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Theorem eulerpartlemgh 31522
Description: Lemma for eulerpart 31526: The 𝐹 function is a bijection on the 𝑈 subsets. (Contributed by Thierry Arnoux, 15-Aug-2018.)
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))))))
eulerpartlemgh.1 𝑈 = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))
Assertion
Ref Expression
eulerpartlemgh (𝐴 ∈ (𝑇𝑅) → (𝐹𝑈):𝑈1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
Distinct variable groups:   𝑧,𝑡   𝑓,𝑔,𝑘,𝑛,𝑡,𝐴   𝑓,𝐽,𝑛,𝑡   𝑓,𝑁,𝑘,𝑛,𝑡   𝑛,𝑂,𝑡   𝑃,𝑔,𝑘   𝑅,𝑓,𝑘,𝑛,𝑡   𝑇,𝑛,𝑡   𝑥,𝑡,𝑦,𝑧   𝑓,𝑚,𝑥,𝑔,𝑘,𝑛,𝑡,𝐴   𝑛,𝐹,𝑡,𝑥   𝑦,𝑓,𝑛   𝑥,𝐽,𝑦   𝑡,𝑃
Allowed substitution hints:   𝐴(𝑦,𝑧,𝑜,𝑟)   𝐷(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑚,𝑛,𝑜,𝑟)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑚,𝑛,𝑜,𝑟)   𝑅(𝑥,𝑦,𝑧,𝑔,𝑚,𝑜,𝑟)   𝑇(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑚,𝑜,𝑟)   𝑈(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑚,𝑛,𝑜,𝑟)   𝐹(𝑦,𝑧,𝑓,𝑔,𝑘,𝑚,𝑜,𝑟)   𝐺(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑚,𝑛,𝑜,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑚,𝑛,𝑜,𝑟)   𝐽(𝑧,𝑔,𝑘,𝑚,𝑜,𝑟)   𝑀(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑚,𝑛,𝑜,𝑟)   𝑁(𝑥,𝑦,𝑧,𝑔,𝑚,𝑜,𝑟)   𝑂(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑚,𝑜,𝑟)

Proof of Theorem eulerpartlemgh
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 eulerpart.j . . . . 5 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
2 eulerpart.f . . . . 5 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
31, 2oddpwdc 31498 . . . 4 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4 f1of1 6611 . . . 4 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:(𝐽 × ℕ0)–1-1→ℕ)
53, 4ax-mp 5 . . 3 𝐹:(𝐽 × ℕ0)–1-1→ℕ
6 eulerpartlemgh.1 . . . 4 𝑈 = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))
7 iunss 4966 . . . . 5 ( 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0) ↔ ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8 inss2 4210 . . . . . . . 8 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
98sseli 3967 . . . . . . 7 (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) → 𝑡𝐽)
109snssd 4741 . . . . . 6 (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) → {𝑡} ⊆ 𝐽)
11 bitsss 15765 . . . . . 6 (bits‘(𝐴𝑡)) ⊆ ℕ0
12 xpss12 5569 . . . . . 6 (({𝑡} ⊆ 𝐽 ∧ (bits‘(𝐴𝑡)) ⊆ ℕ0) → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
1310, 11, 12sylancl 586 . . . . 5 (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
147, 13mprgbir 3158 . . . 4 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0)
156, 14eqsstri 4005 . . 3 𝑈 ⊆ (𝐽 × ℕ0)
16 f1ores 6626 . . 3 ((𝐹:(𝐽 × ℕ0)–1-1→ℕ ∧ 𝑈 ⊆ (𝐽 × ℕ0)) → (𝐹𝑈):𝑈1-1-onto→(𝐹𝑈))
175, 15, 16mp2an 688 . 2 (𝐹𝑈):𝑈1-1-onto→(𝐹𝑈)
18 simpr 485 . . . . . . . . . 10 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → ((2↑𝑛) · 𝑡) = 𝑝)
19 2nn 11699 . . . . . . . . . . . . . 14 2 ∈ ℕ
2019a1i 11 . . . . . . . . . . . . 13 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → 2 ∈ ℕ)
2111sseli 3967 . . . . . . . . . . . . . 14 (𝑛 ∈ (bits‘(𝐴𝑡)) → 𝑛 ∈ ℕ0)
2221adantl 482 . . . . . . . . . . . . 13 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → 𝑛 ∈ ℕ0)
2320, 22nnexpcld 13596 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → (2↑𝑛) ∈ ℕ)
24 simplr 765 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → 𝑡 ∈ ℕ)
2523, 24nnmulcld 11679 . . . . . . . . . . 11 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → ((2↑𝑛) · 𝑡) ∈ ℕ)
2625adantr 481 . . . . . . . . . 10 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → ((2↑𝑛) · 𝑡) ∈ ℕ)
2718, 26eqeltrrd 2919 . . . . . . . . 9 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → 𝑝 ∈ ℕ)
2827rexlimdva2 3292 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝𝑝 ∈ ℕ))
