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Theorem eulerpartlemt 34352
Description: Lemma for eulerpart 34363. (Contributed by Thierry Arnoux, 19-Sep-2017.)
Hypotheses
Ref Expression
eulerpart.p 𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
eulerpart.o 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
eulerpart.j 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
eulerpart.f 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
eulerpart.h 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
eulerpart.m 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
eulerpart.r 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
eulerpart.t 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
Assertion
Ref Expression
eulerpartlemt ((ℕ0m 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
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.
StepHypRef Expression
1 elmapi 8887 . . . . . . . . . 10 (𝑜 ∈ (ℕ0m 𝐽) → 𝑜:𝐽⟶ℕ0)
21adantr 480 . . . . . . . . 9 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → 𝑜:𝐽⟶ℕ0)
3 c0ex 11252 . . . . . . . . . . 11 0 ∈ V
43fconst 6794 . . . . . . . . . 10 ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0}
54a1i 11 . . . . . . . . 9 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0})
6 disjdif 4477 . . . . . . . . . 10 (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅
76a1i 11 . . . . . . . . 9 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅)
8 fun 6770 . . . . . . . . 9 (((𝑜:𝐽⟶ℕ0 ∧ ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0}) ∧ (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0 ∪ {0}))
92, 5, 7, 8syl21anc 838 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0 ∪ {0}))
10 eulerpart.j . . . . . . . . . . 11 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
11 ssrab2 4089 . . . . . . . . . . 11 {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
1210, 11eqsstri 4029 . . . . . . . . . 10 𝐽 ⊆ ℕ
13 undif 4487 . . . . . . . . . 10 (𝐽 ⊆ ℕ ↔ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ)
1412, 13mpbi 230 . . . . . . . . 9 (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ
15 0nn0 12538 . . . . . . . . . . 11 0 ∈ ℕ0
16 snssi 4812 . . . . . . . . . . 11 (0 ∈ ℕ0 → {0} ⊆ ℕ0)
1715, 16ax-mp 5 . . . . . . . . . 10 {0} ⊆ ℕ0
18 ssequn2 4198 . . . . . . . . . 10 ({0} ⊆ ℕ0 ↔ (ℕ0 ∪ {0}) = ℕ0)
1917, 18mpbi 230 . . . . . . . . 9 (ℕ0 ∪ {0}) = ℕ0
2014, 19feq23i 6730 . . . . . . . 8 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0 ∪ {0}) ↔ (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
219, 20sylib 218 . . . . . . 7 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
22 nn0ex 12529 . . . . . . . 8 0 ∈ V
23 nnex 12269 . . . . . . . 8 ℕ ∈ V
2422, 23elmap 8909 . . . . . . 7 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0m ℕ) ↔ (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
2521, 24sylibr 234 . . . . . 6 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0m ℕ))
26 cnvun 6164 . . . . . . . . 9 (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) = (𝑜((ℕ ∖ 𝐽) × {0}))
2726imaeq1i 6076 . . . . . . . 8 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜((ℕ ∖ 𝐽) × {0})) “ ℕ)
28 imaundir 6172 . . . . . . . 8 ((𝑜((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ))
2927, 28eqtri 2762 . . . . . . 7 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ))
30 vex 3481 . . . . . . . . . . 11 𝑜 ∈ V
31 cnveq 5886 . . . . . . . . . . . . 13 (𝑓 = 𝑜𝑓 = 𝑜)
