Theorem eulerpartlemgv 34631

Description: Lemma for eulerpart 34640: 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 5957	. . . . . 6 ⊢ (𝑜 = 𝐴 → (𝑜 ↾ 𝐽) = (𝐴 ↾ 𝐽))
2	1	coeq2d 5832	. . . . 5 ⊢ (𝑜 = 𝐴 → (bits ∘ (𝑜 ↾ 𝐽)) = (bits ∘ (𝐴 ↾ 𝐽)))
3	2	fveq2d 6867	. . . 4 ⊢ (𝑜 = 𝐴 → (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))) = (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))
4	3	imaeq2d 6046	. . 3 ⊢ (𝑜 = 𝐴 → (𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))) = (𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽)))))
5	4	fveq2d 6867	. 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 6876	. 2 ⊢ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))) ∈ V
8	5, 6, 7	fvmpt 6971	1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075  {crab 3413   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4554   class class class wbr 5099  {copab 5161   ↦ cmpt 5180  ^◡ccnv 5644   ↾ cres 5647   “ cima 5648   ∘ ccom 5649  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394   supp csupp 8135   ↑_m cmap 8803  Fincfn 8923  1c1 11071   · cmul 11075   ≤ cle 11214  𝟭cind 12192  ℕcn 12207  2c2 12269  ℕ₀cn0 12478  ↑cexp 14071  Σcsu 15696   ∥ cdvds 16269  bitscbits 16436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fv 6525

This theorem is referenced by:  eulerpartlemgvv  34634  eulerpartlemgf  34637  eulerpartlemn  34639