Theorem eulerpartlemt 34882

Description: Lemma for eulerpart 34893. (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 8848	. . . . . . . . . 10 ⊢ (𝑜 ∈ (ℕ0 ↑m 𝐽) → 𝑜:𝐽⟶ℕ0)
2	1	adantr 486	. . . . . . . . 9 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → 𝑜:𝐽⟶ℕ0)
3		c0ex 11224	. . . . . . . . . . 11 ⊢ 0 ∈ V
4	3	fconst 6761	. . . . . . . . . 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 4426	. . . . . . . . . 10 ⊢ (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅
7	6	a1i 11	. . . . . . . . 9 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅)
8		fun 6737	. . . . . . . . 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 851	. . . . . . . 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 4028	. . . . . . . . . . 11 ⊢ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
12	10, 11	eqsstri 3977	. . . . . . . . . 10 ⊢ 𝐽 ⊆ ℕ
13		undif 4438	. . . . . . . . . 10 ⊢ (𝐽 ⊆ ℕ ↔ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ)
14	12, 13	mpbi 233	. . . . . . . . 9 ⊢ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ
15		0nn0 12543	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
16		snssi 4746	. . . . . . . . . . 11 ⊢ (0 ∈ ℕ0 → {0} ⊆ ℕ0)
17	15, 16	ax-mp 5	. . . . . . . . . 10 ⊢ {0} ⊆ ℕ0
18		ssequn2 4135	. . . . . . . . . 10 ⊢ ({0} ⊆ ℕ0 ↔ (ℕ0 ∪ {0}) = ℕ0)
19	17, 18	mpbi 233	. . . . . . . . 9 ⊢ (ℕ0 ∪ {0}) = ℕ0
20	14, 19	feq23i 6696	. . . . . . . 8 ⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0 ∪ {0}) ↔ (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
21	9, 20	sylib 221	. . . . . . 7 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
22		nn0ex 12534	. . . . . . . 8 ⊢ ℕ0 ∈ V
23		nnex 12263	. . . . . . . 8 ⊢ ℕ ∈ V
24	22, 23	elmap 8878	. . . . . . 7 ⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0 ↑m ℕ) ↔ (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
25	21, 24	sylibr 237	. . . . . 6 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0 ↑m ℕ))
26		cnvun 6133	. . . . . . . . 9 ⊢ ◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) = (◡𝑜 ∪ ◡((ℕ ∖ 𝐽) × {0}))
27	26	imaeq1i 6053	. . . . . . . 8 ⊢ (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((◡𝑜 ∪ ◡((ℕ ∖ 𝐽) × {0})) “ ℕ)
28		imaundir 6142	. . . . . . . 8 ⊢ ((◡𝑜 ∪ ◡((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ))
29	27, 28	eqtri 2783	. . . . . . 7 ⊢ (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ))
30		vex 3454	. . . . . . . . . 10 ⊢ 𝑜 ∈ V
31		cnveq 5853	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑜 → ◡𝑓 = ◡𝑜)
32	31	imaeq1d 6055	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑜 → (◡𝑓 “ ℕ) = (◡𝑜 “ ℕ))
33	32	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑓 = 𝑜 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝑜 “ ℕ) ∈ Fin))
34		eulerpart.r	. . . . . . . . . 10 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
35	30, 33, 34	elab2 3636	. . . . . . . . 9 ⊢ (𝑜 ∈ 𝑅 ↔ (◡𝑜 “ ℕ) ∈ Fin)
36	35	bilani 510	. . . . . . . 8 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡𝑜 “ ℕ) ∈ Fin)
37		cnvxp 6148	. . . . . . . . . . . . . 14 ⊢ ◡((ℕ ∖ 𝐽) × {0}) = ({0} × (ℕ ∖ 𝐽))
38	37	dmeqi 5888	. . . . . . . . . . . . 13 ⊢ dom ◡((ℕ ∖ 𝐽) × {0}) = dom ({0} × (ℕ ∖ 𝐽))
39		2nn 12338	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
40		2z 12650	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