2928rexlimdva 3289 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝𝑝 ∈ ℕ))
3029pm4.71rd 563 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ (𝑝 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)))
31 rex0 4321 . . . . . . . . . . . . . . 15 ¬ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝
32 simplr 765 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → 𝑡 ∈ ℕ)
33 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ 𝑡 ∈ (𝐴 “ ℕ))
34 eulerpart.p . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
35 eulerpart.o . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
36 eulerpart.d . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
37 eulerpart.h . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
38 eulerpart.m . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
39 eulerpart.r . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
40 eulerpart.t . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
4134, 35, 36, 1, 2, 37, 38, 39, 40eulerpartlemt0 31513 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
4241simp1bi 1139 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0m ℕ))
43 elmapi 8418 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ (ℕ0m ℕ) → 𝐴:ℕ⟶ℕ0)
4442, 43syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
4544ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → 𝐴:ℕ⟶ℕ0)
46 ffn 6511 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴:ℕ⟶ℕ0𝐴 Fn ℕ)
47 elpreima 6824 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 Fn ℕ → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴𝑡) ∈ ℕ)))
4845, 46, 473syl 18 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴𝑡) ∈ ℕ)))
4933, 48mtbid 325 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ (𝑡 ∈ ℕ ∧ (𝐴𝑡) ∈ ℕ))
50 imnan 400 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℕ → ¬ (𝐴𝑡) ∈ ℕ) ↔ ¬ (𝑡 ∈ ℕ ∧ (𝐴𝑡) ∈ ℕ))
5149, 50sylibr 235 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (𝑡 ∈ ℕ → ¬ (𝐴𝑡) ∈ ℕ))
5232, 51mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ (𝐴𝑡) ∈ ℕ)
5345, 32ffvelrnd 6848 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (𝐴𝑡) ∈ ℕ0)
54 elnn0 11888 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑡) ∈ ℕ0 ↔ ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
5553, 54sylib 219 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
56 orel1 884 . . . . . . . . . . . . . . . . . . 19 (¬ (𝐴𝑡) ∈ ℕ → (((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0) → (𝐴𝑡) = 0))
5752, 55, 56sylc 65 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (𝐴𝑡) = 0)
5857fveq2d 6671 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (bits‘(𝐴𝑡)) = (bits‘0))
59 0bits 15778 . . . . . . . . . . . . . . . . 17 (bits‘0) = ∅
6058, 59syl6eq 2877 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (bits‘(𝐴𝑡)) = ∅)
6160rexeqdv 3422 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝))
6231, 61mtbiri 328 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
6362ex 413 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (¬ 𝑡 ∈ (𝐴 “ ℕ) → ¬ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
6463con4d 115 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝𝑡 ∈ (𝐴 “ ℕ)))
6564impr 455 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ (𝐴 “ ℕ))
66 eldif 3950 . . . . . . . . . . . . . . . . . . . 20 (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽))
6734, 35, 36, 1, 2, 37, 38, 39, 40eulerpartlemf 31514 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴𝑡) = 0)
6866, 67sylan2br 594 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽)) → (𝐴𝑡) = 0)
6968anassrs 468 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (𝐴𝑡) = 0)
7069fveq2d 6671 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (bits‘(𝐴𝑡)) = (bits‘0))
