Theorem eulerpartlemt 32338

Description: Lemma for eulerpart 32349. (Contributed by Thierry Arnoux, 19-Sep-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}

Assertion

Ref	Expression
eulerpartlemt	⊢ ((ℕ0 ↑m 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽))

Distinct variable groups:   𝑓,𝑚,𝐽   𝑅,𝑚   𝑇,𝑚

Allowed substitution hints:   𝐷(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑚,𝑛,𝑟)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑚,𝑛,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,𝑚,𝑛,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑚,𝑛,𝑟)   𝐽(𝑥,𝑦,𝑧,𝑔,𝑘,𝑛,𝑟)   𝑀(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑚,𝑛,𝑟)   𝑁(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑚,𝑛,𝑟)   𝑂(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑚,𝑛,𝑟)

Proof of Theorem eulerpartlemt

Dummy variable 𝑜 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 8637	. . . . . . . . . 10 ⊢ (𝑜 ∈ (ℕ0 ↑m 𝐽) → 𝑜:𝐽⟶ℕ0)
2	1	adantr 481	. . . . . . . . 9 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → 𝑜:𝐽⟶ℕ0)
3		c0ex 10969	. . . . . . . . . . 11 ⊢ 0 ∈ V
4	3	fconst 6660	. . . . . . . . . 10 ⊢ ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0}
5	4	a1i 11	. . . . . . . . 9 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0})
6		disjdif 4405	. . . . . . . . . 10 ⊢ (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅
7	6	a1i 11	. . . . . . . . 9 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅)
8		fun 6636	. . . . . . . . 9 ⊢ (((𝑜:𝐽⟶ℕ0 ∧ ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0}) ∧ (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0 ∪ {0}))
9	2, 5, 7, 8	syl21anc 835	. . . . . . . 8 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0 ∪ {0}))
10		eulerpart.j	. . . . . . . . . . 11 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
11		ssrab2 4013	. . . . . . . . . . 11 ⊢ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
12	10, 11	eqsstri 3955	. . . . . . . . . 10 ⊢ 𝐽 ⊆ ℕ
13		undif 4415	. . . . . . . . . 10 ⊢ (𝐽 ⊆ ℕ ↔ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ)
14	12, 13	mpbi 229	. . . . . . . . 9 ⊢ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ
15		0nn0 12248	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
16		snssi 4741	. . . . . . . . . . 11 ⊢ (0 ∈ ℕ0 → {0} ⊆ ℕ0)
17	15, 16	ax-mp 5	. . . . . . . . . 10 ⊢ {0} ⊆ ℕ0
18		ssequn2 4117	. . . . . . . . . 10 ⊢ ({0} ⊆ ℕ0 ↔ (ℕ0 ∪ {0}) = ℕ0)
19	17, 18	mpbi 229	. . . . . . . . 9 ⊢ (ℕ0 ∪ {0}) = ℕ0
20	14, 19	feq23i 6594	. . . . . . . 8 ⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0 ∪ {0}) ↔ (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
21	9, 20	sylib 217	. . . . . . 7 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
22		nn0ex 12239	. . . . . . . 8 ⊢ ℕ0 ∈ V
23		nnex 11979	. . . . . . . 8 ⊢ ℕ ∈ V
24	22, 23	elmap 8659	. . . . . . 7 ⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0 ↑m ℕ) ↔ (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
25	21, 24	sylibr 233	. . . . . 6 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0 ↑m ℕ))
26		cnvun 6046	. . . . . . . . 9 ⊢ ◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) = (◡𝑜 ∪ ◡((ℕ ∖ 𝐽) × {0}))
27	26	imaeq1i 5966	. . . . . . . 8 ⊢ (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((◡𝑜 ∪ ◡((ℕ ∖ 𝐽) × {0})) “ ℕ)
28		imaundir 6054	. . . . . . . 8 ⊢ ((◡𝑜 ∪ ◡((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ))
29	27, 28	eqtri 2766	. . . . . . 7 ⊢ (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ))
30		vex 3436	. . . . . . . . . . 11 ⊢ 𝑜 ∈ V
31		cnveq 5782	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑜 → ◡𝑓 = ◡𝑜)