41		iddvds 16359	. . . . . . . . . . . . . . . . 17 ⊢ (2 ∈ ℤ → 2 ∥ 2)
42	40, 41	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ 2 ∥ 2
43		breq2 5107	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 2 → (2 ∥ 𝑧 ↔ 2 ∥ 2))
44	43	notbid 321	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 2 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 2))
45	44, 10	elrab2 3649	. . . . . . . . . . . . . . . . 17 ⊢ (2 ∈ 𝐽 ↔ (2 ∈ ℕ ∧ ¬ 2 ∥ 2))
46	45	simprbi 503	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ 𝐽 → ¬ 2 ∥ 2)
47	42, 46	mt2 203	. . . . . . . . . . . . . . 15 ⊢ ¬ 2 ∈ 𝐽
48		eldif 3909	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ (ℕ ∖ 𝐽) ↔ (2 ∈ ℕ ∧ ¬ 2 ∈ 𝐽))
49	39, 47, 48	mpbir2an 724	. . . . . . . . . . . . . 14 ⊢ 2 ∈ (ℕ ∖ 𝐽)
50		ne0i 4287	. . . . . . . . . . . . . 14 ⊢ (2 ∈ (ℕ ∖ 𝐽) → (ℕ ∖ 𝐽) ≠ ∅)
51		dmxp 5913	. . . . . . . . . . . . . 14 ⊢ ((ℕ ∖ 𝐽) ≠ ∅ → dom ({0} × (ℕ ∖ 𝐽)) = {0})
52	49, 50, 51	mp2b 10	. . . . . . . . . . . . 13 ⊢ dom ({0} × (ℕ ∖ 𝐽)) = {0}
53	38, 52	eqtri 2783	. . . . . . . . . . . 12 ⊢ dom ◡((ℕ ∖ 𝐽) × {0}) = {0}
54	53	ineq1i 4162	. . . . . . . . . . 11 ⊢ (dom ◡((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ({0} ∩ ℕ)
55		incom 4155	. . . . . . . . . . 11 ⊢ (ℕ ∩ {0}) = ({0} ∩ ℕ)
56		0nnn 12296	. . . . . . . . . . . 12 ⊢ ¬ 0 ∈ ℕ
57		disjsn 4672	. . . . . . . . . . . 12 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
58	56, 57	mpbir 234	. . . . . . . . . . 11 ⊢ (ℕ ∩ {0}) = ∅
59	54, 55, 58	3eqtr2i 2789	. . . . . . . . . 10 ⊢ (dom ◡((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅
60		imadisj 6076	. . . . . . . . . 10 ⊢ ((◡((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅ ↔ (dom ◡((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅)
61	59, 60	mpbir 234	. . . . . . . . 9 ⊢ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅
62		0fi 9049	. . . . . . . . 9 ⊢ ∅ ∈ Fin
63	61, 62	eqeltri 2856	. . . . . . . 8 ⊢ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈ Fin
64		unfi 9165	. . . . . . . 8 ⊢ (((◡𝑜 “ ℕ) ∈ Fin ∧ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈ Fin) → ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ)) ∈ Fin)
65	36, 63, 64	sylancl 598	. . . . . . 7 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ)) ∈ Fin)
66	29, 65	eqeltrid 2864	. . . . . 6 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ∈ Fin)
67		cnvimass 6078	. . . . . . . . 9 ⊢ (◡𝑜 “ ℕ) ⊆ dom 𝑜
68	67, 2	fssdm 6722	. . . . . . . 8 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡𝑜 “ ℕ) ⊆ 𝐽)
69		0ss 4350	. . . . . . . . . 10 ⊢ ∅ ⊆ 𝐽
70	61, 69	eqsstri 3977	. . . . . . . . 9 ⊢ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽
71	70	a1i 11	. . . . . . . 8 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽)
72	68, 71	unssd 4138	. . . . . . 7 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ)) ⊆ 𝐽)
73	29, 72	eqsstrid 3969	. . . . . 6 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ⊆ 𝐽)
74		eulerpart.p	. . . . . . 7 ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
75		eulerpart.o	. . . . . . 7 ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
76		eulerpart.d	. . . . . . 7 ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
77		eulerpart.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
78		eulerpart.h	. . . . . . 7 ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
79		eulerpart.m	. . . . . . 7 ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})
