Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eulerpartgbij Structured version   Visualization version   GIF version

Theorem eulerpartgbij 29761
 Description: Lemma for eulerpart 29771: The 𝐺 function is a bijection. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
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}
eulerpart.g 𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
Assertion
Ref Expression
eulerpartgbij 𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅)
Distinct variable groups:   𝑓,𝑔,𝑘,𝑛,𝑜,𝑥,𝑦,𝑧   𝑜,𝐹   𝑓,𝑟,𝐽,𝑜,𝑥,𝑦   𝑜,𝑀,𝑟   𝑓,𝑁,𝑔,𝑥   𝑃,𝑔   𝑅,𝑓,𝑜   𝑜,𝐻,𝑟   𝑇,𝑓,𝑜
Allowed substitution hints:   𝐷(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑘,𝑛,𝑜,𝑟)   𝑅(𝑥,𝑦,𝑧,𝑔,𝑘,𝑛,𝑟)   𝑇(𝑥,𝑦,𝑧,𝑔,𝑘,𝑛,𝑟)   𝐹(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐺(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛)   𝐽(𝑧,𝑔,𝑘,𝑛)   𝑀(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛)   𝑁(𝑦,𝑧,𝑘,𝑛,𝑜,𝑟)   𝑂(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)

Proof of Theorem eulerpartgbij
Dummy variables 𝑎 𝑚 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnex 10903 . . . . 5 ℕ ∈ V
2 indf1ofs 29415 . . . . 5 (ℕ ∈ V → ((𝟭‘ℕ) ↾ Fin):(𝒫 ℕ ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑𝑚 ℕ) ∣ (𝑓 “ {1}) ∈ Fin})
31, 2ax-mp 5 . . . 4 ((𝟭‘ℕ) ↾ Fin):(𝒫 ℕ ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑𝑚 ℕ) ∣ (𝑓 “ {1}) ∈ Fin}
4 incom 3767 . . . . . . 7 (({0, 1} ↑𝑚 ℕ) ∩ {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}) = ({𝑓 ∣ (𝑓 “ ℕ) ∈ Fin} ∩ ({0, 1} ↑𝑚 ℕ))
5 eulerpart.r . . . . . . . 8 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
65ineq2i 3773 . . . . . . 7 (({0, 1} ↑𝑚 ℕ) ∩ 𝑅) = (({0, 1} ↑𝑚 ℕ) ∩ {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin})
7 dfrab2 3862 . . . . . . 7 {𝑓 ∈ ({0, 1} ↑𝑚 ℕ) ∣ (𝑓 “ ℕ) ∈ Fin} = ({𝑓 ∣ (𝑓 “ ℕ) ∈ Fin} ∩ ({0, 1} ↑𝑚 ℕ))
84, 6, 73eqtr4i 2642 . . . . . 6 (({0, 1} ↑𝑚 ℕ) ∩ 𝑅) = {𝑓 ∈ ({0, 1} ↑𝑚 ℕ) ∣ (𝑓 “ ℕ) ∈ Fin}
9 elmapfun 7767 . . . . . . . . 9 (𝑓 ∈ ({0, 1} ↑𝑚 ℕ) → Fun 𝑓)
10 elmapi 7765 . . . . . . . . . 10 (𝑓 ∈ ({0, 1} ↑𝑚 ℕ) → 𝑓:ℕ⟶{0, 1})
11 frn 5966 . . . . . . . . . 10 (𝑓:ℕ⟶{0, 1} → ran 𝑓 ⊆ {0, 1})
1210, 11syl 17 . . . . . . . . 9 (𝑓 ∈ ({0, 1} ↑𝑚 ℕ) → ran 𝑓 ⊆ {0, 1})
13 fimacnvinrn2 6257 . . . . . . . . . 10 ((Fun 𝑓 ∧ ran 𝑓 ⊆ {0, 1}) → (𝑓 “ ℕ) = (𝑓 “ (ℕ ∩ {0, 1})))
14 df-pr 4128 . . . . . . . . . . . . . 14 {0, 1} = ({0} ∪ {1})
1514ineq2i 3773 . . . . . . . . . . . . 13 (ℕ ∩ {0, 1}) = (ℕ ∩ ({0} ∪ {1}))
16 indi 3832 . . . . . . . . . . . . 13 (ℕ ∩ ({0} ∪ {1})) = ((ℕ ∩ {0}) ∪ (ℕ ∩ {1}))
17 0nnn 10929 . . . . . . . . . . . . . . 15 ¬ 0 ∈ ℕ
18 disjsn 4192 . . . . . . . . . . . . . . 15 ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
1917, 18mpbir 220 . . . . . . . . . . . . . 14 (ℕ ∩ {0}) = ∅
20 1nn 10908 . . . . . . . . . . . . . . . . 17 1 ∈ ℕ
21 1ex 9914 . . . . . . . . . . . . . . . . . 18 1 ∈ V
2221snss 4259 . . . . . . . . . . . . . . . . 17 (1 ∈ ℕ ↔ {1} ⊆ ℕ)
2320, 22mpbi 219 . . . . . . . . . . . . . . . 16 {1} ⊆ ℕ
24 dfss 3555 . . . . . . . . . . . . . . . 16 ({1} ⊆ ℕ ↔ {1} = ({1} ∩ ℕ))
2523, 24mpbi 219 . . . . . . . . . . . . . . 15 {1} = ({1} ∩ ℕ)
26 incom 3767 . . . . . . . . . . . . . . 15 ({1} ∩ ℕ) = (ℕ ∩ {1})
2725, 26eqtr2i 2633 . . . . . . . . . . . . . 14 (ℕ ∩ {1}) = {1}
2819, 27uneq12i 3727 . . . . . . . . . . . . 13 ((ℕ ∩ {0}) ∪ (ℕ ∩ {1})) = (∅ ∪ {1})
2915, 16, 283eqtri 2636 . . . . . . . . . . . 12 (ℕ ∩ {0, 1}) = (∅ ∪ {1})
30 uncom 3719 . . . . . . . . . . . 12 (∅ ∪ {1}) = ({1} ∪ ∅)
31 un0 3919 . . . . . . . . . . . 12 ({1} ∪ ∅) = {1}
3229, 30, 313eqtri 2636 . . . . . . . . . . 11 (ℕ ∩ {0, 1}) = {1}
3332imaeq2i 5383 . . . . . . . . . 10 (𝑓 “ (ℕ ∩ {0, 1})) = (𝑓 “ {1})
3413, 33syl6eq 2660 . . . . . . . . 9 ((Fun 𝑓 ∧ ran 𝑓 ⊆ {0, 1}) → (𝑓 “ ℕ) = (𝑓 “ {1}))
359, 12, 34syl2anc 691 . . . . . . . 8 (𝑓 ∈ ({0, 1} ↑𝑚 ℕ) → (𝑓 “ ℕ) = (𝑓 “ {1}))
3635eleq1d 2672 . . . . . . 7 (𝑓 ∈ ({0, 1} ↑𝑚 ℕ) → ((𝑓 “ ℕ) ∈ Fin ↔ (𝑓 “ {1}) ∈ Fin))
