Theorem eulerpartlemgv 34887

Description: Lemma for eulerpart 34896: 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 5966	. . . . . 6 ⊢ (𝑜 = 𝐴 → (𝑜 ↾ 𝐽) = (𝐴 ↾ 𝐽))
2	1	coeq2d 5842	. . . . 5 ⊢ (𝑜 = 𝐴 → (bits ∘ (𝑜 ↾ 𝐽)) = (bits ∘ (𝐴 ↾ 𝐽)))
3	2	fveq2d 6883	. . . 4 ⊢ (𝑜 = 𝐴 → (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))) = (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))
4	3	imaeq2d 6056	. . 3 ⊢ (𝑜 = 𝐴 → (𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))) = (𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽)))))
5	4	fveq2d 6883	. 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 6892	. 2 ⊢ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))) ∈ V
8	5, 6, 7	fvmpt 6987	1 ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  {crab 3412   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557   class class class wbr 5103  {copab 5167   ↦ cmpt 5186  ^◡ccnv 5654   ↾ cres 5657   “ cima 5658   ∘ ccom 5659  ‘cfv 6533  (class class class)co 7414   ∈ cmpo 7416   supp csupp 8159   ↑_m cmap 8829  Fincfn 8955  1c1 11128   · cmul 11132   ≤ cle 11271  𝟭cind 12245  ℕcn 12260  2c2 12322  ℕ₀cn0 12531  ↑cexp 14128  Σcsu 15776   ∥ cdvds 16345  bitscbits 16512

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fv 6541

This theorem is used by:  eulerpartlemgvv  34890  eulerpartlemgf  34893  eulerpartlemn  34895