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Theorem eulerpartlemt 31528
Description: Lemma for eulerpart 31539. (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 8417 . . . . . . . . . 10 (𝑜 ∈ (ℕ0m 𝐽) → 𝑜:𝐽⟶ℕ0)
21adantr 481 . . . . . . . . 9 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → 𝑜:𝐽⟶ℕ0)
3 c0ex 10623 . . . . . . . . . . 11 0 ∈ V
43fconst 6558 . . . . . . . . . 10 ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0}
54a1i 11 . . . . . . . . 9 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0})
6 disjdif 4417 . . . . . . . . . 10 (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅
76a1i 11 . . . . . . . . 9 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅)
8 fun 6533 . . . . . . . . 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 833 . . . . . . . 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 4053 . . . . . . . . . . 11 {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
1210, 11eqsstri 3998 . . . . . . . . . 10 𝐽 ⊆ ℕ
13 undif 4426 . . . . . . . . . 10 (𝐽 ⊆ ℕ ↔ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ)
1412, 13mpbi 231 . . . . . . . . 9 (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ
15 0nn0 11900 . . . . . . . . . . 11 0 ∈ ℕ0
16 snssi 4733 . . . . . . . . . . 11 (0 ∈ ℕ0 → {0} ⊆ ℕ0)
1715, 16ax-mp 5 . . . . . . . . . 10 {0} ⊆ ℕ0
18 ssequn2 4156 . . . . . . . . . 10 ({0} ⊆ ℕ0 ↔ (ℕ0 ∪ {0}) = ℕ0)
1917, 18mpbi 231 . . . . . . . . 9 (ℕ0 ∪ {0}) = ℕ0
2014, 19feq23i 6501 . . . . . . . 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 11891 . . . . . . . 8 0 ∈ V
23 nnex 11632 . . . . . . . 8 ℕ ∈ V
2422, 23elmap 8424 . . . . . . 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 5994 . . . . . . . . 9 (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) = (𝑜((ℕ ∖ 𝐽) × {0}))
2726imaeq1i 5919 . . . . . . . 8 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜((ℕ ∖ 𝐽) × {0})) “ ℕ)
28 imaundir 6002 . . . . . . . 8 ((𝑜((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ))
2927, 28eqtri 2841 . . . . . . 7 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ))
30 vex 3495 . . . . . . . . . . 11 𝑜 ∈ V
31 cnveq 5737 . . . . . . . . . . . . 13 (𝑓 = 𝑜𝑓 = 𝑜)
3231imaeq1d 5921 . . . . . . . . . . . 12 (𝑓 = 𝑜 → (𝑓 “ ℕ) = (𝑜 “ ℕ))
3332eleq1d 2894 . . . . . . . . . . 11 (𝑓 = 𝑜 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝑜 “ ℕ) ∈ Fin))
34 eulerpart.r . . . . . . . . . . 11 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
3530, 33, 34elab2 3667 . . . . . . . . . 10 (𝑜𝑅 ↔ (𝑜 “ ℕ) ∈ Fin)
3635biimpi 217 . . . . . . . . 9 (𝑜𝑅 → (𝑜 “ ℕ) ∈ Fin)
3736adantl 482 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 “ ℕ) ∈ Fin)
38 cnvxp 6007 . . . . . . . . . . . . . 14 ((ℕ ∖ 𝐽) × {0}) = ({0} × (ℕ ∖ 𝐽))
3938dmeqi 5766 . . . . . . . . . . . . 13 dom ((ℕ ∖ 𝐽) × {0}) = dom ({0} × (ℕ ∖ 𝐽))
40 2nn 11698 . . . . . . . . . . . . . . 15 2 ∈ ℕ
41 2z 12002 . . . . . . . . . . . . . . . . 17 2 ∈ ℤ
42 iddvds 15611 . . . . . . . . . . . . . . . . 17 (2 ∈ ℤ → 2 ∥ 2)
4341, 42ax-mp 5 . . . . . . . . . . . . . . . 16 2 ∥ 2
44 breq2 5061 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 2 → (2 ∥ 𝑧 ↔ 2 ∥ 2))
4544notbid 319 . . . . . . . . . . . . . . . . . 18 (𝑧 = 2 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 2))
4645, 10elrab2 3680 . . . . . . . . . . . . . . . . 17 (2 ∈ 𝐽 ↔ (2 ∈ ℕ ∧ ¬ 2 ∥ 2))
4746simprbi 497 . . . . . . . . . . . . . . . 16 (2 ∈ 𝐽 → ¬ 2 ∥ 2)