3736rabbiia 3161 . . . . . 6 {𝑓 ∈ ({0, 1} ↑𝑚 ℕ) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ ({0, 1} ↑𝑚 ℕ) ∣ (𝑓 “ {1}) ∈ Fin}
388, 37eqtr2i 2633 . . . . 5 {𝑓 ∈ ({0, 1} ↑𝑚 ℕ) ∣ (𝑓 “ {1}) ∈ Fin} = (({0, 1} ↑𝑚 ℕ) ∩ 𝑅)
39 f1oeq3 6042 . . . . 5 ({𝑓 ∈ ({0, 1} ↑𝑚 ℕ) ∣ (𝑓 “ {1}) ∈ Fin} = (({0, 1} ↑𝑚 ℕ) ∩ 𝑅) → (((𝟭‘ℕ) ↾ Fin):(𝒫 ℕ ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑𝑚 ℕ) ∣ (𝑓 “ {1}) ∈ Fin} ↔ ((𝟭‘ℕ) ↾ Fin):(𝒫 ℕ ∩ Fin)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅)))
4038, 39ax-mp 5 . . . 4 (((𝟭‘ℕ) ↾ Fin):(𝒫 ℕ ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑𝑚 ℕ) ∣ (𝑓 “ {1}) ∈ Fin} ↔ ((𝟭‘ℕ) ↾ Fin):(𝒫 ℕ ∩ Fin)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅))
413, 40mpbi 219 . . 3 ((𝟭‘ℕ) ↾ Fin):(𝒫 ℕ ∩ Fin)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅)
42 eulerpart.j . . . . . . 7 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
43 eulerpart.f . . . . . . 7 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
4442, 43oddpwdc 29743 . . . . . 6 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
45 f1opwfi 8153 . . . . . 6 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → (𝑎 ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin) ↦ (𝐹𝑎)):(𝒫 (𝐽 × ℕ0) ∩ Fin)–1-1-onto→(𝒫 ℕ ∩ Fin))
4644, 45ax-mp 5 . . . . 5 (𝑎 ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin) ↦ (𝐹𝑎)):(𝒫 (𝐽 × ℕ0) ∩ Fin)–1-1-onto→(𝒫 ℕ ∩ Fin)
47 eulerpart.p . . . . . . . 8 𝑃 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
48 eulerpart.o . . . . . . . 8 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
49 eulerpart.d . . . . . . . 8 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
50 eulerpart.h . . . . . . . 8 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
51 eulerpart.m . . . . . . . 8 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
5247, 48, 49, 42, 43, 50, 51eulerpartlem1 29756 . . . . . . 7 𝑀:𝐻1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)
53 bitsf1o 15005 . . . . . . . . . . . . . 14 (bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin)
5453a1i 11 . . . . . . . . . . . . 13 (⊤ → (bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin))
5542, 1rabex2 4742 . . . . . . . . . . . . . 14 𝐽 ∈ V
5655a1i 11 . . . . . . . . . . . . 13 (⊤ → 𝐽 ∈ V)
57 nn0ex 11175 . . . . . . . . . . . . . 14 0 ∈ V
5857a1i 11 . . . . . . . . . . . . 13 (⊤ → ℕ0 ∈ V)
5957pwex 4774 . . . . . . . . . . . . . . 15 𝒫 ℕ0 ∈ V
6059inex1 4727 . . . . . . . . . . . . . 14 (𝒫 ℕ0 ∩ Fin) ∈ V
6160a1i 11 . . . . . . . . . . . . 13 (⊤ → (𝒫 ℕ0 ∩ Fin) ∈ V)
62 0nn0 11184 . . . . . . . . . . . . . 14 0 ∈ ℕ0
6362a1i 11 . . . . . . . . . . . . 13 (⊤ → 0 ∈ ℕ0)
64 fvres 6117 . . . . . . . . . . . . . . 15 (0 ∈ ℕ0 → ((bits ↾ ℕ0)‘0) = (bits‘0))
6562, 64ax-mp 5 . . . . . . . . . . . . . 14 ((bits ↾ ℕ0)‘0) = (bits‘0)
66 0bits 14999 . . . . . . . . . . . . . 14 (bits‘0) = ∅
6765, 66eqtr2i 2633 . . . . . . . . . . . . 13 ∅ = ((bits ↾ ℕ0)‘0)
68 elmapi 7765 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ (ℕ0𝑚 𝐽) → 𝑓:𝐽⟶ℕ0)
69 frnnn0supp 11226 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ V ∧ 𝑓:𝐽⟶ℕ0) → (𝑓 supp 0) = (𝑓 “ ℕ))
7055, 68, 69sylancr 694 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (ℕ0𝑚 𝐽) → (𝑓 supp 0) = (𝑓 “ ℕ))
7170eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑓 ∈ (ℕ0𝑚 𝐽) → ((𝑓 supp 0) ∈ Fin ↔ (𝑓 “ ℕ) ∈ Fin))
7271rabbiia 3161 . . . . . . . . . . . . . 14 {𝑓 ∈ (ℕ0𝑚 𝐽) ∣ (𝑓 supp 0) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐽) ∣ (𝑓 “ ℕ) ∈ Fin}
73 elmapfun 7767 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (ℕ0𝑚 𝐽) → Fun 𝑓)
74 vex 3176 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
75 funisfsupp 8163 . . . . . . . . . . . . . . . . 17 ((Fun 𝑓𝑓 ∈ V ∧ 0 ∈ ℕ0) → (𝑓 finSupp 0 ↔ (𝑓 supp 0) ∈ Fin))
7674, 62, 75mp3an23 1408 . . . . . . . . . . . . . . . 16 (Fun 𝑓 → (𝑓 finSupp 0 ↔ (𝑓 supp 0) ∈ Fin))
7773, 76syl 17 . . . . . . . . . . . . . . 15 (𝑓 ∈ (ℕ0𝑚 𝐽) → (𝑓 finSupp 0 ↔ (𝑓 supp 0) ∈ Fin))
