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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eulerpartlemgvv Structured version   Visualization version   GIF version

Theorem eulerpartlemgvv 33932
Description: Lemma for eulerpart 33938: value of the function 𝐺 evaluated. (Contributed by Thierry Arnoux, 10-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))))))
Assertion
Ref Expression
eulerpartlemgvv ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → ((𝐺𝐴)‘𝐵) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵, 1, 0))
Distinct variable groups:   𝑓,𝑘,𝑛,𝑡,𝑥,𝑦,𝑧   𝑓,𝑜,𝑟,𝐴   𝑜,𝐹   𝐻,𝑟   𝑓,𝐽   𝑛,𝑜,𝑟,𝐽,𝑥,𝑦   𝑜,𝑀   𝑓,𝑁   𝑔,𝑛,𝑃   𝑅,𝑜   𝑇,𝑜   𝑡,𝐴,𝑛,𝑥,𝑦   𝐵,𝑛,𝑡,𝑥,𝑦   𝑛,𝐹,𝑡,𝑥,𝑦   𝑡,𝐽   𝑛,𝑀,𝑡,𝑥,𝑦   𝑅,𝑛   𝑡,𝑟,𝑅,𝑥,𝑦   𝑇,𝑛,𝑟,𝑡,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑧,𝑔,𝑘)   𝐵(𝑧,𝑓,𝑔,𝑘,𝑜,𝑟)   𝐷(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝑃(𝑥,𝑦,𝑧,𝑡,𝑓,𝑘,𝑜,𝑟)   𝑅(𝑧,𝑓,𝑔,𝑘)   𝑇(𝑧,𝑓,𝑔,𝑘)   𝐹(𝑧,𝑓,𝑔,𝑘,𝑟)   𝐺(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑜)   𝐽(𝑧,𝑔,𝑘)   𝑀(𝑧,𝑓,𝑔,𝑘,𝑟)   𝑁(𝑥,𝑦,𝑧,𝑡,𝑔,𝑘,𝑛,𝑜,𝑟)   𝑂(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑘,𝑛,𝑜,𝑟)

Proof of Theorem eulerpartlemgvv
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eulerpart.p . . . . 5 𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
2 eulerpart.o . . . . 5 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
3 eulerpart.d . . . . 5 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
4 eulerpart.j . . . . 5 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
5 eulerpart.f . . . . 5 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
6 eulerpart.h . . . . 5 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
7 eulerpart.m . . . . 5 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
8 eulerpart.r . . . . 5 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
9 eulerpart.t . . . . 5 𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}
10 eulerpart.g . . . . 5 𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10eulerpartlemgv 33929 . . . 4 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽))))))
1211fveq1d 6893 . . 3 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴)‘𝐵) = (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵))
1312adantr 480 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → ((𝐺𝐴)‘𝐵) = (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵))
14 nnex 12240 . . 3 ℕ ∈ V
15 imassrn 6068 . . . 4 (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ⊆ ran 𝐹
164, 5oddpwdc 33910 . . . . 5 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
17 f1of 6833 . . . . 5 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:(𝐽 × ℕ0)⟶ℕ)
18 frn 6723 . . . . 5 (𝐹:(𝐽 × ℕ0)⟶ℕ → ran 𝐹 ⊆ ℕ)
1916, 17, 18mp2b 10 . . . 4 ran 𝐹 ⊆ ℕ
2015, 19sstri 3987 . . 3 (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ⊆ ℕ
21 simpr 484 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ)
22 indfval 33571 . . 3 ((ℕ ∈ V ∧ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ⊆ ℕ ∧ 𝐵 ∈ ℕ) → (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵) = if(𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))), 1, 0))
2314, 20, 21, 22mp3an12i 1462 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵) = if(𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))), 1, 0))
24 ffn 6716 . . . . . 6 (𝐹:(𝐽 × ℕ0)⟶ℕ → 𝐹 Fn (𝐽 × ℕ0))
2516, 17, 24mp2b 10 . . . . 5 𝐹 Fn (𝐽 × ℕ0)
261, 2, 3, 4, 5, 6, 7, 8, 9, 10eulerpartlemmf 33931 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (bits ∘ (𝐴𝐽)) ∈ 𝐻)
271, 2, 3, 4, 5, 6, 7eulerpartlem1 33923 . . . . . . . . . . 11 𝑀:𝐻1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)
28 f1of 6833 . . . . . . . . . . 11 (𝑀:𝐻1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) → 𝑀:𝐻⟶(𝒫 (𝐽 × ℕ0) ∩ Fin))
2927, 28ax-mp 5 . . . . . . . . . 10 𝑀:𝐻⟶(𝒫 (𝐽 × ℕ0) ∩ Fin)
3029ffvelcdmi 7087 . . . . . . . . 9 ((bits ∘ (𝐴𝐽)) ∈ 𝐻 → (𝑀‘(bits ∘ (𝐴𝐽))) ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin))
3126, 30syl 17 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin))
3231elin1d 4194 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) ∈ 𝒫 (𝐽 × ℕ0))
3332adantr 480 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝑀‘(bits ∘ (𝐴𝐽))) ∈ 𝒫 (𝐽 × ℕ0))
3433elpwid 4607 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝑀‘(bits ∘ (𝐴𝐽))) ⊆ (𝐽 × ℕ0))