3231imaeq1d 6078 . . . . . . . . . . . 12 (𝑓 = 𝑜 → (𝑓 “ ℕ) = (𝑜 “ ℕ))
3332eleq1d 2823 . . . . . . . . . . 11 (𝑓 = 𝑜 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝑜 “ ℕ) ∈ Fin))
34 eulerpart.r . . . . . . . . . . 11 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
3530, 33, 34elab2 3684 . . . . . . . . . 10 (𝑜𝑅 ↔ (𝑜 “ ℕ) ∈ Fin)
3635biimpi 216 . . . . . . . . 9 (𝑜𝑅 → (𝑜 “ ℕ) ∈ Fin)
3736adantl 481 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 “ ℕ) ∈ Fin)
38 cnvxp 6178 . . . . . . . . . . . . . 14 ((ℕ ∖ 𝐽) × {0}) = ({0} × (ℕ ∖ 𝐽))
3938dmeqi 5917 . . . . . . . . . . . . 13 dom ((ℕ ∖ 𝐽) × {0}) = dom ({0} × (ℕ ∖ 𝐽))
40 2nn 12336 . . . . . . . . . . . . . . 15 2 ∈ ℕ
41 2z 12646 . . . . . . . . . . . . . . . . 17 2 ∈ ℤ
42 iddvds 16303 . . . . . . . . . . . . . . . . 17 (2 ∈ ℤ → 2 ∥ 2)
4341, 42ax-mp 5 . . . . . . . . . . . . . . . 16 2 ∥ 2
44 breq2 5151 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 2 → (2 ∥ 𝑧 ↔ 2 ∥ 2))
4544notbid 318 . . . . . . . . . . . . . . . . . 18 (𝑧 = 2 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 2))
4645, 10elrab2 3697 . . . . . . . . . . . . . . . . 17 (2 ∈ 𝐽 ↔ (2 ∈ ℕ ∧ ¬ 2 ∥ 2))
4746simprbi 496 . . . . . . . . . . . . . . . 16 (2 ∈ 𝐽 → ¬ 2 ∥ 2)
4843, 47mt2 200 . . . . . . . . . . . . . . 15 ¬ 2 ∈ 𝐽
49 eldif 3972 . . . . . . . . . . . . . . 15 (2 ∈ (ℕ ∖ 𝐽) ↔ (2 ∈ ℕ ∧ ¬ 2 ∈ 𝐽))
5040, 48, 49mpbir2an 711 . . . . . . . . . . . . . 14 2 ∈ (ℕ ∖ 𝐽)
51 ne0i 4346 . . . . . . . . . . . . . 14 (2 ∈ (ℕ ∖ 𝐽) → (ℕ ∖ 𝐽) ≠ ∅)
52 dmxp 5941 . . . . . . . . . . . . . 14 ((ℕ ∖ 𝐽) ≠ ∅ → dom ({0} × (ℕ ∖ 𝐽)) = {0})
5350, 51, 52mp2b 10 . . . . . . . . . . . . 13 dom ({0} × (ℕ ∖ 𝐽)) = {0}
5439, 53eqtri 2762 . . . . . . . . . . . 12 dom ((ℕ ∖ 𝐽) × {0}) = {0}
5554ineq1i 4223 . . . . . . . . . . 11 (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ({0} ∩ ℕ)
56 incom 4216 . . . . . . . . . . 11 (ℕ ∩ {0}) = ({0} ∩ ℕ)
57 0nnn 12299 . . . . . . . . . . . 12 ¬ 0 ∈ ℕ
58 disjsn 4715 . . . . . . . . . . . 12 ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
5957, 58mpbir 231 . . . . . . . . . . 11 (ℕ ∩ {0}) = ∅
6055, 56, 593eqtr2i 2768 . . . . . . . . . 10 (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅
61 imadisj 6099 . . . . . . . . . 10 ((((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅ ↔ (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅)
6260, 61mpbir 231 . . . . . . . . 9 (((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅
63 0fi 9080 . . . . . . . . 9 ∅ ∈ Fin
6462, 63eqeltri 2834 . . . . . . . 8 (((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈ Fin
65 unfi 9209 . . . . . . . 8 (((𝑜 “ ℕ) ∈ Fin ∧ (((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈ Fin) → ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ)) ∈ Fin)
6637, 64, 65sylancl 586 . . . . . . 7 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ)) ∈ Fin)
6729, 66eqeltrid 2842 . . . . . 6 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ∈ Fin)
68 cnvimass 6101 . . . . . . . . 9 (𝑜 “ ℕ) ⊆ dom 𝑜
6968, 2fssdm 6755 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 “ ℕ) ⊆ 𝐽)
70 0ss 4405 . . . . . . . . . 10 ∅ ⊆ 𝐽
7162, 70eqsstri 4029 . . . . . . . . 9 (((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽
7271a1i 11 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽)
7369, 72unssd 4201 . . . . . . 7 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ)) ⊆ 𝐽)
