Theorem eulerpartlemgv 34407

Description: Lemma for eulerpart 34416: value of the function 𝐺. (Contributed by Thierry Arnoux, 13-Nov-2017.)

Hypotheses

Ref	Expression
eulerpart.p	⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
eulerpart.o	⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d	⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
eulerpart.j	⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
eulerpart.f	⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
eulerpart.h	⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
eulerpart.m	⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
eulerpart.r	⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
eulerpart.t	⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}
eulerpart.g	⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))

Assertion

Ref	Expression
eulerpartlemgv	⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))))

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

Step	Hyp	Ref	Expression
1		reseq1 5926	. . . . . 6 ⊢ (𝑜 = 𝐴 → (𝑜 ↾ 𝐽) = (𝐴 ↾ 𝐽))
2	1	coeq2d 5806	. . . . 5 ⊢ (𝑜 = 𝐴 → (bits ∘ (𝑜 ↾ 𝐽)) = (bits ∘ (𝐴 ↾ 𝐽)))
3	2	fveq2d 6832	. . . 4 ⊢ (𝑜 = 𝐴 → (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))) = (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))
4	3	imaeq2d 6013	. . 3 ⊢ (𝑜 = 𝐴 → (𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))) = (𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽)))))
5	4	fveq2d 6832	. 2 ⊢ (𝑜 = 𝐴 → ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))))
6		eulerpart.g	. 2 ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))
7		fvex 6841	. 2 ⊢ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))) ∈ V
8	5, 6, 7	fvmpt 6935	1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  ∀wral 3048  {crab 3396   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549   class class class wbr 5093  {copab 5155   ↦ cmpt 5174  ^◡ccnv 5618   ↾ cres 5621   “ cima 5622   ∘ ccom 5623  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354   supp csupp 8096   ↑_m cmap 8756  Fincfn 8875  1c1 11014   · cmul 11018   ≤ cle 11154  ℕcn 12132  2c2 12187  ℕ₀cn0 12388  ↑cexp 13970  Σcsu 15595   ∥ cdvds 16165  bitscbits 16332  𝟭cind 32836

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fv 6494

This theorem is referenced by:  eulerpartlemgvv  34410  eulerpartlemgf  34413  eulerpartlemn  34415