7877rabbiia 3161 . . . . . . . . . . . . . 14 {𝑓 ∈ (ℕ0𝑚 𝐽) ∣ 𝑓 finSupp 0} = {𝑓 ∈ (ℕ0𝑚 𝐽) ∣ (𝑓 supp 0) ∈ Fin}
79 incom 3767 . . . . . . . . . . . . . . 15 ({𝑓 ∣ (𝑓 “ ℕ) ∈ Fin} ∩ (ℕ0𝑚 𝐽)) = ((ℕ0𝑚 𝐽) ∩ {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin})
80 dfrab2 3862 . . . . . . . . . . . . . . 15 {𝑓 ∈ (ℕ0𝑚 𝐽) ∣ (𝑓 “ ℕ) ∈ Fin} = ({𝑓 ∣ (𝑓 “ ℕ) ∈ Fin} ∩ (ℕ0𝑚 𝐽))
815ineq2i 3773 . . . . . . . . . . . . . . 15 ((ℕ0𝑚 𝐽) ∩ 𝑅) = ((ℕ0𝑚 𝐽) ∩ {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin})
8279, 80, 813eqtr4ri 2643 . . . . . . . . . . . . . 14 ((ℕ0𝑚 𝐽) ∩ 𝑅) = {𝑓 ∈ (ℕ0𝑚 𝐽) ∣ (𝑓 “ ℕ) ∈ Fin}
8372, 78, 823eqtr4ri 2643 . . . . . . . . . . . . 13 ((ℕ0𝑚 𝐽) ∩ 𝑅) = {𝑓 ∈ (ℕ0𝑚 𝐽) ∣ 𝑓 finSupp 0}
84 elmapfun 7767 . . . . . . . . . . . . . . 15 (𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) → Fun 𝑟)
85 vex 3176 . . . . . . . . . . . . . . . . 17 𝑟 ∈ V
86 0ex 4718 . . . . . . . . . . . . . . . . 17 ∅ ∈ V
87 funisfsupp 8163 . . . . . . . . . . . . . . . . 17 ((Fun 𝑟𝑟 ∈ V ∧ ∅ ∈ V) → (𝑟 finSupp ∅ ↔ (𝑟 supp ∅) ∈ Fin))
8885, 86, 87mp3an23 1408 . . . . . . . . . . . . . . . 16 (Fun 𝑟 → (𝑟 finSupp ∅ ↔ (𝑟 supp ∅) ∈ Fin))
8988bicomd 212 . . . . . . . . . . . . . . 15 (Fun 𝑟 → ((𝑟 supp ∅) ∈ Fin ↔ 𝑟 finSupp ∅))
9084, 89syl 17 . . . . . . . . . . . . . 14 (𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) → ((𝑟 supp ∅) ∈ Fin ↔ 𝑟 finSupp ∅))
9190rabbiia 3161 . . . . . . . . . . . . 13 {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ 𝑟 finSupp ∅}
9254, 56, 58, 61, 63, 67, 83, 91fcobijfs 28889 . . . . . . . . . . . 12 (⊤ → (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ ((bits ↾ ℕ0) ∘ 𝑓)):((ℕ0𝑚 𝐽) ∩ 𝑅)–1-1-onto→{𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin})
93 elinel1 3761 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) → 𝑓 ∈ (ℕ0𝑚 𝐽))
94 frn 5966 . . . . . . . . . . . . . . . . 17 (𝑓:𝐽⟶ℕ0 → ran 𝑓 ⊆ ℕ0)
95 cores 5555 . . . . . . . . . . . . . . . . 17 (ran 𝑓 ⊆ ℕ0 → ((bits ↾ ℕ0) ∘ 𝑓) = (bits ∘ 𝑓))
9668, 94, 953syl 18 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (ℕ0𝑚 𝐽) → ((bits ↾ ℕ0) ∘ 𝑓) = (bits ∘ 𝑓))
9793, 96syl 17 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) → ((bits ↾ ℕ0) ∘ 𝑓) = (bits ∘ 𝑓))
9897mpteq2ia 4668 . . . . . . . . . . . . . 14 (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ ((bits ↾ ℕ0) ∘ 𝑓)) = (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓))
9998eqcomi 2619 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)) = (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ ((bits ↾ ℕ0) ∘ 𝑓))
100 f1oeq1 6040 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)) = (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ ((bits ↾ ℕ0) ∘ 𝑓)) → ((𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)):((ℕ0𝑚 𝐽) ∩ 𝑅)–1-1-onto→{𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} ↔ (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ ((bits ↾ ℕ0) ∘ 𝑓)):((ℕ0𝑚 𝐽) ∩ 𝑅)–1-1-onto→{𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}))
10199, 100mp1i 13 . . . . . . . . . . . 12 (⊤ → ((𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)):((ℕ0𝑚 𝐽) ∩ 𝑅)–1-1-onto→{𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} ↔ (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ ((bits ↾ ℕ0) ∘ 𝑓)):((ℕ0𝑚 𝐽) ∩ 𝑅)–1-1-onto→{𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}))
10292, 101mpbird 246 . . . . . . . . . . 11 (⊤ → (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)):((ℕ0𝑚 𝐽) ∩ 𝑅)–1-1-onto→{𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin})
103102trud 1484 . . . . . . . . . 10 (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)):((ℕ0𝑚 𝐽) ∩ 𝑅)–1-1-onto→{𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
104 ssrab2 3650 . . . . . . . . . . . . . . . 16 {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
10542, 104eqsstri 3598 . . . . . . . . . . . . . . 15 𝐽 ⊆ ℕ
1061, 57, 1053pm3.2i 1232 . . . . . . . . . . . . . 14 (ℕ ∈ V ∧ ℕ0 ∈ V ∧ 𝐽 ⊆ ℕ)
107 eulerpart.t . . . . . . . . . . . . . . . 16 𝑇 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
