Theorem eulerpart 32982

Description: Euler's theorem on partitions, also known as a special case of Glaisher's theorem. Let 𝑃 be the set of all partitions of 𝑁, represented as multisets of positive integers, which is to say functions from ℕ to ℕ₀ where the value of the function represents the number of repetitions of an individual element, and the sum of all the elements with repetition equals 𝑁. Then the set 𝑂 of all partitions that only consist of odd numbers and the set 𝐷 of all partitions which have no repeated elements have the same cardinality. This is Metamath 100 proof #45. (Contributed by Thierry Arnoux, 14-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)

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}

Assertion

Ref	Expression
eulerpart	⊢ (♯‘𝑂) = (♯‘𝐷)

Distinct variable groups:   𝑓,𝑔,𝑘,𝑛   𝐷,𝑔   𝑓,𝑁,𝑔,𝑘,𝑛   𝑔,𝑂,𝑛   𝑃,𝑔,𝑘,𝑛

Allowed substitution hints:   𝐷(𝑓,𝑘,𝑛)   𝑃(𝑓)   𝑂(𝑓,𝑘)

Proof of Theorem eulerpart

Dummy variables 𝑎 𝑏 ℎ 𝑚 𝑜 𝑞 𝑟 𝑠 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eulerpart.p	. . 3 ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
2		eulerpart.o	. . 3 ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
3		eulerpart.d	. . 3 ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
4		eqid 2736	. . 3 ⊢ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
5		oveq2 7365	. . . 4 ⊢ (𝑎 = 𝑥 → ((2↑𝑏) · 𝑎) = ((2↑𝑏) · 𝑥))
6		oveq2 7365	. . . . 5 ⊢ (𝑏 = 𝑦 → (2↑𝑏) = (2↑𝑦))
7	6	oveq1d 7372	. . . 4 ⊢ (𝑏 = 𝑦 → ((2↑𝑏) · 𝑥) = ((2↑𝑦) · 𝑥))
8	5, 7	cbvmpov 7452	. . 3 ⊢ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) = (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
9		oveq1 7364	. . . . . 6 ⊢ (𝑟 = 𝑚 → (𝑟 supp ∅) = (𝑚 supp ∅))
10	9	eleq1d 2822	. . . . 5 ⊢ (𝑟 = 𝑚 → ((𝑟 supp ∅) ∈ Fin ↔ (𝑚 supp ∅) ∈ Fin))
11	10	cbvrabv 3417	. . . 4 ⊢ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} = {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin}
12	11	eqcomi 2745	. . 3 ⊢ {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin} = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin}
13		fveq1 6841	. . . . . . . 8 ⊢ (𝑡 = 𝑟 → (𝑡‘𝑎) = (𝑟‘𝑎))
14	13	eleq2d 2823	. . . . . . 7 ⊢ (𝑡 = 𝑟 → (𝑏 ∈ (𝑡‘𝑎) ↔ 𝑏 ∈ (𝑟‘𝑎)))
15	14	anbi2d 629	. . . . . 6 ⊢ (𝑡 = 𝑟 → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎)) ↔ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))))
16	15	opabbidv 5171	. . . . 5 ⊢ (𝑡 = 𝑟 → {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))})
17	16	cbvmptv 5218	. . . 4 ⊢ (𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))}) = (𝑟 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))})
18		oveq1 7364	. . . . . . . 8 ⊢ (𝑚 = 𝑠 → (𝑚 supp ∅) = (𝑠 supp ∅))
19	18	eleq1d 2822	. . . . . . 7 ⊢ (𝑚 = 𝑠 → ((𝑚 supp ∅) ∈ Fin ↔ (𝑠 supp ∅) ∈ Fin))
20	19	cbvrabv 3417	. . . . . 6 ⊢ {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin} = {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin}
21	20	eqcomi 2745	. . . . 5 ⊢ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} = {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin}
22		simpl 483	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 𝑎 = 𝑥)
23	22	eleq1d 2822	. . . . . . 7 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ↔ 𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
24		simpr 485	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 𝑏 = 𝑦)
25	22	fveq2d 6846	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑟‘𝑎) = (𝑟‘𝑥))
26	24, 25	eleq12d 2832	. . . . . . 7 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑏 ∈ (𝑟‘𝑎) ↔ 𝑦 ∈ (𝑟‘𝑥)))
27	23, 26	anbi12d 631	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎)) ↔ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))))
28	27	cbvopabv 5178	. . . . 5 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))}
29	21, 28	mpteq12i 5211	. . . 4 ⊢ (𝑟 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))}) = (𝑟 ∈ {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})
30	17, 29	eqtri 2764	. . 3 ⊢ (𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))}) = (𝑟 ∈ {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})
31		cnveq 5829	. . . . . 6 ⊢ (ℎ = 𝑓 → ◡ℎ = ◡𝑓)
32	31	imaeq1d 6012	. . . . 5 ⊢ (ℎ = 𝑓 → (◡ℎ “ ℕ) = (◡𝑓 “ ℕ))
33	32	eleq1d 2822	. . . 4 ⊢ (ℎ = 𝑓 → ((◡ℎ “ ℕ) ∈ Fin ↔ (◡𝑓 “ ℕ) ∈ Fin))
34	33	cbvabv 2809	. . 3 ⊢ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin} = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
35	32	sseq1d 3975	. . . 4 ⊢ (ℎ = 𝑓 → ((◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ↔ (◡𝑓 “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
36	35	cbvrabv 3417	. . 3 ⊢ {ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}}
37		reseq1 5931	. . . . . . . . 9 ⊢ (𝑜 = 𝑞 → (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) = (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
