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Theorem eulerpartlemt 31242
Description: Lemma for eulerpart 31253. (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 8285 . . . . . . . . . 10 (𝑜 ∈ (ℕ0𝑚 𝐽) → 𝑜:𝐽⟶ℕ0)
21adantr 481 . . . . . . . . 9 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → 𝑜:𝐽⟶ℕ0)
3 c0ex 10488 . . . . . . . . . . 11 0 ∈ V
43fconst 6440 . . . . . . . . . 10 ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0}
54a1i 11 . . . . . . . . 9 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0})
6 disjdif 4341 . . . . . . . . . 10 (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅
76a1i 11 . . . . . . . . 9 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅)
8 fun 6415 . . . . . . . . 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 834 . . . . . . . 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 3983 . . . . . . . . . . 11 {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
1210, 11eqsstri 3928 . . . . . . . . . 10 𝐽 ⊆ ℕ
13 undif 4350 . . . . . . . . . 10 (𝐽 ⊆ ℕ ↔ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ)
1412, 13mpbi 231 . . . . . . . . 9 (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ
15 0nn0 11766 . . . . . . . . . . 11 0 ∈ ℕ0
16 snssi 4654 . . . . . . . . . . 11 (0 ∈ ℕ0 → {0} ⊆ ℕ0)
1715, 16ax-mp 5 . . . . . . . . . 10 {0} ⊆ ℕ0
18 ssequn2 4086 . . . . . . . . . 10 ({0} ⊆ ℕ0 ↔ (ℕ0 ∪ {0}) = ℕ0)
1917, 18mpbi 231 . . . . . . . . 9 (ℕ0 ∪ {0}) = ℕ0
2014, 19feq23i 6383 . . . . . . . 8 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0 ∪ {0}) ↔ (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
219, 20sylib 219 . . . . . . 7 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
22 nn0ex 11757 . . . . . . . 8 0 ∈ V
23 nnex 11498 . . . . . . . 8 ℕ ∈ V
2422, 23elmap 8292 . . . . . . 7 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0𝑚 ℕ) ↔ (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
2521, 24sylibr 235 . . . . . 6 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0𝑚 ℕ))
26 cnvun 5884 . . . . . . . . 9 (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) = (𝑜((ℕ ∖ 𝐽) × {0}))
2726imaeq1i 5810 . . . . . . . 8 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜((ℕ ∖ 𝐽) × {0})) “ ℕ)
28 imaundir 5892 . . . . . . . 8 ((𝑜((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ))
2927, 28eqtri 2821 . . . . . . 7 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ))
30 vex 3443 . . . . . . . . . . 11 𝑜 ∈ V
31 cnveq 5637 . . . . . . . . . . . . 13 (𝑓 = 𝑜𝑓 = 𝑜)
3231imaeq1d 5812 . . . . . . . . . . . 12 (𝑓 = 𝑜 → (𝑓 “ ℕ) = (𝑜 “ ℕ))
3332eleq1d 2869 . . . . . . . . . . 11 (𝑓 = 𝑜 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝑜 “ ℕ) ∈ Fin))
34 eulerpart.r . . . . . . . . . . 11 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
3530, 33, 34elab2 3611 . . . . . . . . . 10 (𝑜𝑅 ↔ (𝑜 “ ℕ) ∈ Fin)
3635biimpi 217 . . . . . . . . 9 (𝑜𝑅 → (𝑜 “ ℕ) ∈ Fin)
3736adantl 482 . . . . . . . 8 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → (𝑜 “ ℕ) ∈ Fin)