7170, 59syl6eq 2877 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (bits‘(𝐴𝑡)) = ∅)
7271rexeqdv 3422 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝))
7331, 72mtbiri 328 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → ¬ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
7473ex 413 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (¬ 𝑡𝐽 → ¬ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
7574con4d 115 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝𝑡𝐽))
7675impr 455 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡𝐽)
7765, 76elind 4175 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽))
78 simprr 769 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
7977, 78jca 512 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
8079ex 413 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → ((𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝) → (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)))
8180reximdv2 3276 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 → ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
82 ssrab2 4060 . . . . . . . . . 10 {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
831, 82eqsstri 4005 . . . . . . . . 9 𝐽 ⊆ ℕ
848, 83sstri 3980 . . . . . . . 8 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ
85 ssrexv 4038 . . . . . . . 8 (((𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ → (∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
8684, 85mp1i 13 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
8781, 86impbid 213 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
8830, 87bitr3d 282 . . . . 5 (𝐴 ∈ (𝑇𝑅) → ((𝑝 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝) ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
89 eqeq2 2838 . . . . . . . 8 (𝑚 = 𝑝 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
90892rexbidv 3305 . . . . . . 7 (𝑚 = 𝑝 → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
9190elrab 3684 . . . . . 6 (𝑝 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ↔ (𝑝 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
9291a1i 11 . . . . 5 (𝐴 ∈ (𝑇𝑅) → (𝑝 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ↔ (𝑝 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)))
936imaeq2i 5925 . . . . . . . . 9 (𝐹𝑈) = (𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))
94 imaiun 6998 . . . . . . . . 9 (𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡))))
9593, 94eqtri 2849 . . . . . . . 8 (𝐹𝑈) = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡))))
9695eleq2i 2909 . . . . . . 7 (𝑝 ∈ (𝐹𝑈) ↔ 𝑝 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))))
97 eliun 4921 . . . . . . 7 (𝑝 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))))
98 f1ofn 6613 . . . . . . . . . . . . 13 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹 Fn (𝐽 × ℕ0))
993, 98ax-mp 5 . . . . . . . . . . . 12 𝐹 Fn (𝐽 × ℕ0)
100 snssi 4740 . . . . . . . . . . . . 13 (𝑡𝐽 → {𝑡} ⊆ 𝐽)
101100, 11, 12sylancl 586 . . . . . . . . . . . 12 (𝑡𝐽 → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
102 ovelimab 7316 . . . . . . . . . . . 12 ((𝐹 Fn (𝐽 × ℕ0) ∧ ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0)) → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑥𝐹𝑛)))
10399, 101, 102sylancr 587 . . . . . . . . . . 11 (𝑡𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑥𝐹𝑛)))
104 vex 3503 . . . . . . . . . . . 12 𝑡 ∈ V
105 oveq1 7155 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (𝑥𝐹𝑛) = (𝑡𝐹𝑛))
106105eqeq2d 2837 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝑝 = (𝑥𝐹𝑛) ↔ 𝑝 = (𝑡𝐹𝑛)))
107106rexbidv 3302 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛)))
108104, 107rexsn 4619 . . . . . . . . . . 11 (∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛))
