 𝐴 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐴 “ ℕ) ∈ Fin))
10		eulerpartlems.r	. . . 4 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
11	9, 10	elab2g 3634	. . 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 1541   ∈ wcel 2110  {cab 2708   ∩ cin 3899   ↦ cmpt 5170  ^◡ccnv 5613   “ cima 5617  ⟶wf 6473  ‘cfv 6477  (class class class)co 7341   ↑_m cmap 8745  Fincfn 8864   · cmul 11003  ℕcn 12117  ℕ₀cn0 12373  Σcsu 15585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-map 8747

This theorem is referenced by:  eulerpartlemsv2  34361  eulerpartlemsf  34362  eulerpartlems  34363  eulerpartlemsv3  34364  eulerpartlemgc  34365