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Theorem eulerpartlemt 31529
Description: Lemma for eulerpart 31540. (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 8418 . . . . . . . . . 10 (𝑜 ∈ (ℕ0m 𝐽) → 𝑜:𝐽⟶ℕ0)
21adantr 481 . . . . . . . . 9 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → 𝑜:𝐽⟶ℕ0)
3 c0ex 10624 . . . . . . . . . . 11 0 ∈ V
43fconst 6559 . . . . . . . . . 10 ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0}
54a1i 11 . . . . . . . . 9 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0})
6 disjdif 4419 . . . . . . . . . 10 (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅
76a1i 11 . . . . . . . . 9 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅)
8 fun 6534 . . . . . . . . 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 4055 . . . . . . . . . . 11 {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
1210, 11eqsstri 4000 . . . . . . . . . 10 𝐽 ⊆ ℕ
13 undif 4428 . . . . . . . . . 10 (𝐽 ⊆ ℕ ↔ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ)
1412, 13mpbi 231 . . . . . . . . 9 (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ
15 0nn0 11901 . . . . . . . . . . 11 0 ∈ ℕ0
16 snssi 4735 . . . . . . . . . . 11 (0 ∈ ℕ0 → {0} ⊆ ℕ0)
1715, 16ax-mp 5 . . . . . . . . . 10 {0} ⊆ ℕ0
18 ssequn2 4158 . . . . . . . . . 10 ({0} ⊆ ℕ0 ↔ (ℕ0 ∪ {0}) = ℕ0)
1917, 18mpbi 231 . . . . . . . . 9 (ℕ0 ∪ {0}) = ℕ0
2014, 19feq23i 6502 . . . . . . . 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 11892 . . . . . . . 8 0 ∈ V
23 nnex 11633 . . . . . . . 8 ℕ ∈ V
2422, 23elmap 8425 . . . . . . 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 5995 . . . . . . . . 9 (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) = (𝑜((ℕ ∖ 𝐽) × {0}))
2726imaeq1i 5920 . . . . . . . 8 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜((ℕ ∖ 𝐽) × {0})) “ ℕ)
28 imaundir 6003 . . . . . . . 8 ((𝑜((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ))
2927, 28eqtri 2844 . . . . . . 7 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ))
30 vex 3498 . . . . . . . . . . 11 𝑜 ∈ V
31 cnveq 5738 . . . . . . . . . . . . 13 (𝑓 = 𝑜𝑓 = 𝑜)
3231imaeq1d 5922 . . . . . . . . . . . 12 (𝑓 = 𝑜 → (𝑓 “ ℕ) = (𝑜 “ ℕ))
3332eleq1d 2897 . . . . . . . . . . 11 (𝑓 = 𝑜 → ((𝑓 “ ℕ) ∈ Fin ↔ (𝑜 “ ℕ) ∈ Fin))
34 eulerpart.r . . . . . . . . . . 11 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
3530, 33, 34elab2 3669 . . . . . . . . . 10 (𝑜𝑅 ↔ (𝑜 “ ℕ) ∈ Fin)
3635biimpi 217 . . . . . . . . 9 (𝑜𝑅 → (𝑜 “ ℕ) ∈ Fin)
3736adantl 482 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 “ ℕ) ∈ Fin)
38 cnvxp 6008 . . . . . . . . . . . . . 14 ((ℕ ∖ 𝐽) × {0}) = ({0} × (ℕ ∖ 𝐽))
3938dmeqi 5767 . . . . . . . . . . . . 13 dom ((ℕ ∖ 𝐽) × {0}) = dom ({0} × (ℕ ∖ 𝐽))
40 2nn 11699 . . . . . . . . . . . . . . 15 2 ∈ ℕ
41 2z 12003 . . . . . . . . . . . . . . . . 17 2 ∈ ℤ
42 iddvds 15613 . . . . . . . . . . . . . . . . 17 (2 ∈ ℤ → 2 ∥ 2)
4341, 42ax-mp 5 . . . . . . . . . . . . . . . 16 2 ∥ 2
44 breq2 5062 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 2 → (2 ∥ 𝑧 ↔ 2 ∥ 2))
4544notbid 319 . . . . . . . . . . . . . . . . . 18 (𝑧 = 2 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 2))
4645, 10elrab2 3682 . . . . . . . . . . . . . . . . 17 (2 ∈ 𝐽 ↔ (2 ∈ ℕ ∧ ¬ 2 ∥ 2))
4746simprbi 497 . . . . . . . . . . . . . . . 16 (2 ∈ 𝐽 → ¬ 2 ∥ 2)
4843, 47mt2 201 . . . . . . . . . . . . . . 15 ¬ 2 ∈ 𝐽
49 eldif 3945 . . . . . . . . . . . . . . 15 (2 ∈ (ℕ ∖ 𝐽) ↔ (2 ∈ ℕ ∧ ¬ 2 ∈ 𝐽))
