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Theorem eulerpartlemgh 34359
Description: Lemma for eulerpart 34363: 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 34335 . . . 4 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4 f1of1 6847 . . . 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 5049 . . . . 5 ( 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0) ↔ ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8 inss2 4245 . . . . . . . 8 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
98sseli 3990 . . . . . . 7 (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) → 𝑡𝐽)
109snssd 4813 . . . . . 6 (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) → {𝑡} ⊆ 𝐽)
11 bitsss 16459 . . . . . 6 (bits‘(𝐴𝑡)) ⊆ ℕ0
12 xpss12 5703 . . . . . 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 3065 . . . 4 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0)
156, 14eqsstri 4029 . . 3 𝑈 ⊆ (𝐽 × ℕ0)
16 f1ores 6862 . . 3 ((𝐹:(𝐽 × ℕ0)–1-1→ℕ ∧ 𝑈 ⊆ (𝐽 × ℕ0)) → (𝐹𝑈):𝑈1-1-onto→(𝐹𝑈))
175, 15, 16mp2an 692 . 2 (𝐹𝑈):𝑈1-1-onto→(𝐹𝑈)
18 simpr 484 . . . . . . . . . 10 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → ((2↑𝑛) · 𝑡) = 𝑝)
19 2nn 12336 . . . . . . . . . . . . . 14 2 ∈ ℕ
2019a1i 11 . . . . . . . . . . . . 13 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → 2 ∈ ℕ)
2111sseli 3990 . . . . . . . . . . . . . 14 (𝑛 ∈ (bits‘(𝐴𝑡)) → 𝑛 ∈ ℕ0)
2221adantl 481 . . . . . . . . . . . . 13 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → 𝑛 ∈ ℕ0)
2320, 22nnexpcld 14280 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → (2↑𝑛) ∈ ℕ)
24 simplr 769 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → 𝑡 ∈ ℕ)
2523, 24nnmulcld 12316 . . . . . . . . . . 11 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → ((2↑𝑛) · 𝑡) ∈ ℕ)
2625adantr 480 . . . . . . . . . 10 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → ((2↑𝑛) · 𝑡) ∈ ℕ)
2718, 26eqeltrrd 2839 . . . . . . . . 9 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → 𝑝 ∈ ℕ)
2827rexlimdva2 3154 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝𝑝 ∈ ℕ))
2928rexlimdva 3152 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝𝑝 ∈ ℕ))
3029pm4.71rd 562 . . . . . 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 4365 . . . . . . . . . . . . . . 15 ¬ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝
32 simplr 769 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → 𝑡 ∈ ℕ)
33 simpr 484 . . . . . . . . . . . . . . . . . . . . . 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 34350 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
4241simp1bi 1144 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0m ℕ))
43 elmapi 8887 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ (ℕ0m ℕ) → 𝐴:ℕ⟶ℕ0)
4442, 43syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
4544ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → 𝐴:ℕ⟶ℕ0)
46 ffn 6736 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴:ℕ⟶ℕ0𝐴 Fn ℕ)
47 elpreima 7077 . . . . . . . . . . . . . . . . . . . . . . 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 324 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ (𝑡 ∈ ℕ ∧ (𝐴𝑡) ∈ ℕ))
50 imnan 399 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℕ → ¬ (𝐴𝑡) ∈ ℕ) ↔ ¬ (𝑡 ∈ ℕ ∧ (𝐴𝑡) ∈ ℕ))
5149, 50sylibr 234 . . . . . . . . . . . . . . . . . . . 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, 32ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (𝐴𝑡) ∈ ℕ0)
54 elnn0 12525 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑡) ∈ ℕ0 ↔ ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
5553, 54sylib 218 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
56 orel1 888 . . . . . . . . . . . . . . . . . . 19 (¬ (𝐴𝑡) ∈ ℕ → (((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0) → (𝐴𝑡) = 0))
5752, 55, 56sylc 65 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (𝐴𝑡) = 0)
5857fveq2d 6910 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (bits‘(𝐴𝑡)) = (bits‘0))
59 0bits 16472 . . . . . . . . . . . . . . . . 17 (bits‘0) = ∅
6058, 59eqtrdi 2790 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (bits‘(𝐴𝑡)) = ∅)
6160rexeqdv 3324 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝))
6231, 61mtbiri 327 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
6362ex 412 . . . . . . . . . . . . 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 454 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ (𝐴 “ ℕ))
66 eldif 3972 . . . . . . . . . . . . . . . . . . . 20 (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽))
6734, 35, 36, 1, 2, 37, 38, 39, 40eulerpartlemf 34351 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴𝑡) = 0)
6866, 67sylan2br 595 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽)) → (𝐴𝑡) = 0)
6968anassrs 467 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (𝐴𝑡) = 0)
7069fveq2d 6910 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (bits‘(𝐴𝑡)) = (bits‘0))
7170, 59eqtrdi 2790 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (bits‘(𝐴𝑡)) = ∅)
7271rexeqdv 3324 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝))
7331, 72mtbiri 327 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → ¬ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
7473ex 412 . . . . . . . . . . . . 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 454 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡𝐽)
7765, 76elind 4209 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽))
78 simprr 773 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
