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Theorem eulerpartlemt0 29564
Description: Lemma for eulerpart 29577. (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 5206 . . . . . 6 (𝑓 = 𝐴𝑓 = 𝐴)
21imaeq1d 5371 . . . . 5 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
32sseq1d 3594 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ (𝐴 “ ℕ) ⊆ 𝐽))
4 eulerpart.t . . . 4 𝑇 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
53, 4elrab2 3332 . . 3 (𝐴𝑇 ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽))
62eleq1d 2671 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
7 eulerpart.r . . . 4 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
86, 7elab4g 3323 . . 3 (𝐴𝑅 ↔ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin))
95, 8anbi12i 728 . 2 ((𝐴𝑇𝐴𝑅) ↔ ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin)))
10 elin 3757 . 2 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴𝑇𝐴𝑅))
11 elex 3184 . . . . 5 (𝐴 ∈ (ℕ0𝑚 ℕ) → 𝐴 ∈ V)
1211pm4.71i 661 . . . 4 (𝐴 ∈ (ℕ0𝑚 ℕ) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ 𝐴 ∈ V))
1312anbi1i 726 . . 3 ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)) ↔ ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ 𝐴 ∈ V) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)))
14 3anass 1034 . . 3 ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)))
15 an42 861 . . 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 290 . 2 ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽) ↔ ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin)))
179, 10, 163bitr4i 290 1 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  {cab 2595  wral 2895  {crab 2899  Vcvv 3172  cin 3538  wss 3539  c0 3873  𝒫 cpw 4107   class class class wbr 4577  {copab 4636  cmpt 4637  ccnv 5027  cima 5031  cfv 5790  (class class class)co 6527  cmpt2 6529   supp csupp 7159  𝑚 cmap 7721  Fincfn 7818  1c1 9793   · cmul 9797  cle 9931  cn 10867  2c2 10917  0cn0 11139  cexp 12677  Σcsu 14210  cdvds 14767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-cnv 5036  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041
This theorem is referenced by:  eulerpartlemf  29565  eulerpartlemt  29566  eulerpartlemmf  29570  eulerpartlemgvv  29571  eulerpartlemgu  29572  eulerpartlemgh  29573  eulerpartlemgs2  29575  eulerpartlemn  29576
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