5040, 48, 49mpbir2an 707 . . . . . . . . . . . . . 14 2 ∈ (ℕ ∖ 𝐽)
51 ne0i 4299 . . . . . . . . . . . . . 14 (2 ∈ (ℕ ∖ 𝐽) → (ℕ ∖ 𝐽) ≠ ∅)
52 dmxp 5793 . . . . . . . . . . . . . 14 ((ℕ ∖ 𝐽) ≠ ∅ → dom ({0} × (ℕ ∖ 𝐽)) = {0})
5350, 51, 52mp2b 10 . . . . . . . . . . . . 13 dom ({0} × (ℕ ∖ 𝐽)) = {0}
5439, 53eqtri 2844 . . . . . . . . . . . 12 dom ((ℕ ∖ 𝐽) × {0}) = {0}
5554ineq1i 4184 . . . . . . . . . . 11 (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ({0} ∩ ℕ)
56 incom 4177 . . . . . . . . . . 11 (ℕ ∩ {0}) = ({0} ∩ ℕ)
57 0nnn 11662 . . . . . . . . . . . 12 ¬ 0 ∈ ℕ
58 disjsn 4641 . . . . . . . . . . . 12 ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
5957, 58mpbir 232 . . . . . . . . . . 11 (ℕ ∩ {0}) = ∅
6055, 56, 593eqtr2i 2850 . . . . . . . . . 10 (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅
61 imadisj 5942 . . . . . . . . . 10 ((((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅ ↔ (dom ((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ∅)
6260, 61mpbir 232 . . . . . . . . 9 (((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅
63 0fin 8735 . . . . . . . . 9 ∅ ∈ Fin
6462, 63eqeltri 2909 . . . . . . . 8 (((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈ Fin
65 unfi 8774 . . . . . . . 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 2917 . . . . . 6 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ∈ Fin)
68 cnvimass 5943 . . . . . . . . 9 (𝑜 “ ℕ) ⊆ dom 𝑜
6968, 2fssdm 6524 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (𝑜 “ ℕ) ⊆ 𝐽)
70 0ss 4349 . . . . . . . . . 10 ∅ ⊆ 𝐽
7162, 70eqsstri 4000 . . . . . . . . 9 (((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽
7271a1i 11 . . . . . . . 8 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → (((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆ 𝐽)
7369, 72unssd 4161 . . . . . . 7 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜 “ ℕ) ∪ (((ℕ ∖ 𝐽) × {0}) “ ℕ)) ⊆ 𝐽)
7429, 73eqsstrid 4014 . . . . . 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 31527 . . . . . 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 5862 . . . . . 6 ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽) = ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽))
85 ffn 6508 . . . . . . . 8 (𝑜:𝐽⟶ℕ0𝑜 Fn 𝐽)
86 fnresdm 6460 . . . . . . . . 9 (𝑜 Fn 𝐽 → (𝑜𝐽) = 𝑜)
87 incom 4177 . . . . . . . . . . . 12 ((ℕ ∖ 𝐽) ∩ 𝐽) = (𝐽 ∩ (ℕ ∖ 𝐽))
8887, 6eqtri 2844 . . . . . . . . . . 11 ((ℕ ∖ 𝐽) ∩ 𝐽) = ∅
89 fnconstg 6561 . . . . . . . . . . . 12 (0 ∈ ℕ0 → ((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽))
90 fnresdisj 6461 . . . . . . . . . . . 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 4139 . . . . . . . 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 4343 . . . . . . 7 (𝑜 ∪ ∅) = 𝑜
9795, 96syl6eq 2872 . . . . . 6 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → ((𝑜𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = 𝑜)
9884, 97syl5req 2869 . . . . 5 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) → 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
99 reseq1 5841 . . . . . 6 (𝑚 = (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) → (𝑚𝐽) = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽))
10099rspceeqv 3637 . . . . 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 31527 . . . . . . . . . . . 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 8425 . . . . . . . . . 10 (𝑚 ∈ (ℕ0m ℕ) ↔ 𝑚:ℕ⟶ℕ0)