7977, 78jca 511 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
8079ex 412 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → ((𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝) → (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)))
8180reximdv2 3161 . . . . . . 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 4089 . . . . . . . . . 10 {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
831, 82eqsstri 4029 . . . . . . . . 9 𝐽 ⊆ ℕ
848, 83sstri 4004 . . . . . . . 8 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ
85 ssrexv 4064 . . . . . . . 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 212 . . . . . 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 281 . . . . 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 2746 . . . . . . . 8 (𝑚 = 𝑝 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
90892rexbidv 3219 . . . . . . 7 (𝑚 = 𝑝 → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
9190elrab 3694 . . . . . 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 6077 . . . . . . . . 9 (𝐹𝑈) = (𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))
94 imaiun 7264 . . . . . . . . 9 (𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡))))
9593, 94eqtri 2762 . . . . . . . 8 (𝐹𝑈) = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡))))
9695eleq2i 2830 . . . . . . 7 (𝑝 ∈ (𝐹𝑈) ↔ 𝑝 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))))
97 eliun 4999 . . . . . . 7 (𝑝 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))))
98 f1ofn 6849 . . . . . . . . . . . . 13 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹 Fn (𝐽 × ℕ0))
993, 98ax-mp 5 . . . . . . . . . . . 12 𝐹 Fn (𝐽 × ℕ0)
100 snssi 4812 . . . . . . . . . . . . 13 (𝑡𝐽 → {𝑡} ⊆ 𝐽)
101100, 11, 12sylancl 586 . . . . . . . . . . . 12 (𝑡𝐽 → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
102 ovelimab 7610 . . . . . . . . . . . 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 3481 . . . . . . . . . . . 12 𝑡 ∈ V
105 oveq1 7437 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (𝑥𝐹𝑛) = (𝑡𝐹𝑛))
106105eqeq2d 2745 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝑝 = (𝑥𝐹𝑛) ↔ 𝑝 = (𝑡𝐹𝑛)))
107106rexbidv 3176 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛)))
108104, 107rexsn 4686 . . . . . . . . . . 11 (∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛))
109103, 108bitrdi 287 . . . . . . . . . 10 (𝑡𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛)))
110 df-ov 7433 . . . . . . . . . . . . . . 15 (𝑡𝐹𝑛) = (𝐹‘⟨𝑡, 𝑛⟩)
111110eqeq1i 2739 . . . . . . . . . . . . . 14 ((𝑡𝐹𝑛) = 𝑝 ↔ (𝐹‘⟨𝑡, 𝑛⟩) = 𝑝)
112 eqcom 2741 . . . . . . . . . . . . . 14 ((𝑡𝐹𝑛) = 𝑝𝑝 = (𝑡𝐹𝑛))
113111, 112bitr3i 277 . . . . . . . . . . . . 13 ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝𝑝 = (𝑡𝐹𝑛))
114 opelxpi 5725 . . . . . . . . . . . . . . 15 ((𝑡𝐽𝑛 ∈ ℕ0) → ⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0))
1151, 2oddpwdcv 34336 . . . . . . . . . . . . . . . 16 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)))
116 vex 3481 . . . . . . . . . . . . . . . . . . 19 𝑛 ∈ V
117104, 116op2nd 8021 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨𝑡, 𝑛⟩) = 𝑛
118117oveq2i 7441 . . . . . . . . . . . . . . . . 17 (2↑(2nd ‘⟨𝑡, 𝑛⟩)) = (2↑𝑛)
119104, 116op1st 8020 . . . . . . . . . . . . . . . . 17 (1st ‘⟨𝑡, 𝑛⟩) = 𝑡
120118, 119oveq12i 7442 . . . . . . . . . . . . . . . 16 ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)) = ((2↑𝑛) · 𝑡)
121115, 120eqtrdi 2790 . . . . . . . . . . . . . . 15 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
122114, 121syl 17 . . . . . . . . . . . . . 14 ((𝑡𝐽𝑛 ∈ ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
123122eqeq1d 2736 . . . . . . . . . . . . 13 ((𝑡𝐽𝑛 ∈ ℕ0) → ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
124113, 123bitr3id 285 . . . . . . . . . . . 12 ((𝑡𝐽𝑛 ∈ ℕ0) → (𝑝 = (𝑡𝐹𝑛) ↔ ((2↑𝑛) · 𝑡) = 𝑝))
12521, 124sylan2 593 . . . . . . . . . . 11 ((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) → (𝑝 = (𝑡𝐹𝑛) ↔ ((2↑𝑛) · 𝑡) = 𝑝))
126125rexbidva 3174 . . . . . . . . . 10 (𝑡𝐽 → (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
127109, 126bitrd 279 . . . . . . . . 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 3089 . . . . . . 7 (∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
13096, 97, 1293bitri 297 . . . . . 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 312 . . . 4 (𝐴 ∈ (𝑇𝑅) → (𝑝 ∈ (𝐹𝑈) ↔ 𝑝 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
133132eqrdv 2732 . . 3 (𝐴 ∈ (𝑇𝑅) → (𝐹𝑈) = {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
134 f1oeq3 6838 . . 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 233 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 206  wa 395  wo 847   = wceq 1536  wcel 2105  {cab 2711  wral 3058  wrex 3067  {crab 3432  cdif 3959  cin 3961  wss 3962  c0 4338  𝒫 cpw 4604  {csn 4630  cop 4636   ciun 4995   class class class wbr 5147  {copab 5209  cmpt 5230   × cxp 5686  ccnv 5687  cres 5690  cima 5691  ccom 5692   Fn wfn 6557  wf 6558  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  cmpo 7432  1st c1st 8010  2nd c2nd 8011   supp csupp 8183  m cmap 8864  Fincfn 8983  0cc0 11152  1c1 11153   · cmul 11157  cle 11293  cn 12263  2c2 12318  0cn0 12523  cexp 14098  Σcsu 15718  cdvds 16286  bitscbits 16452  𝟭cind 33990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-dvds 16287  df-bits 16455
This theorem is referenced by:  eulerpartlemgs2  34361
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