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Theorem eulerpartlemgh 30263
 Description: Lemma for eulerpart 30267: 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 30239 . . . 4 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4 f1of1 6103 . . . 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 4534 . . . . 5 ( 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0) ↔ ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8 inss2 3818 . . . . . . . 8 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
98sseli 3584 . . . . . . 7 (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) → 𝑡𝐽)
109snssd 4316 . . . . . 6 (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) → {𝑡} ⊆ 𝐽)
11 bitsss 15091 . . . . . 6 (bits‘(𝐴𝑡)) ⊆ ℕ0
12 xpss12 5196 . . . . . 6 (({𝑡} ⊆ 𝐽 ∧ (bits‘(𝐴𝑡)) ⊆ ℕ0) → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
1310, 11, 12sylancl 693 . . . . 5 (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
147, 13mprgbir 2923 . . . 4 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0)
156, 14eqsstri 3620 . . 3 𝑈 ⊆ (𝐽 × ℕ0)
16 f1ores 6118 . . 3 ((𝐹:(𝐽 × ℕ0)–1-1→ℕ ∧ 𝑈 ⊆ (𝐽 × ℕ0)) → (𝐹𝑈):𝑈1-1-onto→(𝐹𝑈))
175, 15, 16mp2an 707 . 2 (𝐹𝑈):𝑈1-1-onto→(𝐹𝑈)
18 simpr 477 . . . . . . . . . . 11 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → ((2↑𝑛) · 𝑡) = 𝑝)
19 2nn 11145 . . . . . . . . . . . . . . 15 2 ∈ ℕ
2019a1i 11 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → 2 ∈ ℕ)
2111sseli 3584 . . . . . . . . . . . . . . 15 (𝑛 ∈ (bits‘(𝐴𝑡)) → 𝑛 ∈ ℕ0)
2221adantl 482 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → 𝑛 ∈ ℕ0)
2320, 22nnexpcld 12986 . . . . . . . . . . . . 13 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → (2↑𝑛) ∈ ℕ)
24 simplr 791 . . . . . . . . . . . . 13 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → 𝑡 ∈ ℕ)
2523, 24nnmulcld 11028 . . . . . . . . . . . 12 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → ((2↑𝑛) · 𝑡) ∈ ℕ)
2625adantr 481 . . . . . . . . . . 11 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → ((2↑𝑛) · 𝑡) ∈ ℕ)
2718, 26eqeltrrd 2699 . . . . . . . . . 10 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) ∧ ((2↑𝑛) · 𝑡) = 𝑝) → 𝑝 ∈ ℕ)
2827exp31 629 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (𝑛 ∈ (bits‘(𝐴𝑡)) → (((2↑𝑛) · 𝑡) = 𝑝𝑝 ∈ ℕ)))
2928rexlimdv 3025 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝𝑝 ∈ ℕ))
3029rexlimdva 3026 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝𝑝 ∈ ℕ))
3130pm4.71rd 666 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ (𝑝 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)))
32 rex0 3920 . . . . . . . . . . . . . . 15 ¬ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝
33 simplr 791 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → 𝑡 ∈ ℕ)
34 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ 𝑡 ∈ (𝐴 “ ℕ))
35 eulerpart.p . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑃 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
36 eulerpart.o . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
37 eulerpart.d . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
38 eulerpart.h . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
39 eulerpart.m . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
40 eulerpart.r . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
41 eulerpart.t . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑇 = {𝑓 ∈ (ℕ0𝑚 ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
4235, 36, 37, 1, 2, 38, 39, 40, 41eulerpartlemt0 30254 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
4342simp1bi 1074 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0𝑚 ℕ))
44 elmapi 7839 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ (ℕ0𝑚 ℕ) → 𝐴:ℕ⟶ℕ0)
4543, 44syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
4645ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → 𝐴:ℕ⟶ℕ0)
47 ffn 6012 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴:ℕ⟶ℕ0𝐴 Fn ℕ)
48 elpreima 6303 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 Fn ℕ → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴𝑡) ∈ ℕ)))
4946, 47, 483syl 18 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (𝑡 ∈ (𝐴 “ ℕ) ↔ (𝑡 ∈ ℕ ∧ (𝐴𝑡) ∈ ℕ)))
5034, 49mtbid 314 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ (𝑡 ∈ ℕ ∧ (𝐴𝑡) ∈ ℕ))
51 imnan 438 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℕ → ¬ (𝐴𝑡) ∈ ℕ) ↔ ¬ (𝑡 ∈ ℕ ∧ (𝐴𝑡) ∈ ℕ))
5250, 51sylibr 224 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (𝑡 ∈ ℕ → ¬ (𝐴𝑡) ∈ ℕ))
5333, 52mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ (𝐴𝑡) ∈ ℕ)