4843, 47mt2 201 . . . . . . . . . . . . . . 15 ¬ 2 ∈ 𝐽
49 eldif 3943 . . . . . . . . . . . . . . 15 (2 ∈ (ℕ ∖ 𝐽) ↔ (2 ∈ ℕ ∧ ¬ 2 ∈ 𝐽))
5040, 48, 49mpbir2an 707 . . . . . . . . . . . . . 14 2 ∈ (ℕ ∖ 𝐽)
51 ne0i 4297 . . . . . . . . . . . . . 14 (2 ∈ (ℕ ∖ 𝐽) → (ℕ ∖ 𝐽) ≠ ∅)
52 dmxp 5792 . . . . . . . . . . . . . 14 ((ℕ ∖ 𝐽) ≠ ∅ → dom ({0} × (ℕ ∖ 𝐽)) = {0})
5350, 51, 52mp2b 10 . . . . . . . . . . . . 13 dom ({0} × (ℕ ∖ 𝐽)) = {0}
5439, 53eqtri 2841 . . . . . . . . . . . 12 dom ((ℕ ∖ 𝐽) × {0}) = {0}
5554ineq1i 4182 . . . . . . . . . . 11 (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ({0} ∩ ℕ)
56 incom 4175 . . . . . . . . . . 11 (ℕ ∩ {0}) = ({0} ∩ ℕ)
57 0nnn 11661 . . . . . . . . . . . 12 ¬ 0 ∈ ℕ
58 disjsn 4639 . . . . . . . . . . . 12 ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
5957, 58mpbir 232 . . . . . . . . . . 11 (ℕ ∩ {0}) = ∅
6055, 56, 593eqtr2i 2847 . . . . . . . . . 10 (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅
61 imadisj 5941 . . . . . . . . . 10 ((((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅ ↔ (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅)
6260, 61mpbir 232 . . . . . . . . 9 (((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅
63 0fin 8734 . . . . . . . . 9 ∅ ∈ Fin
6462, 63eqeltri 2906 . . . . . . . 8 (((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈ Fin
65 unfi 8773 . . . . . . . 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 2914 . . . . . 6 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ∈ Fin)
68 cnvimass 5942 . . . . . . . . 9 (𝑜 “ ℕ) ⊆ dom 𝑜
6968, 2fssdm 6523 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 “ ℕ) ⊆ 𝐽)
70 0ss 4347 . . . . . . . . . 10 ∅ ⊆ 𝐽
7162, 70eqsstri 3998 . . . . . . . . 9 (((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽
7271a1i 11 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽)
7369, 72unssd 4159 . . . . . . 7 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ)) ⊆ 𝐽)
7429, 73eqsstrid 4012 . . . . . 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 31526 . . . . . 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 1335 . . . . 5 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇𝑅))
84 resundir 5861 . . . . . 6 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽) = ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽))
85 ffn 6507 . . . . . . . 8 (𝑜:𝐽⟶ℕ0𝑜 Fn 𝐽)
86 fnresdm 6459 . . . . . . . . 9 (𝑜 Fn 𝐽 → (𝑜𝐽) = 𝑜)
87 incom 4175 . . . . . . . . . . . 12 ((ℕ ∖ 𝐽) ∩ 𝐽) = (𝐽 ∩ (ℕ ∖ 𝐽))
8887, 6eqtri 2841 . . . . . . . . . . 11 ((ℕ ∖ 𝐽) ∩ 𝐽) = ∅
89 fnconstg 6560 . . . . . . . . . . . 12 (0 ∈ ℕ0 → ((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽))
90 fnresdisj 6460 . . . . . . . . . . . 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 4137 . . . . . . . 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 4341 . . . . . . 7 (𝑜 ∪ ∅) = 𝑜
9795, 96syl6eq 2869 . . . . . 6 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = 𝑜)
9884, 97syl5req 2866 . . . . 5 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
99 reseq1 5840 . . . . . 6 (𝑚 = (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) → (𝑚𝐽) = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
10099rspceeqv 3635 . . . . 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 31526 . . . . . . . . . . . 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 1134 . . . . . . . . . 10 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑚 ∈ (ℕ0m ℕ))