108106, 107sylib 219 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑚:ℕ⟶ℕ0)
109 fssres 6538 . . . . . . . . 9 ((𝑚:ℕ⟶ℕ0𝐽 ⊆ ℕ) → (𝑚𝐽):𝐽⟶ℕ0)
110108, 12, 109sylancl 586 . . . . . . . 8 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽):𝐽⟶ℕ0)
11110, 23rabex2 5229 . . . . . . . . 9 𝐽 ∈ V
11222, 111elmap 8425 . . . . . . . 8 ((𝑚𝐽) ∈ (ℕ0m 𝐽) ↔ (𝑚𝐽):𝐽⟶ℕ0)
113110, 112sylibr 235 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽) ∈ (ℕ0m 𝐽))
114102, 113eqeltrd 2913 . . . . . 6 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜 ∈ (ℕ0m 𝐽))
115 ffun 6511 . . . . . . . . . 10 (𝑚:ℕ⟶ℕ0 → Fun 𝑚)
116 respreima 6829 . . . . . . . . . 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 8731 . . . . . . . . . 10 ((𝑚 “ ℕ) ∈ Fin → ((𝑚 “ ℕ) ∩ 𝐽) ∈ Fin)
120118, 119syl 17 . . . . . . . . 9 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → ((𝑚 “ ℕ) ∩ 𝐽) ∈ Fin)
121117, 120eqeltrd 2913 . . . . . . . 8 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → ((𝑚𝐽) “ ℕ) ∈ Fin)
122 vex 3498 . . . . . . . . . 10 𝑚 ∈ V
123122resex 5893 . . . . . . . . 9 (𝑚𝐽) ∈ V
124 cnveq 5738 . . . . . . . . . . 11 (𝑓 = (𝑚𝐽) → 𝑓 = (𝑚𝐽))
125124imaeq1d 5922 . . . . . . . . . 10 (𝑓 = (𝑚𝐽) → (𝑓 “ ℕ) = ((𝑚𝐽) “ ℕ))
126125eleq1d 2897 . . . . . . . . 9 (𝑓 = (𝑚𝐽) → ((𝑓 “ ℕ) ∈ Fin ↔ ((𝑚𝐽) “ ℕ) ∈ Fin))
127123, 126, 34elab2 3669 . . . . . . . 8 ((𝑚𝐽) ∈ 𝑅 ↔ ((𝑚𝐽) “ ℕ) ∈ Fin)
128121, 127sylibr 235 . . . . . . 7 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑚𝐽) ∈ 𝑅)
129102, 128eqeltrd 2913 . . . . . 6 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → 𝑜𝑅)
130114, 129jca 512 . . . . 5 ((𝑚 ∈ (𝑇𝑅) ∧ 𝑜 = (𝑚𝐽)) → (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅))
131130rexlimiva 3281 . . . 4 (∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽) → (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅))
132101, 131impbii 210 . . 3 ((𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅) ↔ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽))
133132abbii 2886 . 2 {𝑜 ∣ (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅)} = {𝑜 ∣ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽)}
134 df-in 3942 . 2 ((ℕ0m 𝐽) ∩ 𝑅) = {𝑜 ∣ (𝑜 ∈ (ℕ0m 𝐽) ∧ 𝑜𝑅)}
135 eqid 2821 . . 3 (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)) = (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
136135rnmpt 5821 . 2 ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)) = {𝑜 ∣ ∃𝑚 ∈ (𝑇𝑅)𝑜 = (𝑚𝐽)}
137133, 134, 1363eqtr4i 2854 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 2799  wne 3016  wral 3138  wrex 3139  {crab 3142  cdif 3932  cun 3933  cin 3934  wss 3935  c0 4290  𝒫 cpw 4537  {csn 4559   class class class wbr 5058  {copab 5120  cmpt 5138   × cxp 5547  ccnv 5548  dom cdm 5549  ran crn 5550  cres 5551  cima 5552  Fun wfun 6343   Fn wfn 6344  wf 6345  cfv 6349  (class class class)co 7145  cmpo 7147   supp csupp 7821  m cmap 8396  Fincfn 8498  0cc0 10526  1c1 10527   · cmul 10531  cle 10665  cn 11627  2c2 11681  0cn0 11886  cz 11970  cexp 13419  Σcsu 15032  cdvds 15597
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 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602
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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-neg 10862  df-nn 11628  df-2 11689  df-n0 11887  df-z 11971  df-dvds 15598
This theorem is referenced by:  eulerpartgbij  31530
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