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Theorem eulerpartlemgh 30414
Description: Lemma for eulerpart 30418: 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 30390 . . . 4 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4 f1of1 6123 . . . 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 4552 . . . . 5 ( 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0) ↔ ∀𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
8 inss2 3826 . . . . . . . 8 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ 𝐽
98sseli 3591 . . . . . . 7 (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) → 𝑡𝐽)
109snssd 4331 . . . . . 6 (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) → {𝑡} ⊆ 𝐽)
11 bitsss 15129 . . . . . 6 (bits‘(𝐴𝑡)) ⊆ ℕ0
12 xpss12 5215 . . . . . 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 2924 . . . 4 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0)
156, 14eqsstri 3627 . . 3 𝑈 ⊆ (𝐽 × ℕ0)
16 f1ores 6138 . . 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 11170 . . . . . . . . . . . . . . 15 2 ∈ ℕ
2019a1i 11 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → 2 ∈ ℕ)
2111sseli 3591 . . . . . . . . . . . . . . 15 (𝑛 ∈ (bits‘(𝐴𝑡)) → 𝑛 ∈ ℕ0)
2221adantl 482 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → 𝑛 ∈ ℕ0)
2320, 22nnexpcld 13013 . . . . . . . . . . . . 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 11053 . . . . . . . . . . . 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 2700 . . . . . . . . . 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 3026 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝𝑝 ∈ ℕ))
3029rexlimdva 3027 . . . . . . 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 3930 . . . . . . . . . . . . . . 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 30405 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
4342simp1bi 1074 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0𝑚 ℕ))
44 elmapi 7864 . . . . . . . . . . . . . . . . . . . . . . . . 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 6032 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴:ℕ⟶ℕ0𝐴 Fn ℕ)
48 elpreima 6323 . . . . . . . . . . . . . . . . . . . . . . 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 6346 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (𝐴𝑡) ∈ ℕ0)
55 elnn0 11279 . . . . . . . . . . . . . . . . . . . 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 6182 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (bits‘(𝐴𝑡)) = (bits‘0))
60 0bits 15142 . . . . . . . . . . . . . . . . 17 (bits‘0) = ∅
6159, 60syl6eq 2670 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (𝐴 “ ℕ)) → (bits‘(𝐴𝑡)) = ∅)
6261rexeqdv 3140 . . . . . . . . . . . . . . 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 3577 . . . . . . . . . . . . . . . . . . . 20 (𝑡 ∈ (ℕ ∖ 𝐽) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡𝐽))
6835, 36, 37, 1, 2, 38, 39, 40, 41eulerpartlemf 30406 . . . . . . . . . . . . . . . . . . . 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 6182 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (bits‘(𝐴𝑡)) = (bits‘0))
7271, 60syl6eq 2670 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡𝐽) → (bits‘(𝐴𝑡)) = ∅)
7372rexeqdv 3140 . . . . . . . . . . . . . . 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 3790 . . . . . . . . . 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 3011 . . . . . . 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 3679 . . . . . . . . . 10 {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
841, 83eqsstri 3627 . . . . . . . . 9 𝐽 ⊆ ℕ
858, 84sstri 3604 . . . . . . . 8 ((𝐴 “ ℕ) ∩ 𝐽) ⊆ ℕ
86 ssrexv 3659 . . . . . . . 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 2631 . . . . . . . 8 (𝑚 = 𝑝 → (((2↑𝑛) · 𝑡) = 𝑚 ↔ ((2↑𝑛) · 𝑡) = 𝑝))
91902rexbidv 3053 . . . . . . 7 (𝑚 = 𝑝 → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑝))
9291elrab 3357 . . . . . 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 5452 . . . . . . . . 9 (𝐹𝑈) = (𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡))))