109103, 108syl6bb 288 . . . . . . . . . 10 (𝑡𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛)))
110 df-ov 7151 . . . . . . . . . . . . . . 15 (𝑡𝐹𝑛) = (𝐹‘⟨𝑡, 𝑛⟩)
111110eqeq1i 2831 . . . . . . . . . . . . . 14 ((𝑡𝐹𝑛) = 𝑝 ↔ (𝐹‘⟨𝑡, 𝑛⟩) = 𝑝)
112 eqcom 2833 . . . . . . . . . . . . . 14 ((𝑡𝐹𝑛) = 𝑝𝑝 = (𝑡𝐹𝑛))
113111, 112bitr3i 278 . . . . . . . . . . . . 13 ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝𝑝 = (𝑡𝐹𝑛))
114 opelxpi 5591 . . . . . . . . . . . . . . 15 ((𝑡𝐽𝑛 ∈ ℕ0) → ⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0))
1151, 2oddpwdcv 31499 . . . . . . . . . . . . . . . 16 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)))
116 vex 3503 . . . . . . . . . . . . . . . . . . 19 𝑛 ∈ V
117104, 116op2nd 7689 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨𝑡, 𝑛⟩) = 𝑛
118117oveq2i 7159 . . . . . . . . . . . . . . . . 17 (2↑(2nd ‘⟨𝑡, 𝑛⟩)) = (2↑𝑛)
119104, 116op1st 7688 . . . . . . . . . . . . . . . . 17 (1st ‘⟨𝑡, 𝑛⟩) = 𝑡
120118, 119oveq12i 7160 . . . . . . . . . . . . . . . 16 ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)) = ((2↑𝑛) · 𝑡)
121115, 120syl6eq 2877 . . . . . . . . . . . . . . 15 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
122114, 121syl 17 . . . . . . . . . . . . . 14 ((𝑡𝐽𝑛 ∈ ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
123122eqeq1d 2828 . . . . . . . . . . . . 13 ((𝑡𝐽𝑛 ∈ ℕ0) → ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
124113, 123syl5bbr 286 . . . . . . . . . . . 12 ((𝑡𝐽𝑛 ∈ ℕ0) → (𝑝 = (𝑡𝐹𝑛) ↔ ((2↑𝑛) · 𝑡) = 𝑝))
12521, 124sylan2 592 . . . . . . . . . . 11 ((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) → (𝑝 = (𝑡𝐹𝑛) ↔ ((2↑𝑛) · 𝑡) = 𝑝))
126125rexbidva 3301 . . . . . . . . . 10 (𝑡𝐽 → (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
127109, 126bitrd 280 . . . . . . . . 9 (𝑡𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
1289, 127syl 17 . . . . . . . 8 (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
129128rexbiia 3251 . . . . . . 7 (∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
13096, 97, 1293bitri 298 . . . . . 6 (𝑝 ∈ (𝐹𝑈) ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
131130a1i 11 . . . . 5 (𝐴 ∈ (𝑇𝑅) → (𝑝 ∈ (𝐹𝑈) ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
13288, 92, 1313bitr4rd 313 . . . 4 (𝐴 ∈ (𝑇𝑅) → (𝑝 ∈ (𝐹𝑈) ↔ 𝑝 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
133132eqrdv 2824 . . 3 (𝐴 ∈ (𝑇𝑅) → (𝐹𝑈) = {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
134 f1oeq3 6603 . . 3 ((𝐹𝑈) = {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} → ((𝐹𝑈):𝑈1-1-onto→(𝐹𝑈) ↔ (𝐹𝑈):𝑈1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
135133, 134syl 17 . 2 (𝐴 ∈ (𝑇𝑅) → ((𝐹𝑈):𝑈1-1-onto→(𝐹𝑈) ↔ (𝐹𝑈):𝑈1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
13617, 135mpbii 234 1 (𝐴 ∈ (𝑇𝑅) → (𝐹𝑈):𝑈1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wcel 2107  {cab 2804  wral 3143  wrex 3144  {crab 3147  cdif 3937  cin 3939  wss 3940  c0 4295  𝒫 cpw 4542  {csn 4564  cop 4570   ciun 4917   class class class wbr 5063  {copab 5125  cmpt 5143   × cxp 5552  ccnv 5553  cres 5556  cima 5557  ccom 5558   Fn wfn 6347  wf 6348  1-1wf1 6349  1-1-ontowf1o 6351  cfv 6352  (class class class)co 7148  cmpo 7150  1st c1st 7678  2nd c2nd 7679   supp csupp 7821  m cmap 8396  Fincfn 8498  0cc0 10526  1c1 10527   · cmul 10531  cle 10665  cn 11627  2c2 11681  0cn0 11886  cexp 13419  Σcsu 15032  cdvds 15597  bitscbits 15758  𝟭cind 31155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  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  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-fz 12883  df-fzo 13024  df-fl 13152  df-seq 13360  df-exp 13420  df-dvds 15598  df-bits 15761
This theorem is referenced by:  eulerpartlemgs2  31524
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