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Theorem eulerpartlemt 32971
Description: Lemma for eulerpart 32982. (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 8787 . . . . . . . . . 10 (𝑜 ∈ (ℕ0m 𝐽) → 𝑜:𝐽⟶ℕ0)
21adantr 481 . . . . . . . . 9 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → 𝑜:𝐽⟶ℕ0)
3 c0ex 11149 . . . . . . . . . . 11 0 ∈ V
43fconst 6728 . . . . . . . . . 10 ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0}
54a1i 11 . . . . . . . . 9 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0})
6 disjdif 4431 . . . . . . . . . 10 (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅
76a1i 11 . . . . . . . . 9 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅)
8 fun 6704 . . . . . . . . 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 836 . . . . . . . 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 4037 . . . . . . . . . . 11 {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
1210, 11eqsstri 3978 . . . . . . . . . 10 𝐽 ⊆ ℕ
13 undif 4441 . . . . . . . . . 10 (𝐽 ⊆ ℕ ↔ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ)
1412, 13mpbi 229 . . . . . . . . 9 (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ
15 0nn0 12428 . . . . . . . . . . 11 0 ∈ ℕ0
16 snssi 4768 . . . . . . . . . . 11 (0 ∈ ℕ0 → {0} ⊆ ℕ0)
1715, 16ax-mp 5 . . . . . . . . . 10 {0} ⊆ ℕ0
18 ssequn2 4143 . . . . . . . . . 10 ({0} ⊆ ℕ0 ↔ (ℕ0 ∪ {0}) = ℕ0)
1917, 18mpbi 229 . . . . . . . . 9 (ℕ0 ∪ {0}) = ℕ0
2014, 19feq23i 6662 . . . . . . . 8 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0 ∪ {0}) ↔ (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
219, 20sylib 217 . . . . . . 7 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
22 nn0ex 12419 . . . . . . . 8 0 ∈ V
23 nnex 12159 . . . . . . . 8 ℕ ∈ V
2422, 23elmap 8809 . . . . . . 7 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0m ℕ) ↔ (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):ℕ⟶ℕ0)
2521, 24sylibr 233 . . . . . 6 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0m ℕ))
26 cnvun 6095 . . . . . . . . 9 (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) = (𝑜((ℕ ∖ 𝐽) × {0}))
2726imaeq1i 6010 . . . . . . . 8 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜((ℕ ∖ 𝐽) × {0})) “ ℕ)
28 imaundir 6103 . . . . . . . 8 ((𝑜((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ))
2927, 28eqtri 2764 . . . . . . 7 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ))
30 vex 3449 . . . . . . . . . . 11 𝑜 ∈ V
31 cnveq 5829 . . . . . . . . . . . . 13 (𝑓 = 𝑜𝑓 = 𝑜)
3231imaeq1d 6012 . . . . . . . . . . . 12 (𝑓 = 𝑜 → (𝑓 “ ℕ) = (𝑜 “ ℕ))
3332eleq1d 2822 . . . . . . . . . . 11 (𝑓 = 𝑜 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝑜 “ ℕ) ∈ Fin))
34 eulerpart.r . . . . . . . . . . 11 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
3530, 33, 34elab2 3634 . . . . . . . . . 10 (𝑜𝑅 ↔ (𝑜 “ ℕ) ∈ Fin)
3635biimpi 215 . . . . . . . . 9 (𝑜𝑅 → (𝑜 “ ℕ) ∈ Fin)
3736adantl 482 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 “ ℕ) ∈ Fin)
38 cnvxp 6109 . . . . . . . . . . . . . 14 ((ℕ ∖ 𝐽) × {0}) = ({0} × (ℕ ∖ 𝐽))
3938dmeqi 5860 . . . . . . . . . . . . 13 dom ((ℕ ∖ 𝐽) × {0}) = dom ({0} × (ℕ ∖ 𝐽))
40 2nn 12226 . . . . . . . . . . . . . . 15 2 ∈ ℕ
41 2z 12535 . . . . . . . . . . . . . . . . 17 2 ∈ ℤ
42 iddvds 16152 . . . . . . . . . . . . . . . . 17 (2 ∈ ℤ → 2 ∥ 2)
4341, 42ax-mp 5 . . . . . . . . . . . . . . . 16 2 ∥ 2
44 breq2 5109 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 2 → (2 ∥ 𝑧 ↔ 2 ∥ 2))