10722, 23elmap 8424 . . . . . . . . . 10 (𝑚 ∈ (ℕ0m ℕ) ↔ 𝑚:ℕ⟶ℕ0)
108106, 107sylib 219 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑚:ℕ⟶ℕ0)
109 fssres 6537 . . . . . . . . 9 ((𝑚:ℕ⟶ℕ0𝐽 ⊆ ℕ) → (𝑚𝐽):𝐽⟶ℕ0)
110108, 12, 109sylancl 586 . . . . . . . 8 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽):𝐽⟶ℕ0)
11110, 23rabex2 5228 . . . . . . . . 9 𝐽 ∈ V
11222, 111elmap 8424 . . . . . . . 8 ((𝑚𝐽) ∈ (ℕ0m 𝐽) ↔ (𝑚𝐽):𝐽⟶ℕ0)
113110, 112sylibr 235 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽) ∈ (ℕ0m 𝐽))
114102, 113eqeltrd 2910 . . . . . 6 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜 ∈ (ℕ0m 𝐽))
115 ffun 6510 . . . . . . . . . 10 (𝑚:ℕ⟶ℕ0 → Fun 𝑚)
116 respreima 6828 . . . . . . . . . 10 (Fun 𝑚 → ((𝑚𝐽) “ ℕ) = ((𝑚 “ ℕ) ∩ 𝐽))
117108, 115, 1163syl 18 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → ((𝑚𝐽) “ ℕ) = ((𝑚 “ ℕ) ∩ 𝐽))
118105simp2d 1135 . . . . . . . . . 10 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚 “ ℕ) ∈ Fin)
119 infi 8730 . . . . . . . . . 10 ((𝑚 “ ℕ) ∈ Fin → ((𝑚 “ ℕ) ∩ 𝐽) ∈ Fin)
120118, 119syl 17 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → ((𝑚 “ ℕ) ∩ 𝐽) ∈ Fin)
121117, 120eqeltrd 2910 . . . . . . . 8 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → ((𝑚𝐽) “ ℕ) ∈ Fin)
122 vex 3495 . . . . . . . . . 10 𝑚 ∈ V
123122resex 5892 . . . . . . . . 9 (𝑚𝐽) ∈ V
124 cnveq 5737 . . . . . . . . . . 11 (𝑓 = (𝑚𝐽) → 𝑓 = (𝑚𝐽))
125124imaeq1d 5921 . . . . . . . . . 10 (𝑓 = (𝑚𝐽) → (𝑓 “ ℕ) = ((𝑚𝐽) “ ℕ))
126125eleq1d 2894 . . . . . . . . 9 (𝑓 = (𝑚𝐽) → ((𝑓 “ ℕ) ∈ Fin ↔ ((𝑚𝐽) “ ℕ) ∈ Fin))
127123, 126, 34elab2 3667 . . . . . . . 8 ((𝑚𝐽) ∈ 𝑅 ↔ ((𝑚𝐽) “ ℕ) ∈ Fin)
128121, 127sylibr 235 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽) ∈ 𝑅)
129102, 128eqeltrd 2910 . . . . . 6 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜𝑅)
130114, 129jca 512 . . . . 5 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅))
131130rexlimiva 3278 . . . 4 (∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽) → (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅))
132101, 131impbii 210 . . 3 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) ↔ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽))
133132abbii 2883 . 2 {𝑜 ∣ (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅)} = {𝑜 ∣ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽)}
134 df-in 3940 . 2 ((ℕ0m 𝐽) ∩ 𝑅) = {𝑜 ∣ (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅)}
135 eqid 2818 . . 3 (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)) = (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
136135rnmpt 5820 . 2 ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)) = {𝑜 ∣ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽)}
137133, 134, 1363eqtr4i 2851 1 ((ℕ0m 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wne 3013  wral 3135  wrex 3136  {crab 3139  cdif 3930  cun 3931  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557   class class class wbr 5057  {copab 5119  cmpt 5137   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  Fun wfun 6342   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147   supp csupp 7819  m cmap 8395  Fincfn 8497  0cc0 10525  1c1 10526   · cmul 10530  cle 10664  cn 11626  2c2 11680  0cn0 11885  cz 11969  cexp 13417  Σcsu 15030  cdvds 15595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-dvds 15596
This theorem is referenced by:  eulerpartgbij  31529
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