38 cnvxp 5897 . . . . . . . . . . . . . 14 ((ℕ ∖ 𝐽) × {0}) = ({0} × (ℕ ∖ 𝐽))
3938dmeqi 5666 . . . . . . . . . . . . 13 dom ((ℕ ∖ 𝐽) × {0}) = dom ({0} × (ℕ ∖ 𝐽))
40 2nn 11564 . . . . . . . . . . . . . . 15 2 ∈ ℕ
41 2z 11868 . . . . . . . . . . . . . . . . 17 2 ∈ ℤ
42 iddvds 15460 . . . . . . . . . . . . . . . . 17 (2 ∈ ℤ → 2 ∥ 2)
4341, 42ax-mp 5 . . . . . . . . . . . . . . . 16 2 ∥ 2
44 breq2 4972 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 2 → (2 ∥ 𝑧 ↔ 2 ∥ 2))
4544notbid 319 . . . . . . . . . . . . . . . . . 18 (𝑧 = 2 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 2))
4645, 10elrab2 3624 . . . . . . . . . . . . . . . . 17 (2 ∈ 𝐽 ↔ (2 ∈ ℕ ∧ ¬ 2 ∥ 2))
4746simprbi 497 . . . . . . . . . . . . . . . 16 (2 ∈ 𝐽 → ¬ 2 ∥ 2)
4843, 47mt2 201 . . . . . . . . . . . . . . 15 ¬ 2 ∈ 𝐽
49 eldif 3875 . . . . . . . . . . . . . . 15 (2 ∈ (ℕ ∖ 𝐽) ↔ (2 ∈ ℕ ∧ ¬ 2 ∈ 𝐽))
5040, 48, 49mpbir2an 707 . . . . . . . . . . . . . 14 2 ∈ (ℕ ∖ 𝐽)
51 ne0i 4226 . . . . . . . . . . . . . 14 (2 ∈ (ℕ ∖ 𝐽) → (ℕ ∖ 𝐽) ≠ ∅)
52 dmxp 5688 . . . . . . . . . . . . . 14 ((ℕ ∖ 𝐽) ≠ ∅ → dom ({0} × (ℕ ∖ 𝐽)) = {0})
5350, 51, 52mp2b 10 . . . . . . . . . . . . 13 dom ({0} × (ℕ ∖ 𝐽)) = {0}
5439, 53eqtri 2821 . . . . . . . . . . . 12 dom ((ℕ ∖ 𝐽) × {0}) = {0}
5554ineq1i 4111 . . . . . . . . . . 11 (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ({0} ∩ ℕ)
56 incom 4105 . . . . . . . . . . 11 (ℕ ∩ {0}) = ({0} ∩ ℕ)
57 0nnn 11527 . . . . . . . . . . . 12 ¬ 0 ∈ ℕ
58 disjsn 4560 . . . . . . . . . . . 12 ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
5957, 58mpbir 232 . . . . . . . . . . 11 (ℕ ∩ {0}) = ∅
6055, 56, 593eqtr2i 2827 . . . . . . . . . 10 (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅
61 imadisj 5831 . . . . . . . . . 10 ((((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅ ↔ (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅)
6260, 61mpbir 232 . . . . . . . . 9 (((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅
63 0fin 8599 . . . . . . . . 9 ∅ ∈ Fin
6462, 63eqeltri 2881 . . . . . . . 8 (((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈ Fin
65 unfi 8638 . . . . . . . 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, 66syl5eqel 2889 . . . . . 6 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ∈ Fin)
68 cnvimass 5832 . . . . . . . . 9 (𝑜 “ ℕ) ⊆ dom 𝑜
6968, 2fssdm 6405 . . . . . . . 8 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → (𝑜 “ ℕ) ⊆ 𝐽)
70 0ss 4276 . . . . . . . . . 10 ∅ ⊆ 𝐽
7162, 70eqsstri 3928 . . . . . . . . 9 (((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽
7271a1i 11 . . . . . . . 8 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → (((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽)
7369, 72unssd 4089 . . . . . . 7 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ)) ⊆ 𝐽)
7429, 73eqsstrid 3942 . . . . . 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 31240 . . . . . 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 1336 . . . . 5 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇𝑅))
