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Theorem eulerpartlemgv 30563
Description: Lemma for eulerpart 30572: value of the function 𝐺. (Contributed by Thierry Arnoux, 13-Nov-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}
eulerpart.g 𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
Assertion
Ref Expression
eulerpartlemgv (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽))))))
Distinct variable groups:   𝐴,𝑜   𝑜,𝐹   𝑜,𝐽   𝑜,𝑀   𝑅,𝑜   𝑇,𝑜
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐷(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,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)   𝐺(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝐽(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑀(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑁(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝑂(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)

Proof of Theorem eulerpartlemgv
StepHypRef Expression
1 reseq1 5422 . . . . . 6 (𝑜 = 𝐴 → (𝑜𝐽) = (𝐴𝐽))
21coeq2d 5317 . . . . 5 (𝑜 = 𝐴 → (bits ∘ (𝑜𝐽)) = (bits ∘ (𝐴𝐽)))
32fveq2d 6233 . . . 4 (𝑜 = 𝐴 → (𝑀‘(bits ∘ (𝑜𝐽))) = (𝑀‘(bits ∘ (𝐴𝐽))))
43imaeq2d 5501 . . 3 (𝑜 = 𝐴 → (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))) = (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))
54fveq2d 6233 . 2 (𝑜 = 𝐴 → ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽))))))
6 eulerpart.g . 2 𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
7 fvex 6239 . 2 ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽))))) ∈ V
85, 6, 7fvmpt 6321 1 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  {crab 2945  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   class class class wbr 4685  {copab 4745  cmpt 4762  ccnv 5142  cres 5145  cima 5146  ccom 5147  cfv 5926  (class class class)co 6690  cmpt2 6692   supp csupp 7340  𝑚 cmap 7899  Fincfn 7997  1c1 9975   · cmul 9979  cle 10113  cn 11058  2c2 11108  0cn0 11330  cexp 12900  Σcsu 14460  cdvds 15027  bitscbits 15188  𝟭cind 30200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  eulerpartlemgvv  30566  eulerpartlemgf  30569  eulerpartlemn  30571
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