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Theorem eulerpartlemt0 33832
Description: Lemma for eulerpart 33845. (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 5873 . . . . . 6 (𝑓 = 𝐴𝑓 = 𝐴)
21imaeq1d 6058 . . . . 5 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
32sseq1d 4013 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ (𝐴 “ ℕ) ⊆ 𝐽))
4 eulerpart.t . . . 4 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
53, 4elrab2 3686 . . 3 (𝐴𝑇 ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽))
62eleq1d 2817 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
7 eulerpart.r . . . 4 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
86, 7elab4g 3673 . . 3 (𝐴𝑅 ↔ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin))
95, 8anbi12i 626 . 2 ((𝐴𝑇𝐴𝑅) ↔ ((𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin)))
10 elin 3964 . 2 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴𝑇𝐴𝑅))
11 elex 3492 . . . . 5 (𝐴 ∈ (ℕ0m ℕ) → 𝐴 ∈ V)
1211pm4.71i 559 . . . 4 (𝐴 ∈ (ℕ0m ℕ) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ 𝐴 ∈ V))
1312anbi1i 623 . . 3 ((𝐴 ∈ (ℕ0m ℕ) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)) ↔ ((𝐴 ∈ (ℕ0m ℕ) ∧ 𝐴 ∈ V) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)))
14 3anass 1094 . . 3 ((𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)))
15 an42 654 . . 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 303 . 2 ((𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽) ↔ ((𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin)))
179, 10, 163bitr4i 303 1 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  {cab 2708  wral 3060  {crab 3431  Vcvv 3473  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602   class class class wbr 5148  {copab 5210  cmpt 5231  ccnv 5675  cima 5679  cfv 6543  (class class class)co 7412  cmpo 7414   supp csupp 8151  m cmap 8826  Fincfn 8945  1c1 11117   · cmul 11121  cle 11256  cn 12219  2c2 12274  0cn0 12479  cexp 14034  Σcsu 15639  cdvds 16204
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  eulerpartlemf  33833  eulerpartlemt  33834  eulerpartlemmf  33838  eulerpartlemgvv  33839  eulerpartlemgu  33840  eulerpartlemgh  33841  eulerpartlemgs2  33843  eulerpartlemn  33844
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