84 resundir 5756 . . . . . 6 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽) = ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽))
85 ffn 6389 . . . . . . . 8 (𝑜:𝐽⟶ℕ0𝑜 Fn 𝐽)
86 fnresdm 6343 . . . . . . . . 9 (𝑜 Fn 𝐽 → (𝑜𝐽) = 𝑜)
87 incom 4105 . . . . . . . . . . . 12 ((ℕ ∖ 𝐽) ∩ 𝐽) = (𝐽 ∩ (ℕ ∖ 𝐽))
8887, 6eqtri 2821 . . . . . . . . . . 11 ((ℕ ∖ 𝐽) ∩ 𝐽) = ∅
89 fnconstg 6442 . . . . . . . . . . . 12 (0 ∈ ℕ0 → ((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽))
90 fnresdisj 6344 . . . . . . . . . . . 12 (((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽) → (((ℕ ∖ 𝐽) ∩ 𝐽) = ∅ ↔ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅))
9115, 89, 90mp2b 10 . . . . . . . . . . 11 (((ℕ ∖ 𝐽) ∩ 𝐽) = ∅ ↔ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅)
9288, 91mpbi 231 . . . . . . . . . 10 (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅
9392a1i 11 . . . . . . . . 9 (𝑜 Fn 𝐽 → (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅)
9486, 93uneq12d 4067 . . . . . . . 8 (𝑜 Fn 𝐽 → ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = (𝑜 ∪ ∅))
952, 85, 943syl 18 . . . . . . 7 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = (𝑜 ∪ ∅))
96 un0 4270 . . . . . . 7 (𝑜 ∪ ∅) = 𝑜
9795, 96syl6eq 2849 . . . . . 6 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = 𝑜)
9884, 97syl5req 2846 . . . . 5 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
99 reseq1 5735 . . . . . 6 (𝑚 = (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) → (𝑚𝐽) = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
10099rspceeqv 3579 . . . . 5 (((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇𝑅) ∧ 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽)) → ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽))
10183, 98, 100syl2anc 584 . . . 4 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) → ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽))
102 simpr 485 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜 = (𝑚𝐽))
103 simpl 483 . . . . . . . . . . . 12 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑚 ∈ (𝑇𝑅))
10475, 76, 77, 10, 78, 79, 80, 34, 81eulerpartlemt0 31240 . . . . . . . . . . . 12 (𝑚 ∈ (𝑇𝑅) ↔ (𝑚 ∈ (ℕ0𝑚 ℕ) ∧ (𝑚 “ ℕ) ∈ Fin ∧ (𝑚 “ ℕ) ⊆ 𝐽))
105103, 104sylib 219 . . . . . . . . . . 11 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚 ∈ (ℕ0𝑚 ℕ) ∧ (𝑚 “ ℕ) ∈ Fin ∧ (𝑚 “ ℕ) ⊆ 𝐽))
106105simp1d 1135 . . . . . . . . . 10 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑚 ∈ (ℕ0𝑚 ℕ))
10722, 23elmap 8292 . . . . . . . . . 10 (𝑚 ∈ (ℕ0𝑚 ℕ) ↔ 𝑚:ℕ⟶ℕ0)
108106, 107sylib 219 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑚:ℕ⟶ℕ0)
109 fssres 6419 . . . . . . . . 9 ((𝑚:ℕ⟶ℕ0𝐽 ⊆ ℕ) → (𝑚𝐽):𝐽⟶ℕ0)
110108, 12, 109sylancl 586 . . . . . . . 8 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽):𝐽⟶ℕ0)
11110, 23rabex2 5135 . . . . . . . . 9 𝐽 ∈ V
11222, 111elmap 8292 . . . . . . . 8 ((𝑚𝐽) ∈ (ℕ0𝑚 𝐽) ↔ (𝑚𝐽):𝐽⟶ℕ0)
113110, 112sylibr 235 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽) ∈ (ℕ0𝑚 𝐽))
114102, 113eqeltrd 2885 . . . . . 6 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜 ∈ (ℕ0𝑚 𝐽))