80		eulerpart.t	. . . . . . 7 ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}
81	74, 75, 76, 10, 77, 78, 79, 34, 80	eulerpartlemt0 34880	. . . . . 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))
82	25, 66, 73, 81	syl3anbrc 1362	. . . . 5 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇 ∩ 𝑅))
83		resundir 5987	. . . . . 6 ⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽) = ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽))
84		ffn 6702	. . . . . . . 8 ⊢ (𝑜:𝐽⟶ℕ0 → 𝑜 Fn 𝐽)
85		fnresdm 6651	. . . . . . . . 9 ⊢ (𝑜 Fn 𝐽 → (𝑜 ↾ 𝐽) = 𝑜)
86		disjdifr 4427	. . . . . . . . . . 11 ⊢ ((ℕ ∖ 𝐽) ∩ 𝐽) = ∅
87		fnconstg 6763	. . . . . . . . . . . 12 ⊢ (0 ∈ ℕ0 → ((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽))
88		fnresdisj 6652	. . . . . . . . . . . 12 ⊢ (((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽) → (((ℕ ∖ 𝐽) ∩ 𝐽) = ∅ ↔ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅))
89	15, 87, 88	mp2b 10	. . . . . . . . . . 11 ⊢ (((ℕ ∖ 𝐽) ∩ 𝐽) = ∅ ↔ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅)
90	86, 89	mpbi 233	. . . . . . . . . 10 ⊢ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅
91	90	a1i 11	. . . . . . . . 9 ⊢ (𝑜 Fn 𝐽 → (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅)
92	85, 91	uneq12d 4116	. . . . . . . 8 ⊢ (𝑜 Fn 𝐽 → ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = (𝑜 ∪ ∅))
93	2, 84, 92	3syl 19	. . . . . . 7 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = (𝑜 ∪ ∅))
94		un0 4344	. . . . . . 7 ⊢ (𝑜 ∪ ∅) = 𝑜
95	93, 94	eqtrdi 2811	. . . . . 6 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = 𝑜)
96	83, 95	eqtr2id 2808	. . . . 5 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
97		reseq1 5966	. . . . . 6 ⊢ (𝑚 = (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) → (𝑚 ↾ 𝐽) = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
98	97	rspceeqv 3599	. . . . 5 ⊢ (((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽)) → ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽))
99	82, 96, 98	syl2anc 596	. . . 4 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) → ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽))
100		simpr 490	. . . . . . 7 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑜 = (𝑚 ↾ 𝐽))
101	74, 75, 76, 10, 77, 78, 79, 34, 80	eulerpartlemt0 34880	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (𝑇 ∩ 𝑅) ↔ (𝑚 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝑚 “ ℕ) ∈ Fin ∧ (◡𝑚 “ ℕ) ⊆ 𝐽))
102	101	birani 509	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝑚 “ ℕ) ∈ Fin ∧ (◡𝑚 “ ℕ) ⊆ 𝐽))
103	102	simp1d 1160	. . . . . . . . . 10 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑚 ∈ (ℕ0 ↑m ℕ))
104	22, 23	elmap 8878	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℕ0 ↑m ℕ) ↔ 𝑚:ℕ⟶ℕ0)
105	103, 104	sylib 221	. . . . . . . . 9 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑚:ℕ⟶ℕ0)
106		fssres 6741	. . . . . . . . 9 ⊢ ((𝑚:ℕ⟶ℕ0 ∧ 𝐽 ⊆ ℕ) → (𝑚 ↾ 𝐽):𝐽⟶ℕ0)
107	105, 12, 106	sylancl 598	. . . . . . . 8 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ↾ 𝐽):𝐽⟶ℕ0)
108	10, 23	rabex2 5305	. . . . . . . . 9 ⊢ 𝐽 ∈ V
109	22, 108	elmap 8878	. . . . . . . 8 ⊢ ((𝑚 ↾ 𝐽) ∈ (ℕ0 ↑m 𝐽) ↔ (𝑚 ↾ 𝐽):𝐽⟶ℕ0)
110	107, 109	sylibr 237	. . . . . . 7 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ↾ 𝐽) ∈ (ℕ0 ↑m 𝐽))