108 cnveq 5218 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑜𝑓 = 𝑜)
109 dfn2 11182 . . . . . . . . . . . . . . . . . . . 20 ℕ = (ℕ0 ∖ {0})
110109a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑜 → ℕ = (ℕ0 ∖ {0}))
111108, 110imaeq12d 5386 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑜 → (𝑓 “ ℕ) = (𝑜 “ (ℕ0 ∖ {0})))
112111sseq1d 3595 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑜 → ((𝑓 “ ℕ) ⊆ 𝐽 ↔ (𝑜 “ (ℕ0 ∖ {0})) ⊆ 𝐽))
113112cbvrabv 3172 . . . . . . . . . . . . . . . 16 {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽} = {𝑜 ∈ (ℕ0𝑚 ℕ) ∣ (𝑜 “ (ℕ0 ∖ {0})) ⊆ 𝐽}
114107, 113eqtri 2632 . . . . . . . . . . . . . . 15 𝑇 = {𝑜 ∈ (ℕ0𝑚 ℕ) ∣ (𝑜 “ (ℕ0 ∖ {0})) ⊆ 𝐽}
115 eqid 2610 . . . . . . . . . . . . . . 15 (𝑜𝑇 ↦ (𝑜𝐽)) = (𝑜𝑇 ↦ (𝑜𝐽))
116114, 115resf1o 28893 . . . . . . . . . . . . . 14 (((ℕ ∈ V ∧ ℕ0 ∈ V ∧ 𝐽 ⊆ ℕ) ∧ 0 ∈ ℕ0) → (𝑜𝑇 ↦ (𝑜𝐽)):𝑇1-1-onto→(ℕ0𝑚 𝐽))
117106, 62, 116mp2an 704 . . . . . . . . . . . . 13 (𝑜𝑇 ↦ (𝑜𝐽)):𝑇1-1-onto→(ℕ0𝑚 𝐽)
118 f1of1 6049 . . . . . . . . . . . . 13 ((𝑜𝑇 ↦ (𝑜𝐽)):𝑇1-1-onto→(ℕ0𝑚 𝐽) → (𝑜𝑇 ↦ (𝑜𝐽)):𝑇1-1→(ℕ0𝑚 𝐽))
119117, 118ax-mp 5 . . . . . . . . . . . 12 (𝑜𝑇 ↦ (𝑜𝐽)):𝑇1-1→(ℕ0𝑚 𝐽)
120 inss1 3795 . . . . . . . . . . . 12 (𝑇𝑅) ⊆ 𝑇
121 f1ores 6064 . . . . . . . . . . . 12 (((𝑜𝑇 ↦ (𝑜𝐽)):𝑇1-1→(ℕ0𝑚 𝐽) ∧ (𝑇𝑅) ⊆ 𝑇) → ((𝑜𝑇 ↦ (𝑜𝐽)) ↾ (𝑇𝑅)):(𝑇𝑅)–1-1-onto→((𝑜𝑇 ↦ (𝑜𝐽)) “ (𝑇𝑅)))
122119, 120, 121mp2an 704 . . . . . . . . . . 11 ((𝑜𝑇 ↦ (𝑜𝐽)) ↾ (𝑇𝑅)):(𝑇𝑅)–1-1-onto→((𝑜𝑇 ↦ (𝑜𝐽)) “ (𝑇𝑅))
123 vex 3176 . . . . . . . . . . . . . . . . . 18 𝑜 ∈ V
124123resex 5363 . . . . . . . . . . . . . . . . 17 (𝑜𝐽) ∈ V
125124, 115fnmpti 5935 . . . . . . . . . . . . . . . 16 (𝑜𝑇 ↦ (𝑜𝐽)) Fn 𝑇
126 fvelimab 6163 . . . . . . . . . . . . . . . 16 (((𝑜𝑇 ↦ (𝑜𝐽)) Fn 𝑇 ∧ (𝑇𝑅) ⊆ 𝑇) → (𝑓 ∈ ((𝑜𝑇 ↦ (𝑜𝐽)) “ (𝑇𝑅)) ↔ ∃𝑚 ∈ (𝑇𝑅)((𝑜𝑇 ↦ (𝑜𝐽))‘𝑚) = 𝑓))
127125, 120, 126mp2an 704 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((𝑜𝑇 ↦ (𝑜𝐽)) “ (𝑇𝑅)) ↔ ∃𝑚 ∈ (𝑇𝑅)((𝑜𝑇 ↦ (𝑜𝐽))‘𝑚) = 𝑓)
128 eqid 2610 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)) = (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
129 vex 3176 . . . . . . . . . . . . . . . . . 18 𝑚 ∈ V
130129resex 5363 . . . . . . . . . . . . . . . . 17 (𝑚𝐽) ∈ V
131128, 130elrnmpti 5297 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)) ↔ ∃𝑚 ∈ (𝑇𝑅)𝑓 = (𝑚𝐽))
13247, 48, 49, 42, 43, 50, 51, 5, 107eulerpartlemt 29760 . . . . . . . . . . . . . . . . 17 ((ℕ0𝑚 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
133132eleq2i 2680 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↔ 𝑓 ∈ ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽)))
134 elinel1 3761 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ (𝑇𝑅) → 𝑚𝑇)
135115fvtresfn 6193 . . . . . . . . . . . . . . . . . . . 20 (𝑚𝑇 → ((𝑜𝑇 ↦ (𝑜𝐽))‘𝑚) = (𝑚𝐽))
136135eqeq1d 2612 . . . . . . . . . . . . . . . . . . 19 (𝑚𝑇 → (((𝑜𝑇 ↦ (𝑜𝐽))‘𝑚) = 𝑓 ↔ (𝑚𝐽) = 𝑓))
137134, 136syl 17 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ (𝑇𝑅) → (((𝑜𝑇 ↦ (𝑜𝐽))‘𝑚) = 𝑓 ↔ (𝑚𝐽) = 𝑓))
138 eqcom 2617 . . . . . . . . . . . . . . . . . 18 ((𝑚𝐽) = 𝑓𝑓 = (𝑚𝐽))
139137, 138syl6bb 275 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (𝑇𝑅) → (((𝑜𝑇 ↦ (𝑜𝐽))‘𝑚) = 𝑓𝑓 = (𝑚𝐽)))
140139rexbiia 3022 . . . . . . . . . . . . . . . 16 (∃𝑚 ∈ (𝑇𝑅)((𝑜𝑇 ↦ (𝑜𝐽))‘𝑚) = 𝑓 ↔ ∃𝑚 ∈ (𝑇𝑅)𝑓 = (𝑚𝐽))
141131, 133, 1403bitr4ri 292 . . . . . . . . . . . . . . 15 (∃𝑚 ∈ (𝑇𝑅)((𝑜𝑇 ↦ (𝑜𝐽))‘𝑚) = 𝑓𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅))
142127, 141bitri 263 . . . . . . . . . . . . . 14 (𝑓 ∈ ((𝑜𝑇 ↦ (𝑜𝐽)) “ (𝑇𝑅)) ↔ 𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅))
143142eqriv 2607 . . . . . . . . . . . . 13 ((𝑜𝑇 ↦ (𝑜𝐽)) “ (𝑇𝑅)) = ((ℕ0𝑚 𝐽) ∩ 𝑅)
144 f1oeq3 6042 . . . . . . . . . . . . 13 (((𝑜𝑇 ↦ (𝑜𝐽)) “ (𝑇𝑅)) = ((ℕ0𝑚 𝐽) ∩ 𝑅) → (((𝑜𝑇 ↦ (𝑜𝐽)) ↾ (𝑇𝑅)):(𝑇𝑅)–1-1-onto→((𝑜𝑇 ↦ (𝑜𝐽)) “ (𝑇𝑅)) ↔ ((𝑜𝑇 ↦ (𝑜𝐽)) ↾ (𝑇𝑅)):(𝑇𝑅)–1-1-onto→((ℕ0𝑚 𝐽) ∩ 𝑅)))
145143, 144ax-mp 5 . . . . . . . . . . . 12 (((𝑜𝑇 ↦ (𝑜𝐽)) ↾ (𝑇𝑅)):(𝑇𝑅)–1-1-onto→((𝑜𝑇 ↦ (𝑜𝐽)) “ (𝑇𝑅)) ↔ ((𝑜𝑇 ↦ (𝑜𝐽)) ↾ (𝑇𝑅)):(𝑇𝑅)–1-1-onto→((ℕ0𝑚 𝐽) ∩ 𝑅))