38	37	coeq2d 5818	. . . . . . . 8 ⊢ (𝑜 = 𝑞 → (bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})) = (bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))
39	38	fveq2d 6846	. . . . . . 7 ⊢ (𝑜 = 𝑞 → ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))) = ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))
40	39	imaeq2d 6013	. . . . . 6 ⊢ (𝑜 = 𝑞 → ((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))) = ((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))
41	40	fveq2d 6846	. . . . 5 ⊢ (𝑜 = 𝑞 → ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))) = ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))))
42	41	cbvmptv 5218	. . . 4 ⊢ (𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) = (𝑞 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))))
43	8	eqcomi 2745	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) = (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎))
44	43	imaeq1i 6010	. . . . . . 7 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))) = ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))
45		eqid 2736	. . . . . . . . . . 11 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))}
46	11, 45	mpteq12i 5211	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))}) = (𝑟 ∈ {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})
47		fveq1 6841	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑡 → (𝑟‘𝑎) = (𝑡‘𝑎))
48	47	eleq2d 2823	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑡 → (𝑏 ∈ (𝑟‘𝑎) ↔ 𝑏 ∈ (𝑡‘𝑎)))
49	48	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑡 → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎)) ↔ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))))
50	49	opabbidv 5171	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑡 → {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})
51	50	cbvmptv 5218	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))}) = (𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})
52	46, 29, 51	3eqtr2i 2770	. . . . . . . . 9 ⊢ (𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))}) = (𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})
53	52	fveq1i 6843	. . . . . . . 8 ⊢ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))) = ((𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))
54	53	imaeq2i 6011	. . . . . . 7 ⊢ ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))) = ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))
55	44, 54	eqtri 2764	. . . . . 6 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))) = ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))
56	55	fveq2i 6845	. . . . 5 ⊢ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))) = ((𝟭‘ℕ)‘((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))
57	56	mpteq2i 5210	. . . 4 ⊢ (𝑞 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) = (𝑞 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))))
58	42, 57	eqtri 2764	. . 3 ⊢ (𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) = (𝑞 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))))
59		eqid 2736	. . 3 ⊢ (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))
60	1, 2, 3, 4, 8, 12, 30, 34, 36, 58, 59	eulerpartlemn 32981	. 2 ⊢ ((𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) ↾ 𝑂):𝑂–1-1-onto→𝐷
61		ovex 7390	. . . . . . 7 ⊢ (ℕ0 ↑m ℕ) ∈ V
62	61	rabex 5289	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∈ V
63	62	inex1 5274	. . . . 5 ⊢ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ∈ V
64	63	mptex 7173	. . . 4 ⊢ (𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) ∈ V
65	64	resex 5985	. . 3 ⊢ ((𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) ↾ 𝑂) ∈ V
66		f1oeq1 6772	. . 3 ⊢ (𝑔 = ((𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) ↾ 𝑂) → (𝑔:𝑂–1-1-onto→𝐷 ↔ ((𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) ↾ 𝑂):𝑂–1-1-onto→𝐷))
67	65, 66	spcev 3565	. 2 ⊢ (((𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) ↾ 𝑂):𝑂–1-1-onto→𝐷 → ∃𝑔 𝑔:𝑂–1-1-onto→𝐷)
68		bren 8893	. . 3 ⊢ (𝑂 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝑂–1-1-onto→𝐷)
69		hasheni 14248	. . 3 ⊢ (𝑂 ≈ 𝐷 → (♯‘𝑂) = (♯‘𝐷))
70	68, 69	sylbir 234	. 2 ⊢ (∃𝑔 𝑔:𝑂–1-1-onto→𝐷 → (♯‘𝑂) = (♯‘𝐷))
71	60, 67, 70	mp2b 10	1 ⊢ (♯‘𝑂) = (♯‘𝐷)

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2713  ∀wral 3064  {crab 3407   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560   class class class wbr 5105  {copab 5167   ↦ cmpt 5188  ^◡ccnv 5632   ↾ cres 5635   “ cima 5636   ∘ ccom 5637  –1-1-onto→wf1o 6495  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359   supp csupp 8092   ↑_m cmap 8765   ≈ cen 8880  Fincfn 8883  1c1 11052   · cmul 11056   ≤ cle 11190  ℕcn 12153  2c2 12208  ℕ₀cn0 12413  ↑cexp 13967  ♯chash 14230  Σcsu 15570   ∥ cdvds 16136  bitscbits 16299  𝟭cind 32609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-dvds 16137  df-bits 16302  df-ind 32610

This theorem is referenced by: (None)