Theorem eulerpart 31750

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 2798	. . 3 ⊢ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
5		oveq2 7143	. . . 4 ⊢ (𝑎 = 𝑥 → ((2↑𝑏) · 𝑎) = ((2↑𝑏) · 𝑥))
6		oveq2 7143	. . . . 5 ⊢ (𝑏 = 𝑦 → (2↑𝑏) = (2↑𝑦))
7	6	oveq1d 7150	. . . 4 ⊢ (𝑏 = 𝑦 → ((2↑𝑏) · 𝑥) = ((2↑𝑦) · 𝑥))
8	5, 7	cbvmpov 7228	. . 3 ⊢ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) = (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
9		oveq1 7142	. . . . . 6 ⊢ (𝑟 = 𝑚 → (𝑟 supp ∅) = (𝑚 supp ∅))
10	9	eleq1d 2874	. . . . 5 ⊢ (𝑟 = 𝑚 → ((𝑟 supp ∅) ∈ Fin ↔ (𝑚 supp ∅) ∈ Fin))
11	10	cbvrabv 3439	. . . 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 2807	. . 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 6644	. . . . . . . 8 ⊢ (𝑡 = 𝑟 → (𝑡‘𝑎) = (𝑟‘𝑎))
14	13	eleq2d 2875	. . . . . . 7 ⊢ (𝑡 = 𝑟 → (𝑏 ∈ (𝑡‘𝑎) ↔ 𝑏 ∈ (𝑟‘𝑎)))
15	14	anbi2d 631	. . . . . 6 ⊢ (𝑡 = 𝑟 → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎)) ↔ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))))
16	15	opabbidv 5096	. . . . 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 5133	. . . 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 7142	. . . . . . . 8 ⊢ (𝑚 = 𝑠 → (𝑚 supp ∅) = (𝑠 supp ∅))
19	18	eleq1d 2874	. . . . . . 7 ⊢ (𝑚 = 𝑠 → ((𝑚 supp ∅) ∈ Fin ↔ (𝑠 supp ∅) ∈ Fin))
20	19	cbvrabv 3439	. . . . . 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 2807	. . . . 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 486	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 𝑎 = 𝑥)
23	22	eleq1d 2874	. . . . . . 7 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ↔ 𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
24		simpr 488	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 𝑏 = 𝑦)
25	22	fveq2d 6649	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑟‘𝑎) = (𝑟‘𝑥))
26	24, 25	eleq12d 2884	. . . . . . 7 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑏 ∈ (𝑟‘𝑎) ↔ 𝑦 ∈ (𝑟‘𝑥)))
27	23, 26	anbi12d 633	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎)) ↔ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))))
28	27	cbvopabv 5102	. . . . 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 5123	. . . 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 2821	. . 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 5708	. . . . . 6 ⊢ (ℎ = 𝑓 → ◡ℎ = ◡𝑓)
32	31	imaeq1d 5895	. . . . 5 ⊢ (ℎ = 𝑓 → (◡ℎ “ ℕ) = (◡𝑓 “ ℕ))
33	32	eleq1d 2874	. . . 4 ⊢ (ℎ = 𝑓 → ((◡ℎ “ ℕ) ∈ Fin ↔ (◡𝑓 “ ℕ) ∈ Fin))
34	33	cbvabv 2866	. . 3 ⊢ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin} = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
35	32	sseq1d 3946	. . . 4 ⊢ (ℎ = 𝑓 → ((◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ↔ (◡𝑓 “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
36	35	cbvrabv 3439	. . 3 ⊢ {ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}}
37		reseq1 5812	. . . . . . . . 9 ⊢ (𝑜 = 𝑞 → (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) = (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
38	37	coeq2d 5697	. . . . . . . 8 ⊢ (𝑜 = 𝑞 → (bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})) = (bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))
39	38	fveq2d 6649	. . . . . . 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 5896	. . . . . 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 6649	. . . . 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 5133	. . . 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 2807	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) = (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎))
44	43	imaeq1i 5893	. . . . . . 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 2798	. . . . . . . . . . 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 5123	. . . . . . . . . 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 6644	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑡 → (𝑟‘𝑎) = (𝑡‘𝑎))
48	47	eleq2d 2875	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑡 → (𝑏 ∈ (𝑟‘𝑎) ↔ 𝑏 ∈ (𝑡‘𝑎)))
49	48	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑡 → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎)) ↔ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))))
50	49	opabbidv 5096	. . . . . . . . . . 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 5133	. . . . . . . . . 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 2827	. . . . . . . . 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 6646	. . . . . . . 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 5894	. . . . . . 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 2821	. . . . . 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 6648	. . . . 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 5122	. . . 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 2821	. . 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 2798	. . 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 31749	. 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 7168	. . . . . . 7 ⊢ (ℕ0 ↑m ℕ) ∈ V
62	61	rabex 5199	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∈ V
63	62	inex1 5185	. . . . 5 ⊢ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ∈ V
64	63	mptex 6963	. . . 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 5866	. . 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 6579	. . 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 3555	. 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 8501	. . 3 ⊢ (𝑂 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝑂–1-1-onto→𝐷)
69		hasheni 13704	. . 3 ⊢ (𝑂 ≈ 𝐷 → (♯‘𝑂) = (♯‘𝐷))
70	68, 69	sylbir 238	. 2 ⊢ (∃𝑔 𝑔:𝑂–1-1-onto→𝐷 → (♯‘𝑂) = (♯‘𝐷))
71	60, 67, 70	mp2b 10	1 ⊢ (♯‘𝑂) = (♯‘𝐷)

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  {crab 3110   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497   class class class wbr 5030  {copab 5092   ↦ cmpt 5110  ^◡ccnv 5518   ↾ cres 5521   “ cima 5522   ∘ ccom 5523  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   supp csupp 7813   ↑_m cmap 8389   ≈ cen 8489  Fincfn 8492  1c1 10527   · cmul 10531   ≤ cle 10665  ℕcn 11625  2c2 11680  ℕ₀cn0 11885  ↑cexp 13425  ♯chash 13686  Σcsu 15034   ∥ cdvds 15599  bitscbits 15758  𝟭cind 31379

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-ac2 9874  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-disj 4996  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-sup 8890  df-inf 8891  df-oi 8958  df-dju 9314  df-card 9352  df-acn 9355  df-ac 9527  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-fl 13157  df-mod 13233  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035  df-dvds 15600  df-bits 15761  df-ind 31380

This theorem is referenced by: (None)