4544notbid 317 . . . . . . . . . . . . . . . . . 18 (𝑧 = 2 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 2))
4645, 10elrab2 3648 . . . . . . . . . . . . . . . . 17 (2 ∈ 𝐽 ↔ (2 ∈ ℕ ∧ ¬ 2 ∥ 2))
4746simprbi 497 . . . . . . . . . . . . . . . 16 (2 ∈ 𝐽 → ¬ 2 ∥ 2)
4843, 47mt2 199 . . . . . . . . . . . . . . 15 ¬ 2 ∈ 𝐽
49 eldif 3920 . . . . . . . . . . . . . . 15 (2 ∈ (ℕ ∖ 𝐽) ↔ (2 ∈ ℕ ∧ ¬ 2 ∈ 𝐽))
5040, 48, 49mpbir2an 709 . . . . . . . . . . . . . 14 2 ∈ (ℕ ∖ 𝐽)
51 ne0i 4294 . . . . . . . . . . . . . 14 (2 ∈ (ℕ ∖ 𝐽) → (ℕ ∖ 𝐽) ≠ ∅)
52 dmxp 5884 . . . . . . . . . . . . . 14 ((ℕ ∖ 𝐽) ≠ ∅ → dom ({0} × (ℕ ∖ 𝐽)) = {0})
5350, 51, 52mp2b 10 . . . . . . . . . . . . 13 dom ({0} × (ℕ ∖ 𝐽)) = {0}
5439, 53eqtri 2764 . . . . . . . . . . . 12 dom ((ℕ ∖ 𝐽) × {0}) = {0}
5554ineq1i 4168 . . . . . . . . . . 11 (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ({0} ∩ ℕ)
56 incom 4161 . . . . . . . . . . 11 (ℕ ∩ {0}) = ({0} ∩ ℕ)
57 0nnn 12189 . . . . . . . . . . . 12 ¬ 0 ∈ ℕ
58 disjsn 4672 . . . . . . . . . . . 12 ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
5957, 58mpbir 230 . . . . . . . . . . 11 (ℕ ∩ {0}) = ∅
6055, 56, 593eqtr2i 2770 . . . . . . . . . 10 (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅
61 imadisj 6032 . . . . . . . . . 10 ((((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅ ↔ (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅)
6260, 61mpbir 230 . . . . . . . . 9 (((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅
63 0fin 9115 . . . . . . . . 9 ∅ ∈ Fin
6462, 63eqeltri 2834 . . . . . . . 8 (((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈ Fin
65 unfi 9116 . . . . . . . 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 6033 . . . . . . . . 9 (𝑜 “ ℕ) ⊆ dom 𝑜
6968, 2fssdm 6688 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 “ ℕ) ⊆ 𝐽)
70 0ss 4356 . . . . . . . . . 10 ∅ ⊆ 𝐽
7162, 70eqsstri 3978 . . . . . . . . 9 (((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽
7271a1i 11 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽)
7369, 72unssd 4146 . . . . . . 7 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ)) ⊆ 𝐽)
7429, 73eqsstrid 3992 . . . . . 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 32969 . . . . . 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 1343 . . . . 5 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇𝑅))
84 resundir 5952 . . . . . 6 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽) = ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽))
85 ffn 6668 . . . . . . . 8 (𝑜:𝐽⟶ℕ0𝑜 Fn 𝐽)
86 fnresdm 6620 . . . . . . . . 9 (𝑜 Fn 𝐽 → (𝑜𝐽) = 𝑜)
87 disjdifr 4432 . . . . . . . . . . 11 ((ℕ ∖ 𝐽) ∩ 𝐽) = ∅
88 fnconstg 6730 . . . . . . . . . . . 12 (0 ∈ ℕ0 → ((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽))
89 fnresdisj 6621 . . . . . . . . . . . 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 229 . . . . . . . . . 10 (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅
9291a1i 11 . . . . . . . . 9 (𝑜 Fn 𝐽 → (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅)
9386, 92uneq12d 4124 . . . . . . . 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 4350 . . . . . . 7 (𝑜 ∪ ∅) = 𝑜
9694, 95eqtrdi 2792 . . . . . 6 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = 𝑜)
9784, 96eqtr2id 2789 . . . . 5 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
98 reseq1 5931 . . . . . 6 (𝑚 = (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) → (𝑚𝐽) = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