146 resmpt 5369 . . . . . . . . . . . . 13 ((𝑇𝑅) ⊆ 𝑇 → ((𝑜𝑇 ↦ (𝑜𝐽)) ↾ (𝑇𝑅)) = (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)))
147 f1oeq1 6040 . . . . . . . . . . . . 13 (((𝑜𝑇 ↦ (𝑜𝐽)) ↾ (𝑇𝑅)) = (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)) → (((𝑜𝑇 ↦ (𝑜𝐽)) ↾ (𝑇𝑅)):(𝑇𝑅)–1-1-onto→((ℕ0𝑚 𝐽) ∩ 𝑅) ↔ (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)):(𝑇𝑅)–1-1-onto→((ℕ0𝑚 𝐽) ∩ 𝑅)))
148120, 146, 147mp2b 10 . . . . . . . . . . . 12 (((𝑜𝑇 ↦ (𝑜𝐽)) ↾ (𝑇𝑅)):(𝑇𝑅)–1-1-onto→((ℕ0𝑚 𝐽) ∩ 𝑅) ↔ (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)):(𝑇𝑅)–1-1-onto→((ℕ0𝑚 𝐽) ∩ 𝑅))
149145, 148bitri 263 . . . . . . . . . . 11 (((𝑜𝑇 ↦ (𝑜𝐽)) ↾ (𝑇𝑅)):(𝑇𝑅)–1-1-onto→((𝑜𝑇 ↦ (𝑜𝐽)) “ (𝑇𝑅)) ↔ (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)):(𝑇𝑅)–1-1-onto→((ℕ0𝑚 𝐽) ∩ 𝑅))
150122, 149mpbi 219 . . . . . . . . . 10 (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)):(𝑇𝑅)–1-1-onto→((ℕ0𝑚 𝐽) ∩ 𝑅)
151 f1oco 6072 . . . . . . . . . 10 (((𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)):((ℕ0𝑚 𝐽) ∩ 𝑅)–1-1-onto→{𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} ∧ (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)):(𝑇𝑅)–1-1-onto→((ℕ0𝑚 𝐽) ∩ 𝑅)) → ((𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽))):(𝑇𝑅)–1-1-onto→{𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin})
152103, 150, 151mp2an 704 . . . . . . . . 9 ((𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽))):(𝑇𝑅)–1-1-onto→{𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
153 f1of 6050 . . . . . . . . . . . . . 14 ((𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)):(𝑇𝑅)–1-1-onto→((ℕ0𝑚 𝐽) ∩ 𝑅) → (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)):(𝑇𝑅)⟶((ℕ0𝑚 𝐽) ∩ 𝑅))
154 eqid 2610 . . . . . . . . . . . . . . . 16 (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)) = (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽))
155154fmpt 6289 . . . . . . . . . . . . . . 15 (∀𝑜 ∈ (𝑇𝑅)(𝑜𝐽) ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↔ (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)):(𝑇𝑅)⟶((ℕ0𝑚 𝐽) ∩ 𝑅))
156155biimpri 217 . . . . . . . . . . . . . 14 ((𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)):(𝑇𝑅)⟶((ℕ0𝑚 𝐽) ∩ 𝑅) → ∀𝑜 ∈ (𝑇𝑅)(𝑜𝐽) ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅))
157150, 153, 156mp2b 10 . . . . . . . . . . . . 13 𝑜 ∈ (𝑇𝑅)(𝑜𝐽) ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅)
158157a1i 11 . . . . . . . . . . . 12 (⊤ → ∀𝑜 ∈ (𝑇𝑅)(𝑜𝐽) ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅))
159 eqidd 2611 . . . . . . . . . . . 12 (⊤ → (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)) = (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽)))
160 eqidd 2611 . . . . . . . . . . . 12 (⊤ → (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)) = (𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)))
161 coeq2 5202 . . . . . . . . . . . 12 (𝑓 = (𝑜𝐽) → (bits ∘ 𝑓) = (bits ∘ (𝑜𝐽)))
162158, 159, 160, 161fmptcof 6304 . . . . . . . . . . 11 (⊤ → ((𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽))) = (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽))))
163162eqcomd 2616 . . . . . . . . . 10 (⊤ → (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽))) = ((𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽))))
164 eqidd 2611 . . . . . . . . . 10 (⊤ → (𝑇𝑅) = (𝑇𝑅))
16550a1i 11 . . . . . . . . . 10 (⊤ → 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin})
166163, 164, 165f1oeq123d 6046 . . . . . . . . 9 (⊤ → ((𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽))):(𝑇𝑅)–1-1-onto𝐻 ↔ ((𝑓 ∈ ((ℕ0𝑚 𝐽) ∩ 𝑅) ↦ (bits ∘ 𝑓)) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝑜𝐽))):(𝑇𝑅)–1-1-onto→{𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}))
167152, 166mpbiri 247 . . . . . . . 8 (⊤ → (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽))):(𝑇𝑅)–1-1-onto𝐻)
168167trud 1484 . . . . . . 7 (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽))):(𝑇𝑅)–1-1-onto𝐻