115 ffun 6392 . . . . . . . . . 10 (𝑚:ℕ⟶ℕ0 → Fun 𝑚)
116 respreima 6708 . . . . . . . . . 10 (Fun 𝑚 → ((𝑚𝐽) “ ℕ) = ((𝑚 “ ℕ) ∩ 𝐽))
117108, 115, 1163syl 18 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → ((𝑚𝐽) “ ℕ) = ((𝑚 “ ℕ) ∩ 𝐽))
118105simp2d 1136 . . . . . . . . . 10 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚 “ ℕ) ∈ Fin)
119 infi 8595 . . . . . . . . . 10 ((𝑚 “ ℕ) ∈ Fin → ((𝑚 “ ℕ) ∩ 𝐽) ∈ Fin)
120118, 119syl 17 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → ((𝑚 “ ℕ) ∩ 𝐽) ∈ Fin)
121117, 120eqeltrd 2885 . . . . . . . 8 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → ((𝑚𝐽) “ ℕ) ∈ Fin)
122 vex 3443 . . . . . . . . . 10 𝑚 ∈ V
123122resex 5787 . . . . . . . . 9 (𝑚𝐽) ∈ V
124 cnveq 5637 . . . . . . . . . . 11 (𝑓 = (𝑚𝐽) → 𝑓 = (𝑚𝐽))
125124imaeq1d 5812 . . . . . . . . . 10 (𝑓 = (𝑚𝐽) → (𝑓 “ ℕ) = ((𝑚𝐽) “ ℕ))
126125eleq1d 2869 . . . . . . . . 9 (𝑓 = (𝑚𝐽) → ((𝑓 “ ℕ) ∈ Fin ↔ ((𝑚𝐽) “ ℕ) ∈ Fin))
127123, 126, 34elab2 3611 . . . . . . . 8 ((𝑚𝐽) ∈ 𝑅 ↔ ((𝑚𝐽) “ ℕ) ∈ Fin)
128121, 127sylibr 235 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽) ∈ 𝑅)
129102, 128eqeltrd 2885 . . . . . 6 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜𝑅)
130114, 129jca 512 . . . . 5 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅))
131130rexlimiva 3246 . . . 4 (∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽) → (𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅))
132101, 131impbii 210 . . 3 ((𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅) ↔ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽))
133132abbii 2863 . 2 {𝑜 ∣ (𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅)} = {𝑜 ∣ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽)}
134 df-in 3872 . 2 ((ℕ0𝑚 𝐽) ∩ 𝑅) = {𝑜 ∣ (𝑜 ∈ (ℕ0𝑚 𝐽) ∧ 𝑜𝑅)}
135 eqid 2797 . . 3 (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)) = (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
136135rnmpt 5716 . 2 ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)) = {𝑜 ∣ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽)}
137133, 134, 1363eqtr4i 2831 1 ((ℕ0𝑚 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396  w3a 1080   = wceq 1525  wcel 2083  {cab 2777  wne 2986  wral 3107  wrex 3108  {crab 3111  cdif 3862  cun 3863  cin 3864  wss 3865  c0 4217  𝒫 cpw 4459  {csn 4478   class class class wbr 4968  {copab 5030  cmpt 5047   × cxp 5448  ccnv 5449  dom cdm 5450  ran crn 5451  cres 5452  cima 5453  Fun wfun 6226   Fn wfn 6227  wf 6228  cfv 6232  (class class class)co 7023  cmpo 7025   supp csupp 7688  𝑚 cmap 8263  Fincfn 8364  0cc0 10390  1c1 10391   · cmul 10395  cle 10529  cn 11492  2c2 11546  0cn0 11751  cz 11835  cexp 13283  Σcsu 14880  cdvds 15444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-oadd 7964  df-er 8146  df-map 8265  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-neg 10726  df-nn 11493  df-2 11554  df-n0 11752  df-z 11836  df-dvds 15445
This theorem is referenced by:  eulerpartgbij  31243
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