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Theorem eulerpartlemt0 30813
Description: Lemma for eulerpart 30826. (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 5464 . . . . . 6 (𝑓 = 𝐴𝑓 = 𝐴)
21imaeq1d 5647 . . . . 5 (𝑓 = 𝐴 → (𝑓 “ ℕ) = (𝐴 “ ℕ))
32sseq1d 3792 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ (𝐴 “ ℕ) ⊆ 𝐽))
4 eulerpart.t . . . 4 𝑇 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
53, 4elrab2 3523 . . 3 (𝐴𝑇 ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽))
62eleq1d 2829 . . . 4 (𝑓 = 𝐴 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝐴 “ ℕ) ∈ Fin))
7 eulerpart.r . . . 4 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
86, 7elab4g 3510 . . 3 (𝐴𝑅 ↔ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin))
95, 8anbi12i 620 . 2 ((𝐴𝑇𝐴𝑅) ↔ ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin)))
10 elin 3958 . 2 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴𝑇𝐴𝑅))
11 elex 3365 . . . . 5 (𝐴 ∈ (ℕ0𝑚 ℕ) → 𝐴 ∈ V)
1211pm4.71i 555 . . . 4 (𝐴 ∈ (ℕ0𝑚 ℕ) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ 𝐴 ∈ V))
1312anbi1i 617 . . 3 ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)) ↔ ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ 𝐴 ∈ V) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)))
14 3anass 1116 . . 3 ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ ((𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽)))
15 an42 647 . . 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 294 . 2 ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽) ↔ ((𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ⊆ 𝐽) ∧ (𝐴 ∈ V ∧ (𝐴 “ ℕ) ∈ Fin)))
179, 10, 163bitr4i 294 1 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  {cab 2751  wral 3055  {crab 3059  Vcvv 3350  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315   class class class wbr 4809  {copab 4871  cmpt 4888  ccnv 5276  cima 5280  cfv 6068  (class class class)co 6842  cmpt2 6844   supp csupp 7497  𝑚 cmap 8060  Fincfn 8160  1c1 10190   · cmul 10194  cle 10329  cn 11274  2c2 11327  0cn0 11538  cexp 13067  Σcsu 14703  cdvds 15267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290
This theorem is referenced by:  eulerpartlemf  30814  eulerpartlemt  30815  eulerpartlemmf  30819  eulerpartlemgvv  30820  eulerpartlemgu  30821  eulerpartlemgh  30822  eulerpartlemgs2  30824  eulerpartlemn  30825
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