7429, 73eqsstrid 4043 . . . . . 6 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ⊆ 𝐽)
75 eulerpart.p . . . . . . 7 𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
76 eulerpart.o . . . . . . 7 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
77 eulerpart.d . . . . . . 7 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
78 eulerpart.f . . . . . . 7 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
79 eulerpart.h . . . . . . 7 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
80 eulerpart.m . . . . . . 7 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
81 eulerpart.t . . . . . . 7 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
8275, 76, 77, 10, 78, 79, 80, 34, 81eulerpartlemt0 34350 . . . . . 6 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇𝑅) ↔ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0m ℕ) ∧ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ∈ Fin ∧ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ⊆ 𝐽))
8325, 67, 74, 82syl3anbrc 1342 . . . . 5 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇𝑅))
84 resundir 6014 . . . . . 6 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽) = ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽))
85 ffn 6736 . . . . . . . 8 (𝑜:𝐽⟶ℕ0𝑜 Fn 𝐽)
86 fnresdm 6687 . . . . . . . . 9 (𝑜 Fn 𝐽 → (𝑜𝐽) = 𝑜)
87 disjdifr 4478 . . . . . . . . . . 11 ((ℕ ∖ 𝐽) ∩ 𝐽) = ∅
88 fnconstg 6796 . . . . . . . . . . . 12 (0 ∈ ℕ0 → ((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽))
89 fnresdisj 6688 . . . . . . . . . . . 12 (((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽) → (((ℕ ∖ 𝐽) ∩ 𝐽) = ∅ ↔ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅))
9015, 88, 89mp2b 10 . . . . . . . . . . 11 (((ℕ ∖ 𝐽) ∩ 𝐽) = ∅ ↔ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅)
9187, 90mpbi 230 . . . . . . . . . 10 (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅
9291a1i 11 . . . . . . . . 9 (𝑜 Fn 𝐽 → (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅)
9386, 92uneq12d 4178 . . . . . . . 8 (𝑜 Fn 𝐽 → ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = (𝑜 ∪ ∅))
942, 85, 933syl 18 . . . . . . 7 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = (𝑜 ∪ ∅))
95 un0 4399 . . . . . . 7 (𝑜 ∪ ∅) = 𝑜
9694, 95eqtrdi 2790 . . . . . 6 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = 𝑜)
9784, 96eqtr2id 2787 . . . . 5 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
98 reseq1 5993 . . . . . 6 (𝑚 = (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) → (𝑚𝐽) = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
9998rspceeqv 3644 . . . . 5 (((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇𝑅) ∧ 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽)) → ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽))
10083, 97, 99syl2anc 584 . . . 4 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽))
101 simpr 484 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜 = (𝑚𝐽))
102 simpl 482 . . . . . . . . . . . 12 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑚 ∈ (𝑇𝑅))
10375, 76, 77, 10, 78, 79, 80, 34, 81eulerpartlemt0 34350 . . . . . . . . . . . 12 (𝑚 ∈ (𝑇𝑅) ↔ (𝑚 ∈ (ℕ0m ℕ) ∧ (𝑚 “ ℕ) ∈ Fin ∧ (𝑚 “ ℕ) ⊆ 𝐽))
104102, 103sylib 218 . . . . . . . . . . 11 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚 ∈ (ℕ0m ℕ) ∧ (𝑚 “ ℕ) ∈ Fin ∧ (𝑚 “ ℕ) ⊆ 𝐽))
105104simp1d 1141 . . . . . . . . . 10 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑚 ∈ (ℕ0m ℕ))
10622, 23elmap 8909 . . . . . . . . . 10 (𝑚 ∈ (ℕ0m ℕ) ↔ 𝑚:ℕ⟶ℕ0)