169 f1oco 6072 . . . . . . 7 ((𝑀:𝐻1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) ∧ (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽))):(𝑇𝑅)–1-1-onto𝐻) → (𝑀 ∘ (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽)))):(𝑇𝑅)–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin))
17052, 168, 169mp2an 704 . . . . . 6 (𝑀 ∘ (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽)))):(𝑇𝑅)–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)
171 eqidd 2611 . . . . . . . . . . 11 (⊤ → (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽))) = (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽))))
172 bitsf 14987 . . . . . . . . . . . . . 14 bits:ℤ⟶𝒫 ℕ0
173 zex 11263 . . . . . . . . . . . . . 14 ℤ ∈ V
174 fex 6394 . . . . . . . . . . . . . 14 ((bits:ℤ⟶𝒫 ℕ0 ∧ ℤ ∈ V) → bits ∈ V)
175172, 173, 174mp2an 704 . . . . . . . . . . . . 13 bits ∈ V
176175, 124coex 7011 . . . . . . . . . . . 12 (bits ∘ (𝑜𝐽)) ∈ V
177176a1i 11 . . . . . . . . . . 11 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅)) → (bits ∘ (𝑜𝐽)) ∈ V)
178171, 177fvmpt2d 6202 . . . . . . . . . 10 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅)) → ((𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽)))‘𝑜) = (bits ∘ (𝑜𝐽)))
179 f1of 6050 . . . . . . . . . . . 12 ((𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽))):(𝑇𝑅)–1-1-onto𝐻 → (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽))):(𝑇𝑅)⟶𝐻)
180167, 179syl 17 . . . . . . . . . . 11 (⊤ → (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽))):(𝑇𝑅)⟶𝐻)
181180ffvelrnda 6267 . . . . . . . . . 10 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅)) → ((𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽)))‘𝑜) ∈ 𝐻)
182178, 181eqeltrrd 2689 . . . . . . . . 9 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅)) → (bits ∘ (𝑜𝐽)) ∈ 𝐻)
183 f1ofn 6051 . . . . . . . . . . . 12 (𝑀:𝐻1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) → 𝑀 Fn 𝐻)
18452, 183ax-mp 5 . . . . . . . . . . 11 𝑀 Fn 𝐻
185 dffn5 6151 . . . . . . . . . . 11 (𝑀 Fn 𝐻𝑀 = (𝑟𝐻 ↦ (𝑀𝑟)))
186184, 185mpbi 219 . . . . . . . . . 10 𝑀 = (𝑟𝐻 ↦ (𝑀𝑟))
187186a1i 11 . . . . . . . . 9 (⊤ → 𝑀 = (𝑟𝐻 ↦ (𝑀𝑟)))
188 fveq2 6103 . . . . . . . . 9 (𝑟 = (bits ∘ (𝑜𝐽)) → (𝑀𝑟) = (𝑀‘(bits ∘ (𝑜𝐽))))
189182, 171, 187, 188fmptco 6303 . . . . . . . 8 (⊤ → (𝑀 ∘ (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽)))) = (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽)))))
190189trud 1484 . . . . . . 7 (𝑀 ∘ (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽)))) = (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽))))
191 f1oeq1 6040 . . . . . . 7 ((𝑀 ∘ (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽)))) = (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽)))) → ((𝑀 ∘ (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽)))):(𝑇𝑅)–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) ↔ (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽)))):(𝑇𝑅)–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)))
192190, 191ax-mp 5 . . . . . 6 ((𝑀 ∘ (𝑜 ∈ (𝑇𝑅) ↦ (bits ∘ (𝑜𝐽)))):(𝑇𝑅)–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) ↔ (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽)))):(𝑇𝑅)–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin))
193170, 192mpbi 219 . . . . 5 (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽)))):(𝑇𝑅)–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)
194 f1oco 6072 . . . . 5 (((𝑎 ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin) ↦ (𝐹𝑎)):(𝒫 (𝐽 × ℕ0) ∩ Fin)–1-1-onto→(𝒫 ℕ ∩ Fin) ∧ (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽)))):(𝑇𝑅)–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)) → ((𝑎 ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin) ↦ (𝐹𝑎)) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽))))):(𝑇𝑅)–1-1-onto→(𝒫 ℕ ∩ Fin))
19546, 193, 194mp2an 704 . . . 4 ((𝑎 ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin) ↦ (𝐹𝑎)) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽))))):(𝑇𝑅)–1-1-onto→(𝒫 ℕ ∩ Fin)