9998rspceeqv 3595 . . . . 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 485 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜 = (𝑚𝐽))
102 simpl 483 . . . . . . . . . . . 12 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑚 ∈ (𝑇𝑅))
10375, 76, 77, 10, 78, 79, 80, 34, 81eulerpartlemt0 32969 . . . . . . . . . . . 12 (𝑚 ∈ (𝑇𝑅) ↔ (𝑚 ∈ (ℕ0m ℕ) ∧ (𝑚 “ ℕ) ∈ Fin ∧ (𝑚 “ ℕ) ⊆ 𝐽))
104102, 103sylib 217 . . . . . . . . . . 11 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚 ∈ (ℕ0m ℕ) ∧ (𝑚 “ ℕ) ∈ Fin ∧ (𝑚 “ ℕ) ⊆ 𝐽))
105104simp1d 1142 . . . . . . . . . 10 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑚 ∈ (ℕ0m ℕ))
10622, 23elmap 8809 . . . . . . . . . 10 (𝑚 ∈ (ℕ0m ℕ) ↔ 𝑚:ℕ⟶ℕ0)
107105, 106sylib 217 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑚:ℕ⟶ℕ0)
108 fssres 6708 . . . . . . . . 9 ((𝑚:ℕ⟶ℕ0𝐽 ⊆ ℕ) → (𝑚𝐽):𝐽⟶ℕ0)
109107, 12, 108sylancl 586 . . . . . . . 8 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽):𝐽⟶ℕ0)
11010, 23rabex2 5291 . . . . . . . . 9 𝐽 ∈ V
11122, 110elmap 8809 . . . . . . . 8 ((𝑚𝐽) ∈ (ℕ0m 𝐽) ↔ (𝑚𝐽):𝐽⟶ℕ0)
112109, 111sylibr 233 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽) ∈ (ℕ0m 𝐽))
113101, 112eqeltrd 2838 . . . . . 6 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜 ∈ (ℕ0m 𝐽))
114 ffun 6671 . . . . . . . . . 10 (𝑚:ℕ⟶ℕ0 → Fun 𝑚)
115 respreima 7016 . . . . . . . . . 10 (Fun 𝑚 → ((𝑚𝐽) “ ℕ) = ((𝑚 “ ℕ) ∩ 𝐽))
116107, 114, 1153syl 18 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → ((𝑚𝐽) “ ℕ) = ((𝑚 “ ℕ) ∩ 𝐽))
117104simp2d 1143 . . . . . . . . . 10 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚 “ ℕ) ∈ Fin)
118 infi 9212 . . . . . . . . . 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 3449 . . . . . . . . . 10 𝑚 ∈ V
122121resex 5985 . . . . . . . . 9 (𝑚𝐽) ∈ V
123 cnveq 5829 . . . . . . . . . . 11 (𝑓 = (𝑚𝐽) → 𝑓 = (𝑚𝐽))
124123imaeq1d 6012 . . . . . . . . . 10 (𝑓 = (𝑚𝐽) → (𝑓 “ ℕ) = ((𝑚𝐽) “ ℕ))
125124eleq1d 2822 . . . . . . . . 9 (𝑓 = (𝑚𝐽) → ((𝑓 “ ℕ) ∈ Fin ↔ ((𝑚𝐽) “ ℕ) ∈ Fin))
126122, 125, 34elab2 3634 . . . . . . . 8 ((𝑚𝐽) ∈ 𝑅 ↔ ((𝑚𝐽) “ ℕ) ∈ Fin)
127120, 126sylibr 233 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽) ∈ 𝑅)
128101, 127eqeltrd 2838 . . . . . 6 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜𝑅)
129113, 128jca 512 . . . . 5 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅))
130129rexlimiva 3144 . . . 4 (∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽) → (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅))
131100, 130impbii 208 . . 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 3917 . 2 ((ℕ0m 𝐽) ∩ 𝑅) = {𝑜 ∣ (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅)}
134 eqid 2736 . . 3 (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)) = (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
135134rnmpt 5910 . 2 ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)) = {𝑜 ∣ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽)}
136132, 133, 1353eqtr4i 2774 1 ((ℕ0m 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  {crab 3407  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  {copab 5167  cmpt 5188   × cxp 5631  ccnv 5632  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359   supp csupp 8092  m cmap 8765  Fincfn 8883  0cc0 11051  1c1 11052   · cmul 11056  cle 11190  cn 12153  2c2 12208  0cn0 12413  cz 12499  cexp 13967  Σcsu 15570  cdvds 16136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-dvds 16137
This theorem is referenced by:  eulerpartgbij  32972
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