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Theorem eulerpartlemt0 30761
 Description: Lemma for eulerpart 30774. (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 5451 . . . . . 6 (𝑓 = 𝐴𝑓 = 𝐴)
21imaeq1d 5623 . . . . 5 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
32sseq1d 3773 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ (𝐴 “ ℕ) ⊆ 𝐽))
4 eulerpart.t . . . 4 𝑇 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
53, 4elrab2 3507 . . 3 (𝐴𝑇 ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽))
62eleq1d 2824 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
7 eulerpart.r . . . 4 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
86, 7elab4g 3495 . . 3 (𝐴𝑅 ↔ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin))
95, 8anbi12i 735 . 2 ((𝐴𝑇𝐴𝑅) ↔ ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin)))
10 elin 3939 . 2 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴𝑇𝐴𝑅))
11 elex 3352 . . . . 5 (𝐴 ∈ (ℕ0𝑚 ℕ) → 𝐴 ∈ V)
1211pm4.71i 667 . . . 4 (𝐴 ∈ (ℕ0𝑚 ℕ) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ 𝐴 ∈ V))
1312anbi1i 733 . . 3 ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)) ↔ ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ 𝐴 ∈ V) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)))
14 3anass 1081 . . 3 ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)))
15 an42 901 . . 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 292 . 2 ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽) ↔ ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin)))
179, 10, 163bitr4i 292 1 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  {cab 2746  ∀wral 3050  {crab 3054  Vcvv 3340   ∩ cin 3714   ⊆ wss 3715  ∅c0 4058  𝒫 cpw 4302   class class class wbr 4804  {copab 4864   ↦ cmpt 4881  ◡ccnv 5265   “ cima 5269  ‘cfv 6049  (class class class)co 6814   ↦ cmpt2 6816   supp csupp 7464   ↑𝑚 cmap 8025  Fincfn 8123  1c1 10149   · cmul 10153   ≤ cle 10287  ℕcn 11232  2c2 11282  ℕ0cn0 11504  ↑cexp 13074  Σcsu 14635   ∥ cdvds 15202 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279 This theorem is referenced by:  eulerpartlemf  30762  eulerpartlemt  30763  eulerpartlemmf  30767  eulerpartlemgvv  30768  eulerpartlemgu  30769  eulerpartlemgh  30770  eulerpartlemgs2  30772  eulerpartlemn  30773
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