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Theorem eulerpartlemt0 31752
 Description: Lemma for eulerpart 31765. (Contributed by Thierry Arnoux, 19-Sep-2017.)
Hypotheses
Ref Expression
eulerpart.p 𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
eulerpart.o 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
eulerpart.j 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
eulerpart.f 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
eulerpart.h 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
eulerpart.m 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
eulerpart.r 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
eulerpart.t 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
Assertion
Ref Expression
eulerpartlemt0 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
Distinct variable groups:   𝐴,𝑓   𝑓,𝐽
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑔,𝑘,𝑛,𝑟)   𝐷(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑅(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑇(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐹(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐽(𝑥,𝑦,𝑧,𝑔,𝑘,𝑛,𝑟)   𝑀(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑁(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑂(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)

Proof of Theorem eulerpartlemt0
StepHypRef Expression
1 cnveq 5709 . . . . . 6 (𝑓 = 𝐴𝑓 = 𝐴)
21imaeq1d 5896 . . . . 5 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
32sseq1d 3946 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ (𝐴 “ ℕ) ⊆ 𝐽))
4 eulerpart.t . . . 4 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
53, 4elrab2 3631 . . 3 (𝐴𝑇 ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽))
62eleq1d 2874 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
7 eulerpart.r . . . 4 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
86, 7elab4g 3619 . . 3 (𝐴𝑅 ↔ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin))
95, 8anbi12i 629 . 2 ((𝐴𝑇𝐴𝑅) ↔ ((𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin)))
10 elin 3897 . 2 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴𝑇𝐴𝑅))
11 elex 3459 . . . . 5 (𝐴 ∈ (ℕ0m ℕ) → 𝐴 ∈ V)
1211pm4.71i 563 . . . 4 (𝐴 ∈ (ℕ0m ℕ) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ 𝐴 ∈ V))
1312anbi1i 626 . . 3 ((𝐴 ∈ (ℕ0m ℕ) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)) ↔ ((𝐴 ∈ (ℕ0m ℕ) ∧ 𝐴 ∈ V) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)))
14 3anass 1092 . . 3 ((𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)))
15 an42 656 . . 3 (((𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin)) ↔ ((𝐴 ∈ (ℕ0m ℕ) ∧ 𝐴 ∈ V) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)))
1613, 14, 153bitr4i 306 . 2 ((𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽) ↔ ((𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin)))
179, 10, 163bitr4i 306 1 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  {crab 3110  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497   class class class wbr 5031  {copab 5093   ↦ cmpt 5111  ◡ccnv 5519   “ cima 5523  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138   supp csupp 7816   ↑m cmap 8392  Fincfn 8495  1c1 10530   · cmul 10534   ≤ cle 10668  ℕcn 11628  2c2 11683  ℕ0cn0 11888  ↑cexp 13428  Σcsu 15037   ∥ cdvds 15602 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-cnv 5528  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533 This theorem is referenced by:  eulerpartlemf  31753  eulerpartlemt  31754  eulerpartlemmf  31758  eulerpartlemgvv  31759  eulerpartlemgu  31760  eulerpartlemgh  31761  eulerpartlemgs2  31763  eulerpartlemn  31764
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