5446, 33ffvelrnd 6326 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (𝐴𝑡) ∈ ℕ0)
55 elnn0 11254 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑡) ∈ ℕ0 ↔ ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
5654, 55sylib 208 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0))
57 orel1 397 . . . . . . . . . . . . . . . . . . 19 (¬ (𝐴𝑡) ∈ ℕ → (((𝐴𝑡) ∈ ℕ ∨ (𝐴𝑡) = 0) → (𝐴𝑡) = 0))
5853, 56, 57sylc 65 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (𝐴𝑡) = 0)
5958fveq2d 6162 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (bits‘(𝐴𝑡)) = (bits‘0))
60 0bits 15104 . . . . . . . . . . . . . . . . 17 (bits‘0) = ∅
6159, 60syl6eq 2671 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (bits‘(𝐴𝑡)) = ∅)
6261rexeqdv 3138 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝))
6332, 62mtbiri 317 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → ¬ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
6463ex 450 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (¬ 𝑡 ∈ (𝐴 “ ℕ) → ¬ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
6564con4d 114 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝𝑡 ∈ (𝐴 “ ℕ)))
6665impr 648 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ (𝐴 “ ℕ))
67 eldif 3570 . . . . . . . . . . . . . . . . . . . 20 (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽))
6835, 36, 37, 1, 2, 38, 39, 40, 41eulerpartlemf 30255 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴𝑡) = 0)
6967, 68sylan2br 493 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽)) → (𝐴𝑡) = 0)
7069anassrs 679 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (𝐴𝑡) = 0)
7170fveq2d 6162 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (bits‘(𝐴𝑡)) = (bits‘0))
7271, 60syl6eq 2671 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (bits‘(𝐴𝑡)) = ∅)
7372rexeqdv 3138 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ ∃𝑛 ∈ ∅ ((2↑𝑛) · 𝑡) = 𝑝))
7432, 73mtbiri 317 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → ¬ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
7574ex 450 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (¬ 𝑡𝐽 → ¬ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
7675con4d 114 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝𝑡𝐽))
7776impr 648 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡𝐽)
7866, 77elind 3782 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽))
79 simprr 795 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
8078, 79jca 554 . . . . . . . . 9 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)) → (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
8180ex 450 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → ((𝑡 ∈ ℕ ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝) → (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)))
8281reximdv2 3010 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 → ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
83 ssrab2 3672 . . . . . . . . . 10 {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
841, 83eqsstri 3620 . . . . . . . . 9 𝐽 ⊆ ℕ
858, 84sstri 3597 . . . . . . . 8 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ
86 ssrexv 3652 . . . . . . . 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))
8785, 86mp1i 13 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
8882, 87impbid 202 . . . . . 6 (𝐴 ∈ (𝑇𝑅) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝 ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
8931, 88bitr3d 270 . . . . 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))
90 eqeq2 2632 . . . . . . . 8 (𝑚 = 𝑝 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
91902rexbidv 3052 . . . . . . 7 (𝑚 = 𝑝 → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
9291elrab 3351 . . . . . 6 (𝑝 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚} ↔ (𝑝 ∈ ℕ ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
9392a1i 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)))
946imaeq2i 5433 . . . . . . . . 9 (𝐹𝑈) = (𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))
95 imaiun 6468 . . . . . . . . 9 (𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡))))
9694, 95eqtri 2643 . . . . . . . 8 (𝐹𝑈) = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡))))
9796eleq2i 2690 . . . . . . 7 (𝑝 ∈ (𝐹𝑈) ↔ 𝑝 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))))
98 eliun 4497 . . . . . . 7 (𝑝 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))))
99 f1ofn 6105 . . . . . . . . . . . . 13 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹 Fn (𝐽 × ℕ0))
1003, 99ax-mp 5 . . . . . . . . . . . 12 𝐹 Fn (𝐽 × ℕ0)