32	31	imaeq1d 5968	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑜 → (◡𝑓 “ ℕ) = (◡𝑜 “ ℕ))
33	32	eleq1d 2823	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑜 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝑜 “ ℕ) ∈ Fin))
34		eulerpart.r	. . . . . . . . . . 11 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
35	30, 33, 34	elab2 3613	. . . . . . . . . 10 ⊢ (𝑜 ∈ 𝑅 ↔ (◡𝑜 “ ℕ) ∈ Fin)
36	35	biimpi 215	. . . . . . . . 9 ⊢ (𝑜 ∈ 𝑅 → (◡𝑜 “ ℕ) ∈ Fin)
37	36	adantl 482	. . . . . . . 8 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡𝑜 “ ℕ) ∈ Fin)
38		cnvxp 6060	. . . . . . . . . . . . . 14 ⊢ ◡((ℕ ∖ 𝐽) × {0}) = ({0} × (ℕ ∖ 𝐽))
39	38	dmeqi 5813	. . . . . . . . . . . . 13 ⊢ dom ◡((ℕ ∖ 𝐽) × {0}) = dom ({0} × (ℕ ∖ 𝐽))
40		2nn 12046	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
41		2z 12352	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
42		iddvds 15979	. . . . . . . . . . . . . . . . 17 ⊢ (2 ∈ ℤ → 2 ∥ 2)
43	41, 42	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ 2 ∥ 2
44		breq2 5078	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 2 → (2 ∥ 𝑧 ↔ 2 ∥ 2))
45	44	notbid 318	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 2 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 2))
46	45, 10	elrab2 3627	. . . . . . . . . . . . . . . . 17 ⊢ (2 ∈ 𝐽 ↔ (2 ∈ ℕ ∧ ¬ 2 ∥ 2))
47	46	simprbi 497	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ 𝐽 → ¬ 2 ∥ 2)
48	43, 47	mt2 199	. . . . . . . . . . . . . . 15 ⊢ ¬ 2 ∈ 𝐽
49		eldif 3897	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ (ℕ ∖ 𝐽) ↔ (2 ∈ ℕ ∧ ¬ 2 ∈ 𝐽))
50	40, 48, 49	mpbir2an 708	. . . . . . . . . . . . . 14 ⊢ 2 ∈ (ℕ ∖ 𝐽)
51		ne0i 4268	. . . . . . . . . . . . . 14 ⊢ (2 ∈ (ℕ ∖ 𝐽) → (ℕ ∖ 𝐽) ≠ ∅)
52		dmxp 5838	. . . . . . . . . . . . . 14 ⊢ ((ℕ ∖ 𝐽) ≠ ∅ → dom ({0} × (ℕ ∖ 𝐽)) = {0})
53	50, 51, 52	mp2b 10	. . . . . . . . . . . . 13 ⊢ dom ({0} × (ℕ ∖ 𝐽)) = {0}
54	39, 53	eqtri 2766	. . . . . . . . . . . 12 ⊢ dom ◡((ℕ ∖ 𝐽) × {0}) = {0}
55	54	ineq1i 4142	. . . . . . . . . . 11 ⊢ (dom ◡((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ({0} ∩ ℕ)
56		incom 4135	. . . . . . . . . . 11 ⊢ (ℕ ∩ {0}) = ({0} ∩ ℕ)
57		0nnn 12009	. . . . . . . . . . . 12 ⊢ ¬ 0 ∈ ℕ
58		disjsn 4647	. . . . . . . . . . . 12 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
59	57, 58	mpbir 230	. . . . . . . . . . 11 ⊢ (ℕ ∩ {0}) = ∅
60	55, 56, 59	3eqtr2i 2772	. . . . . . . . . 10 ⊢ (dom ◡((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅
61		imadisj 5988	. . . . . . . . . 10 ⊢ ((◡((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅ ↔ (dom ◡((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅)
62	60, 61	mpbir 230	. . . . . . . . 9 ⊢ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅
63		0fin 8954	. . . . . . . . 9 ⊢ ∅ ∈ Fin
64	62, 63	eqeltri 2835	. . . . . . . 8 ⊢ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈ Fin
65		unfi 8955	. . . . . . . 8 ⊢ (((◡𝑜 “ ℕ) ∈ Fin ∧ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈ Fin) → ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ)) ∈ Fin)
66	37, 64, 65	sylancl 586	. . . . . . 7 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ)) ∈ Fin)
67	29, 66	eqeltrid 2843	. . . . . 6 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ∈ Fin)
68		cnvimass 5989	. . . . . . . . 9 ⊢ (◡𝑜 “ ℕ) ⊆ dom 𝑜
69	68, 2	fssdm 6620	. . . . . . . 8 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡𝑜 “ ℕ) ⊆ 𝐽)
70		0ss 4330	. . . . . . . . . 10 ⊢ ∅ ⊆ 𝐽
71	62, 70	eqsstri 3955	. . . . . . . . 9 ⊢ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽
72	71	a1i 11	. . . . . . . 8 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽)