95 imaiun 6488 . . . . . . . . 9 (𝐹 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))) = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡))))
9694, 95eqtri 2642 . . . . . . . 8 (𝐹𝑈) = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡))))
9796eleq2i 2691 . . . . . . 7 (𝑝 ∈ (𝐹𝑈) ↔ 𝑝 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))))
98 eliun 4515 . . . . . . 7 (𝑝 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)(𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))) ↔ ∃𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)𝑝 ∈ (𝐹 “ ({𝑡} × (bits‘(𝐴𝑡)))))
99 f1ofn 6125 . . . . . . . . . . . . 13 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹 Fn (𝐽 × ℕ0))
1003, 99ax-mp 5 . . . . . . . . . . . 12 𝐹 Fn (𝐽 × ℕ0)
101 snssi 4330 . . . . . . . . . . . . 13 (𝑡𝐽 → {𝑡} ⊆ 𝐽)
102101, 11, 12sylancl 693 . . . . . . . . . . . 12 (𝑡𝐽 → ({𝑡} × (bits‘(𝐴𝑡))) ⊆ (𝐽 × ℕ0))
103 ovelimab 6797 . . . . . . . . . . . 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 3198 . . . . . . . . . . . 12 𝑡 ∈ V
106 oveq1 6642 . . . . . . . . . . . . . 14 (𝑥 = 𝑡 → (𝑥𝐹𝑛) = (𝑡𝐹𝑛))
107106eqeq2d 2630 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝑝 = (𝑥𝐹𝑛) ↔ 𝑝 = (𝑡𝐹𝑛)))
108107rexbidv 3048 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑥𝐹𝑛) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑝 = (𝑡𝐹𝑛)))
109105, 108rexsn 4214 . . . . . . . . . . 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 6638 . . . . . . . . . . . . . . 15 (𝑡𝐹𝑛) = (𝐹‘⟨𝑡, 𝑛⟩)
112111eqeq1i 2625 . . . . . . . . . . . . . 14 ((𝑡𝐹𝑛) = 𝑝 ↔ (𝐹‘⟨𝑡, 𝑛⟩) = 𝑝)
113 eqcom 2627 . . . . . . . . . . . . . 14 ((𝑡𝐹𝑛) = 𝑝𝑝 = (𝑡𝐹𝑛))
114112, 113bitr3i 266 . . . . . . . . . . . . 13 ((𝐹‘⟨𝑡, 𝑛⟩) = 𝑝𝑝 = (𝑡𝐹𝑛))
115 opelxpi 5138 . . . . . . . . . . . . . . 15 ((𝑡𝐽𝑛 ∈ ℕ0) → ⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0))
1161, 2oddpwdcv 30391 . . . . . . . . . . . . . . . 16 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)))
117 vex 3198 . . . . . . . . . . . . . . . . . . 19 𝑛 ∈ V
118105, 117op2nd 7162 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨𝑡, 𝑛⟩) = 𝑛
119118oveq2i 6646 . . . . . . . . . . . . . . . . 17 (2↑(2nd ‘⟨𝑡, 𝑛⟩)) = (2↑𝑛)
120105, 117op1st 7161 . . . . . . . . . . . . . . . . 17 (1st ‘⟨𝑡, 𝑛⟩) = 𝑡
121119, 120oveq12i 6647 . . . . . . . . . . . . . . . 16 ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)) = ((2↑𝑛) · 𝑡)
122116, 121syl6eq 2670 . . . . . . . . . . . . . . 15 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
123115, 122syl 17 . . . . . . . . . . . . . 14 ((𝑡𝐽𝑛 ∈ ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
124123eqeq1d 2622 . . . . . . . . . . . . 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 3045 . . . . . . . . . 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 3036 . . . . . . 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 2618 . . 3 (𝐴 ∈ (𝑇𝑅) → (𝐹𝑈) = {𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
135 f1oeq3 6116 . . 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 1481  wcel 1988  {cab 2606  wral 2909  wrex 2910  {crab 2913  cdif 3564  cin 3566  wss 3567  c0 3907  𝒫 cpw 4149  {csn 4168  cop 4174   ciun 4511   class class class wbr 4644  {copab 4703  cmpt 4720   × cxp 5102  ccnv 5103  cres 5106  cima 5107  ccom 5108   Fn wfn 5871  wf 5872  1-1wf1 5873  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151  2nd c2nd 7152   supp csupp 7280  𝑚 cmap 7842  Fincfn 7940  0cc0 9921  1c1 9922   · cmul 9926  cle 10060  cn 11005  2c2 11055  0cn0 11277  cexp 12843  Σcsu 14397  cdvds 14964  bitscbits 15122  𝟭cind 30046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-inf 8334  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-n0 11278  df-z 11363  df-uz 11673  df-rp 11818  df-fz 12312  df-fzo 12450  df-fl 12576  df-seq 12785  df-exp 12844  df-dvds 14965  df-bits 15125
This theorem is referenced by:  eulerpartlemgs2  30416
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