196 simpr 476 . . . . . . . . 9 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅)) → 𝑜 ∈ (𝑇𝑅))
197 fvex 6113 . . . . . . . . 9 (𝑀‘(bits ∘ (𝑜𝐽))) ∈ V
198 eqid 2610 . . . . . . . . . 10 (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽)))) = (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽))))
199198fvmpt2 6200 . . . . . . . . 9 ((𝑜 ∈ (𝑇𝑅) ∧ (𝑀‘(bits ∘ (𝑜𝐽))) ∈ V) → ((𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽))))‘𝑜) = (𝑀‘(bits ∘ (𝑜𝐽))))
200196, 197, 199sylancl 693 . . . . . . . 8 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅)) → ((𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽))))‘𝑜) = (𝑀‘(bits ∘ (𝑜𝐽))))
201 f1of 6050 . . . . . . . . . 10 ((𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽)))):(𝑇𝑅)–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) → (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽)))):(𝑇𝑅)⟶(𝒫 (𝐽 × ℕ0) ∩ Fin))
202193, 201mp1i 13 . . . . . . . . 9 (⊤ → (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽)))):(𝑇𝑅)⟶(𝒫 (𝐽 × ℕ0) ∩ Fin))
203202ffvelrnda 6267 . . . . . . . 8 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅)) → ((𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽))))‘𝑜) ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin))
204200, 203eqeltrrd 2689 . . . . . . 7 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅)) → (𝑀‘(bits ∘ (𝑜𝐽))) ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin))
205 eqidd 2611 . . . . . . 7 (⊤ → (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽)))) = (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽)))))
206 eqidd 2611 . . . . . . 7 (⊤ → (𝑎 ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin) ↦ (𝐹𝑎)) = (𝑎 ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin) ↦ (𝐹𝑎)))
207 imaeq2 5381 . . . . . . 7 (𝑎 = (𝑀‘(bits ∘ (𝑜𝐽))) → (𝐹𝑎) = (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))
208204, 205, 206, 207fmptco 6303 . . . . . 6 (⊤ → ((𝑎 ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin) ↦ (𝐹𝑎)) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽))))) = (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
209208trud 1484 . . . . 5 ((𝑎 ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin) ↦ (𝐹𝑎)) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽))))) = (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))
210 f1oeq1 6040 . . . . 5 (((𝑎 ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin) ↦ (𝐹𝑎)) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽))))) = (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))) → (((𝑎 ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin) ↦ (𝐹𝑎)) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽))))):(𝑇𝑅)–1-1-onto→(𝒫 ℕ ∩ Fin) ↔ (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))):(𝑇𝑅)–1-1-onto→(𝒫 ℕ ∩ Fin)))
211209, 210ax-mp 5 . . . 4 (((𝑎 ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin) ↦ (𝐹𝑎)) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝑀‘(bits ∘ (𝑜𝐽))))):(𝑇𝑅)–1-1-onto→(𝒫 ℕ ∩ Fin) ↔ (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))):(𝑇𝑅)–1-1-onto→(𝒫 ℕ ∩ Fin))
212195, 211mpbi 219 . . 3 (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))):(𝑇𝑅)–1-1-onto→(𝒫 ℕ ∩ Fin)
213 f1oco 6072 . . 3 ((((𝟭‘ℕ) ↾ Fin):(𝒫 ℕ ∩ Fin)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))):(𝑇𝑅)–1-1-onto→(𝒫 ℕ ∩ Fin)) → (((𝟭‘ℕ) ↾ Fin) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))):(𝑇𝑅)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅))
21441, 212, 213mp2an 704 . 2 (((𝟭‘ℕ) ↾ Fin) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))):(𝑇𝑅)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅)
215 eulerpart.g . . . 4 𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
21643mpt2exg 7134 . . . . . . . . . 10 ((𝐽 ∈ V ∧ ℕ0 ∈ V) → 𝐹 ∈ V)
21755, 57, 216mp2an 704 . . . . . . . . 9 𝐹 ∈ V
218 imaexg 6995 . . . . . . . . 9 (𝐹 ∈ V → (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))) ∈ V)