111	100, 110	eqeltrd 2860	. . . . . 6 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑜 ∈ (ℕ0 ↑m 𝐽))
112		ffun 6705	. . . . . . . . . 10 ⊢ (𝑚:ℕ⟶ℕ0 → Fun 𝑚)
113		respreima 7058	. . . . . . . . . 10 ⊢ (Fun 𝑚 → (◡(𝑚 ↾ 𝐽) “ ℕ) = ((◡𝑚 “ ℕ) ∩ 𝐽))
114	105, 112, 113	3syl 19	. . . . . . . . 9 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (◡(𝑚 ↾ 𝐽) “ ℕ) = ((◡𝑚 “ ℕ) ∩ 𝐽))
115	102	simp2d 1161	. . . . . . . . . 10 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (◡𝑚 “ ℕ) ∈ Fin)
116		infi 9240	. . . . . . . . . 10 ⊢ ((◡𝑚 “ ℕ) ∈ Fin → ((◡𝑚 “ ℕ) ∩ 𝐽) ∈ Fin)
117	115, 116	syl 18	. . . . . . . . 9 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → ((◡𝑚 “ ℕ) ∩ 𝐽) ∈ Fin)
118	114, 117	eqeltrd 2860	. . . . . . . 8 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (◡(𝑚 ↾ 𝐽) “ ℕ) ∈ Fin)
119		vex 3454	. . . . . . . . . 10 ⊢ 𝑚 ∈ V
120	119	resex 6022	. . . . . . . . 9 ⊢ (𝑚 ↾ 𝐽) ∈ V
121		cnveq 5853	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑚 ↾ 𝐽) → ◡𝑓 = ◡(𝑚 ↾ 𝐽))
122	121	imaeq1d 6055	. . . . . . . . . 10 ⊢ (𝑓 = (𝑚 ↾ 𝐽) → (◡𝑓 “ ℕ) = (◡(𝑚 ↾ 𝐽) “ ℕ))
123	122	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑓 = (𝑚 ↾ 𝐽) → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡(𝑚 ↾ 𝐽) “ ℕ) ∈ Fin))
124	120, 123, 34	elab2 3636	. . . . . . . 8 ⊢ ((𝑚 ↾ 𝐽) ∈ 𝑅 ↔ (◡(𝑚 ↾ 𝐽) “ ℕ) ∈ Fin)
125	118, 124	sylibr 237	. . . . . . 7 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ↾ 𝐽) ∈ 𝑅)
126	100, 125	eqeltrd 2860	. . . . . 6 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑜 ∈ 𝑅)
127	111, 126	jca 521	. . . . 5 ⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅))
128	127	rexlimiva 3155	. . . 4 ⊢ (∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽) → (𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅))
129	99, 128	impbii 212	. . 3 ⊢ ((𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅) ↔ ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽))
130	129	abbii 2827	. 2 ⊢ {𝑜 ∣ (𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅)} = {𝑜 ∣ ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)}
131		df-in 3906	. 2 ⊢ ((ℕ0 ↑m 𝐽) ∩ 𝑅) = {𝑜 ∣ (𝑜 ∈ (ℕ0 ↑m 𝐽) ∧ 𝑜 ∈ 𝑅)}
132		eqid 2760	. . 3 ⊢ (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) = (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽))
133	132	rnmpt 5941	. 2 ⊢ ran (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) = {𝑜 ∣ ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)}
134	130, 131, 133	3eqtr4i 2793	1 ⊢ ((ℕ0 ↑m 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584   class class class wbr 5103  {copab 5167   ↦ cmpt 5186   × cxp 5653  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ↾ cres 5657   “ cima 5658  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415   supp csupp 8158   ↑_m cmap 8826  Fincfn 8952  0cc0 11124  1c1 11125   · cmul 11129   ≤ cle 11268  ℕcn 12257  2c2 12319  ℕ₀cn0 12528  ℤcz 12615  ↑cexp 14125  Σcsu 15773   ∥ cdvds 16342

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-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  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-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  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-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-neg 11468  df-nn 12258  df-2 12327  df-n0 12529  df-z 12616  df-dvds 16343

This theorem is used by:  eulerpartgbij  34883