101 snssi 4315 . . . . . . . . . . . . 13 (𝑡𝐽 → {𝑡} ⊆ 𝐽)
102101, 11, 12sylancl 693 . . . . . . . . . . . 12 (𝑡𝐽 → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
103 ovelimab 6777 . . . . . . . . . . . 12 ((𝐹 Fn (𝐽 × ℕ0) ∧ ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0)) → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑥𝐹𝑛)))
104100, 102, 103sylancr 694 . . . . . . . . . . 11 (𝑡𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑥𝐹𝑛)))
105 vex 3193 . . . . . . . . . . . 12 𝑡 ∈ V
106 oveq1 6622 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (𝑥𝐹𝑛) = (𝑡𝐹𝑛))
107106eqeq2d 2631 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝑝 = (𝑥𝐹𝑛) ↔ 𝑝 = (𝑡𝐹𝑛)))
108107rexbidv 3047 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛)))
109105, 108rexsn 4201 . . . . . . . . . . 11 (∃𝑥 ∈ {𝑡}∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛))
110104, 109syl6bb 276 . . . . . . . . . 10 (𝑡𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛)))
111 df-ov 6618 . . . . . . . . . . . . . . 15 (𝑡𝐹𝑛) = (𝐹‘⟨𝑡, 𝑛⟩)
112111eqeq1i 2626 . . . . . . . . . . . . . 14 ((𝑡𝐹𝑛) = 𝑝 ↔ (𝐹‘⟨𝑡, 𝑛⟩) = 𝑝)
113 eqcom 2628 . . . . . . . . . . . . . 14 ((𝑡𝐹𝑛) = 𝑝𝑝 = (𝑡𝐹𝑛))
114112, 113bitr3i 266 . . . . . . . . . . . . 13 ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝𝑝 = (𝑡𝐹𝑛))
115 opelxpi 5118 . . . . . . . . . . . . . . 15 ((𝑡𝐽𝑛 ∈ ℕ0) → ⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0))
1161, 2oddpwdcv 30240 . . . . . . . . . . . . . . . 16 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)))
117 vex 3193 . . . . . . . . . . . . . . . . . . 19 𝑛 ∈ V
118105, 117op2nd 7137 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨𝑡, 𝑛⟩) = 𝑛
119118oveq2i 6626 . . . . . . . . . . . . . . . . 17 (2↑(2nd ‘⟨𝑡, 𝑛⟩)) = (2↑𝑛)
120105, 117op1st 7136 . . . . . . . . . . . . . . . . 17 (1st ‘⟨𝑡, 𝑛⟩) = 𝑡
121119, 120oveq12i 6627 . . . . . . . . . . . . . . . 16 ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)) = ((2↑𝑛) · 𝑡)
122116, 121syl6eq 2671 . . . . . . . . . . . . . . 15 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
123115, 122syl 17 . . . . . . . . . . . . . 14 ((𝑡𝐽𝑛 ∈ ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
124123eqeq1d 2623 . . . . . . . . . . . . 13 ((𝑡𝐽𝑛 ∈ ℕ0) → ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
125114, 124syl5bbr 274 . . . . . . . . . . . 12 ((𝑡𝐽𝑛 ∈ ℕ0) → (𝑝 = (𝑡𝐹𝑛) ↔ ((2↑𝑛) · 𝑡) = 𝑝))
12621, 125sylan2 491 . . . . . . . . . . 11 ((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) → (𝑝 = (𝑡𝐹𝑛) ↔ ((2↑𝑛) · 𝑡) = 𝑝))
127126rexbidva 3044 . . . . . . . . . 10 (𝑡𝐽 → (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
128110, 127bitrd 268 . . . . . . . . 9 (𝑡𝐽 → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
1299, 128syl 17 . . . . . . . 8 (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) → (𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
130129rexbiia 3035 . . . . . . 7 (∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
13197, 98, 1303bitri 286 . . . . . 6 (𝑝 ∈ (𝐹𝑈) ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝)
132131a1i 11 . . . . 5 (𝐴 ∈ (𝑇𝑅) → (𝑝 ∈ (𝐹𝑈) ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
13389, 93, 1323bitr4rd 301 . . . 4 (𝐴 ∈ (𝑇𝑅) → (𝑝 ∈ (𝐹𝑈) ↔ 𝑝 ∈ {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
134133eqrdv 2619 . . 3 (𝐴 ∈ (𝑇𝑅) → (𝐹𝑈) = {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
135 f1oeq3 6096 . . 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}))
136134, 135syl 17 . 2 (𝐴 ∈ (𝑇𝑅) → ((𝐹𝑈):𝑈1-1-onto→(𝐹𝑈) ↔ (𝐹𝑈):𝑈1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚}))
13717, 136mpbii 223 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 196   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987  {cab 2607  ∀wral 2908  ∃wrex 2909  {crab 2912   ∖ cdif 3557   ∩ cin 3559   ⊆ wss 3560  ∅c0 3897  𝒫 cpw 4136  {csn 4155  ⟨cop 4161  ∪ ciun 4492   class class class wbr 4623  {copab 4682   ↦ cmpt 4683   × cxp 5082  ◡ccnv 5083   ↾ cres 5086   “ cima 5087   ∘ ccom 5088   Fn wfn 5852  ⟶wf 5853  –1-1→wf1 5854  –1-1-onto→wf1o 5856  ‘cfv 5857  (class class class)co 6615   ↦ cmpt2 6617  1st c1st 7126  2nd c2nd 7127   supp csupp 7255   ↑𝑚 cmap 7817  Fincfn 7915  0cc0 9896  1c1 9897   · cmul 9901   ≤ cle 10035  ℕcn 10980  2c2 11030  ℕ0cn0 11252  ↑cexp 12816  Σcsu 14366   ∥ cdvds 14926  bitscbits 15084  𝟭cind 29896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-rp 11793  df-fz 12285  df-fzo 12423  df-fl 12549  df-seq 12758  df-exp 12817  df-dvds 14927  df-bits 15087 This theorem is referenced by:  eulerpartlemgs2  30265
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