219217, 218ax-mp 5 . . . . . . . 8 (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))) ∈ V
220 eqid 2610 . . . . . . . . 9 (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))) = (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))
221220fvmpt2 6200 . . . . . . . 8 ((𝑜 ∈ (𝑇𝑅) ∧ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))) ∈ V) → ((𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))‘𝑜) = (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))
222196, 219, 221sylancl 693 . . . . . . 7 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅)) → ((𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))‘𝑜) = (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))
223 f1of 6050 . . . . . . . . 9 ((𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))):(𝑇𝑅)–1-1-onto→(𝒫 ℕ ∩ Fin) → (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))):(𝑇𝑅)⟶(𝒫 ℕ ∩ Fin))
224212, 223mp1i 13 . . . . . . . 8 (⊤ → (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))):(𝑇𝑅)⟶(𝒫 ℕ ∩ Fin))
225224ffvelrnda 6267 . . . . . . 7 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅)) → ((𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))‘𝑜) ∈ (𝒫 ℕ ∩ Fin))
226222, 225eqeltrrd 2689 . . . . . 6 ((⊤ ∧ 𝑜 ∈ (𝑇𝑅)) → (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))) ∈ (𝒫 ℕ ∩ Fin))
227 eqidd 2611 . . . . . 6 (⊤ → (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))) = (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
228 indf1o 29413 . . . . . . . . . . 11 (ℕ ∈ V → (𝟭‘ℕ):𝒫 ℕ–1-1-onto→({0, 1} ↑𝑚 ℕ))
229 f1ofn 6051 . . . . . . . . . . 11 ((𝟭‘ℕ):𝒫 ℕ–1-1-onto→({0, 1} ↑𝑚 ℕ) → (𝟭‘ℕ) Fn 𝒫 ℕ)
2301, 228, 229mp2b 10 . . . . . . . . . 10 (𝟭‘ℕ) Fn 𝒫 ℕ
231 dffn5 6151 . . . . . . . . . 10 ((𝟭‘ℕ) Fn 𝒫 ℕ ↔ (𝟭‘ℕ) = (𝑏 ∈ 𝒫 ℕ ↦ ((𝟭‘ℕ)‘𝑏)))
232230, 231mpbi 219 . . . . . . . . 9 (𝟭‘ℕ) = (𝑏 ∈ 𝒫 ℕ ↦ ((𝟭‘ℕ)‘𝑏))
233232reseq1i 5313 . . . . . . . 8 ((𝟭‘ℕ) ↾ Fin) = ((𝑏 ∈ 𝒫 ℕ ↦ ((𝟭‘ℕ)‘𝑏)) ↾ Fin)
234 resmpt3 5370 . . . . . . . 8 ((𝑏 ∈ 𝒫 ℕ ↦ ((𝟭‘ℕ)‘𝑏)) ↾ Fin) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ ((𝟭‘ℕ)‘𝑏))
235233, 234eqtri 2632 . . . . . . 7 ((𝟭‘ℕ) ↾ Fin) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ ((𝟭‘ℕ)‘𝑏))
236235a1i 11 . . . . . 6 (⊤ → ((𝟭‘ℕ) ↾ Fin) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ ((𝟭‘ℕ)‘𝑏)))
237 fveq2 6103 . . . . . 6 (𝑏 = (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))) → ((𝟭‘ℕ)‘𝑏) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
238226, 227, 236, 237fmptco 6303 . . . . 5 (⊤ → (((𝟭‘ℕ) ↾ Fin) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))))
239238trud 1484 . . . 4 (((𝟭‘ℕ) ↾ Fin) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
240215, 239eqtr4i 2635 . . 3 𝐺 = (((𝟭‘ℕ) ↾ Fin) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
241 f1oeq1 6040 . . 3 (𝐺 = (((𝟭‘ℕ) ↾ Fin) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))) → (𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅) ↔ (((𝟭‘ℕ) ↾ Fin) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))):(𝑇𝑅)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅)))
242240, 241ax-mp 5 . 2 (𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅) ↔ (((𝟭‘ℕ) ↾ Fin) ∘ (𝑜 ∈ (𝑇𝑅) ↦ (𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽)))))):(𝑇𝑅)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅))
243214, 242mpbir 220 1 𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  ran crn 5039   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   supp csupp 7182   ↑𝑚 cmap 7744  Fincfn 7841   finSupp cfsupp 8158  0cc0 9815  1c1 9816   · cmul 9820   ≤ cle 9954  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ℤcz 11254  ↑cexp 12722  Σcsu 14264   ∥ cdvds 14821  bitscbits 14979  𝟭cind 29400 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822  df-bits 14982  df-ind 29401 This theorem is referenced by:  eulerpartlemgf  29768  eulerpartlemgs2  29769  eulerpartlemn  29770
 Copyright terms: Public domain W3C validator