Theorem eulerpartlemgv 34892

Description: Lemma for eulerpart 34901: 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 5970	. . . . . 6 ⊢ (𝑜 = 𝐴 → (𝑜 ↾ 𝐽) = (𝐴 ↾ 𝐽))
2	1	coeq2d 5846	. . . . 5 ⊢ (𝑜 = 𝐴 → (bits ∘ (𝑜 ↾ 𝐽)) = (bits ∘ (𝐴 ↾ 𝐽)))
3	2	fveq2d 6886	. . . 4 ⊢ (𝑜 = 𝐴 → (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))) = (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))
4	3	imaeq2d 6060	. . 3 ⊢ (𝑜 = 𝐴 → (𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))) = (𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽)))))
5	4	fveq2d 6886	. 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 6895	. 2 ⊢ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))) ∈ V
8	5, 6, 7	fvmpt 6990	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 2740  ∀wral 3078  {crab 3414   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560   class class class wbr 5107  {copab 5171   ↦ cmpt 5190  ^◡ccnv 5658   ↾ cres 5661   “ cima 5662   ∘ ccom 5663  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419   supp csupp 8162   ↑_m cmap 8830  Fincfn 8956  1c1 11129   · cmul 11133   ≤ cle 11272  𝟭cind 12246  ℕcn 12261  2c2 12323  ℕ₀cn0 12532  ↑cexp 14129  Σcsu 15777   ∥ cdvds 16348  bitscbits 16515

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fv 6545

This theorem is used by:  eulerpartlemgvv  34895  eulerpartlemgf  34898  eulerpartlemn  34900