73	69, 72	unssd 4120	. . . . . . 7 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ)) ⊆ 𝐽)
74	29, 73	eqsstrid 3969	. . . . . 6 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ⊆ 𝐽)
75		eulerpart.p	. . . . . . 7 ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
76		eulerpart.o	. . . . . . 7 ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
77		eulerpart.d	. . . . . . 7 ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
78		eulerpart.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
79		eulerpart.h	. . . . . . 7 ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
80		eulerpart.m	. . . . . . 7 ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
81		eulerpart.t	. . . . . . 7 ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}
82	75, 76, 77, 10, 78, 79, 80, 34, 81	eulerpartlemt0 32336	. . . . . 6 ⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇 ∩ 𝑅) ↔ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0 ↑m ℕ) ∧ (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ∈ Fin ∧ (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ⊆ 𝐽))
83	25, 67, 74, 82	syl3anbrc 1342	. . . . 5 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇 ∩ 𝑅))
84		resundir 5906	. . . . . 6 ⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽) = ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽))
85		ffn 6600	. . . . . . . 8 ⊢ (𝑜:𝐽⟶ℕ0 → 𝑜 Fn 𝐽)
86		fnresdm 6551	. . . . . . . . 9 ⊢ (𝑜 Fn 𝐽 → (𝑜 ↾ 𝐽) = 𝑜)
87		disjdifr 4406	. . . . . . . . . . 11 ⊢ ((ℕ ∖ 𝐽) ∩ 𝐽) = ∅
88		fnconstg 6662	. . . . . . . . . . . 12 ⊢ (0 ∈ ℕ0 → ((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽))
89		fnresdisj 6552	. . . . . . . . . . . 12 ⊢ (((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽) → (((ℕ ∖ 𝐽) ∩ 𝐽) = ∅ ↔ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅))
90	15, 88, 89	mp2b 10	. . . . . . . . . . 11 ⊢ (((ℕ ∖ 𝐽) ∩ 𝐽) = ∅ ↔ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅)
91	87, 90	mpbi 229	. . . . . . . . . 10 ⊢ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅
92	91	a1i 11	. . . . . . . . 9 ⊢ (𝑜 Fn 𝐽 → (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅)
93	86, 92	uneq12d 4098	. . . . . . . 8 ⊢ (𝑜 Fn 𝐽 → ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = (𝑜 ∪ ∅))
94	2, 85, 93	3syl 18	. . . . . . 7 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = (𝑜 ∪ ∅))
95		un0 4324	. . . . . . 7 ⊢ (𝑜 ∪ ∅) = 𝑜
96	94, 95	eqtrdi 2794	. . . . . 6 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = 𝑜)
97	84, 96	eqtr2id 2791	. . . . 5 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
98		reseq1 5885	. . . . . 6 ⊢ (𝑚 = (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) → (𝑚 ↾ 𝐽) = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
99	98	rspceeqv 3575	. . . . 5 ⊢ (((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽)) → ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽))
100	83, 97, 99	syl2anc 584	. . . 4 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽))
101		simpr 485	. . . . . . 7 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑜 = (𝑚 ↾ 𝐽))
102		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑚 ∈ (𝑇 ∩ 𝑅))
103	75, 76, 77, 10, 78, 79, 80, 34, 81	eulerpartlemt0 32336	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (𝑇 ∩ 𝑅) ↔ (𝑚 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝑚 “ ℕ) ∈ Fin ∧ (◡𝑚 “ ℕ) ⊆ 𝐽))
104	102, 103	sylib 217	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝑚 “ ℕ) ∈ Fin ∧ (◡𝑚 “ ℕ) ⊆ 𝐽))
105	104	simp1d 1141	. . . . . . . . . 10 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑚 ∈ (ℕ0 ↑m ℕ))
106	22, 23	elmap 8659	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℕ0 ↑m ℕ) ↔ 𝑚:ℕ⟶ℕ0)