107105, 106sylib 218 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑚:ℕ⟶ℕ0)
108 fssres 6774 . . . . . . . . 9 ((𝑚:ℕ⟶ℕ0𝐽 ⊆ ℕ) → (𝑚𝐽):𝐽⟶ℕ0)
109107, 12, 108sylancl 586 . . . . . . . 8 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽):𝐽⟶ℕ0)
11010, 23rabex2 5346 . . . . . . . . 9 𝐽 ∈ V
11122, 110elmap 8909 . . . . . . . 8 ((𝑚𝐽) ∈ (ℕ0m 𝐽) ↔ (𝑚𝐽):𝐽⟶ℕ0)
112109, 111sylibr 234 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽) ∈ (ℕ0m 𝐽))
113101, 112eqeltrd 2838 . . . . . 6 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜 ∈ (ℕ0m 𝐽))
114 ffun 6739 . . . . . . . . . 10 (𝑚:ℕ⟶ℕ0 → Fun 𝑚)
115 respreima 7085 . . . . . . . . . 10 (Fun 𝑚 → ((𝑚𝐽) “ ℕ) = ((𝑚 “ ℕ) ∩ 𝐽))
116107, 114, 1153syl 18 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → ((𝑚𝐽) “ ℕ) = ((𝑚 “ ℕ) ∩ 𝐽))
117104simp2d 1142 . . . . . . . . . 10 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚 “ ℕ) ∈ Fin)
118 infi 9299 . . . . . . . . . 10 ((𝑚 “ ℕ) ∈ Fin → ((𝑚 “ ℕ) ∩ 𝐽) ∈ Fin)
119117, 118syl 17 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → ((𝑚 “ ℕ) ∩ 𝐽) ∈ Fin)
120116, 119eqeltrd 2838 . . . . . . . 8 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → ((𝑚𝐽) “ ℕ) ∈ Fin)
121 vex 3481 . . . . . . . . . 10 𝑚 ∈ V
122121resex 6048 . . . . . . . . 9 (𝑚𝐽) ∈ V
123 cnveq 5886 . . . . . . . . . . 11 (𝑓 = (𝑚𝐽) → 𝑓 = (𝑚𝐽))
124123imaeq1d 6078 . . . . . . . . . 10 (𝑓 = (𝑚𝐽) → (𝑓 “ ℕ) = ((𝑚𝐽) “ ℕ))
125124eleq1d 2823 . . . . . . . . 9 (𝑓 = (𝑚𝐽) → ((𝑓 “ ℕ) ∈ Fin ↔ ((𝑚𝐽) “ ℕ) ∈ Fin))
126122, 125, 34elab2 3684 . . . . . . . 8 ((𝑚𝐽) ∈ 𝑅 ↔ ((𝑚𝐽) “ ℕ) ∈ Fin)
127120, 126sylibr 234 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽) ∈ 𝑅)
128101, 127eqeltrd 2838 . . . . . 6 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜𝑅)
129113, 128jca 511 . . . . 5 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅))
130129rexlimiva 3144 . . . 4 (∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽) → (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅))
131100, 130impbii 209 . . 3 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) ↔ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽))
132131abbii 2806 . 2 {𝑜 ∣ (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅)} = {𝑜 ∣ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽)}
133 df-in 3969 . 2 ((ℕ0m 𝐽) ∩ 𝑅) = {𝑜 ∣ (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅)}
134 eqid 2734 . . 3 (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)) = (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
135134rnmpt 5970 . 2 ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)) = {𝑜 ∣ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽)}
136132, 133, 1353eqtr4i 2772 1 ((ℕ0m 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  {cab 2711  wne 2937  wral 3058  wrex 3067  {crab 3432  cdif 3959  cun 3960  cin 3961  wss 3962  c0 4338  𝒫 cpw 4604  {csn 4630   class class class wbr 5147  {copab 5209  cmpt 5230   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689  cres 5690  cima 5691  Fun wfun 6556   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432   supp csupp 8183  m cmap 8864  Fincfn 8983  0cc0 11152  1c1 11153   · cmul 11157  cle 11293  cn 12263  2c2 12318  0cn0 12523  cz 12610  cexp 14098  Σcsu 15718  cdvds 16286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-dvds 16287
This theorem is referenced by:  eulerpartgbij  34353
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