107	105, 106	sylib 217	. . . . . . . . 9 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑚:ℕ⟶ℕ0)
108		fssres 6640	. . . . . . . . 9 ⊢ ((𝑚:ℕ⟶ℕ0 ∧ 𝐽 ⊆ ℕ) → (𝑚 ↾ 𝐽):𝐽⟶ℕ0)
109	107, 12, 108	sylancl 586	. . . . . . . 8 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ↾ 𝐽):𝐽⟶ℕ0)
110	10, 23	rabex2 5258	. . . . . . . . 9 ⊢ 𝐽 ∈ V
111	22, 110	elmap 8659	. . . . . . . 8 ⊢ ((𝑚 ↾ 𝐽) ∈ (ℕ0 ↑m 𝐽) ↔ (𝑚 ↾ 𝐽):𝐽⟶ℕ0)
112	109, 111	sylibr 233	. . . . . . 7 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ↾ 𝐽) ∈ (ℕ0 ↑m 𝐽))
113	101, 112	eqeltrd 2839	. . . . . 6 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑜 ∈ (ℕ0 ↑m 𝐽))
114		ffun 6603	. . . . . . . . . 10 ⊢ (𝑚:ℕ⟶ℕ0 → Fun 𝑚)
115		respreima 6943	. . . . . . . . . 10 ⊢ (Fun 𝑚 → (◡(𝑚 ↾ 𝐽) “ ℕ) = ((◡𝑚 “ ℕ) ∩ 𝐽))
116	107, 114, 115	3syl 18	. . . . . . . . 9 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (◡(𝑚 ↾ 𝐽) “ ℕ) = ((◡𝑚 “ ℕ) ∩ 𝐽))
117	104	simp2d 1142	. . . . . . . . . 10 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (◡𝑚 “ ℕ) ∈ Fin)
118		infi 9043	. . . . . . . . . 10 ⊢ ((◡𝑚 “ ℕ) ∈ Fin → ((◡𝑚 “ ℕ) ∩ 𝐽) ∈ Fin)
119	117, 118	syl 17	. . . . . . . . 9 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → ((◡𝑚 “ ℕ) ∩ 𝐽) ∈ Fin)
120	116, 119	eqeltrd 2839	. . . . . . . 8 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (◡(𝑚 ↾ 𝐽) “ ℕ) ∈ Fin)
121		vex 3436	. . . . . . . . . 10 ⊢ 𝑚 ∈ V
122	121	resex 5939	. . . . . . . . 9 ⊢ (𝑚 ↾ 𝐽) ∈ V
123		cnveq 5782	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑚 ↾ 𝐽) → ◡𝑓 = ◡(𝑚 ↾ 𝐽))
124	123	imaeq1d 5968	. . . . . . . . . 10 ⊢ (𝑓 = (𝑚 ↾ 𝐽) → (◡𝑓 “ ℕ) = (◡(𝑚 ↾ 𝐽) “ ℕ))
125	124	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑓 = (𝑚 ↾ 𝐽) → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡(𝑚 ↾ 𝐽) “ ℕ) ∈ Fin))
126	122, 125, 34	elab2 3613	. . . . . . . 8 ⊢ ((𝑚 ↾ 𝐽) ∈ 𝑅 ↔ (◡(𝑚 ↾ 𝐽) “ ℕ) ∈ Fin)
127	120, 126	sylibr 233	. . . . . . 7 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ↾ 𝐽) ∈ 𝑅)
128	101, 127	eqeltrd 2839	. . . . . 6 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑜 ∈ 𝑅)
129	113, 128	jca 512	. . . . 5 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅))
130	129	rexlimiva 3210	. . . 4 ⊢ (∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽) → (𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅))
131	100, 130	impbii 208	. . 3 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) ↔ ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽))
132	131	abbii 2808	. 2 ⊢ {𝑜 ∣ (𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅)} = {𝑜 ∣ ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)}
133		df-in 3894	. 2 ⊢ ((ℕ0 ↑m 𝐽) ∩ 𝑅) = {𝑜 ∣ (𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅)}
134		eqid 2738	. . 3 ⊢ (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) = (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽))
135	134	rnmpt 5864	. 2 ⊢ ran (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) = {𝑜 ∣ ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)}
136	132, 133, 135	3eqtr4i 2776	1 ⊢ ((ℕ0 ↑m 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561   class class class wbr 5074  {copab 5136   ↦ cmpt 5157   × cxp 5587  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   ↾ cres 5591   “ cima 5592  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   supp csupp 7977   ↑_m cmap 8615  Fincfn 8733  0cc0 10871  1c1 10872   · cmul 10876   ≤ cle 11010  ℕcn 11973  2c2 12028  ℕ₀cn0 12233  ℤcz 12319  ↑cexp 13782  Σcsu 15397   ∥ cdvds 15963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-dvds 15964

This theorem is referenced by:  eulerpartgbij  32339