35 fvelimab 6965 . . . . 5 ((𝐹 Fn (𝐽 × ℕ0) ∧ (𝑀‘(bits ∘ (𝐴𝐽))) ⊆ (𝐽 × ℕ0)) → (𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ↔ ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵))
3625, 34, 35sylancr 586 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ↔ ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵))
374ssrab3 4076 . . . . . . . . 9 𝐽 ⊆ ℕ
38 fveq1 6890 . . . . . . . . . . . . . . . . . . 19 (𝑟 = (bits ∘ (𝐴𝐽)) → (𝑟𝑥) = ((bits ∘ (𝐴𝐽))‘𝑥))
3938eleq2d 2814 . . . . . . . . . . . . . . . . . 18 (𝑟 = (bits ∘ (𝐴𝐽)) → (𝑦 ∈ (𝑟𝑥) ↔ 𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)))
4039anbi2d 628 . . . . . . . . . . . . . . . . 17 (𝑟 = (bits ∘ (𝐴𝐽)) → ((𝑥𝐽𝑦 ∈ (𝑟𝑥)) ↔ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))))
4140opabbidv 5208 . . . . . . . . . . . . . . . 16 (𝑟 = (bits ∘ (𝐴𝐽)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))})
4214, 37ssexi 5316 . . . . . . . . . . . . . . . . . 18 𝐽 ∈ V
43 abid2 2866 . . . . . . . . . . . . . . . . . . . 20 {𝑦𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)} = ((bits ∘ (𝐴𝐽))‘𝑥)
4443fvexi 6905 . . . . . . . . . . . . . . . . . . 19 {𝑦𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)} ∈ V
4544a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑥𝐽 → {𝑦𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)} ∈ V)
4642, 45opabex3 7965 . . . . . . . . . . . . . . . . 17 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} ∈ V
4746a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑇𝑅) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} ∈ V)
487, 41, 26, 47fvmptd3 7022 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))})
49 simpl 482 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑡𝑦 = 𝑛) → 𝑥 = 𝑡)
5049eleq1d 2813 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑡𝑦 = 𝑛) → (𝑥𝐽𝑡𝐽))
51 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑡𝑦 = 𝑛) → 𝑦 = 𝑛)
5249fveq2d 6895 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑡𝑦 = 𝑛) → ((bits ∘ (𝐴𝐽))‘𝑥) = ((bits ∘ (𝐴𝐽))‘𝑡))
5351, 52eleq12d 2822 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑡𝑦 = 𝑛) → (𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥) ↔ 𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡)))
5450, 53anbi12d 630 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑡𝑦 = 𝑛) → ((𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)) ↔ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))))
5554cbvopabv 5215 . . . . . . . . . . . . . . 15 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} = {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))}
5648, 55eqtrdi 2783 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) = {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))})
5756eleq2d 2814 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ 𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))}))
581, 2, 3, 4, 5, 6, 7, 8, 9eulerpartlemt0 33925 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
5958simp1bi 1143 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0m ℕ))
60 nn0ex 12500 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
6160, 14elmap 8881 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (ℕ0m ℕ) ↔ 𝐴:ℕ⟶ℕ0)
6259, 61sylib 217 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
63 ffun 6719 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴:ℕ⟶ℕ0 → Fun 𝐴)
64 funres 6589 . . . . . . . . . . . . . . . . . . . . . 22 (Fun 𝐴 → Fun (𝐴𝐽))
6562, 63, 643syl 18 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ (𝑇𝑅) → Fun (𝐴𝐽))
66 fssres 6757 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴:ℕ⟶ℕ0𝐽 ⊆ ℕ) → (𝐴𝐽):𝐽⟶ℕ0)
6762, 37, 66sylancl 585 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (𝑇𝑅) → (𝐴𝐽):𝐽⟶ℕ0)
68 fdm 6725 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝐽):𝐽⟶ℕ0 → dom (𝐴𝐽) = 𝐽)
6968eleq2d 2814 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝐽):𝐽⟶ℕ0 → (𝑡 ∈ dom (𝐴𝐽) ↔ 𝑡𝐽))
7067, 69syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (𝑇𝑅) → (𝑡 ∈ dom (𝐴𝐽) ↔ 𝑡𝐽))
7170biimpar 477 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → 𝑡 ∈ dom (𝐴𝐽))
72 fvco 6990 . . . . . . . . . . . . . . . . . . . . 21 ((Fun (𝐴𝐽) ∧ 𝑡 ∈ dom (𝐴𝐽)) → ((bits ∘ (𝐴𝐽))‘𝑡) = (bits‘((𝐴𝐽)‘𝑡)))
7365, 71, 72syl2an2r 684 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → ((bits ∘ (𝐴𝐽))‘𝑡) = (bits‘((𝐴𝐽)‘𝑡)))
74 fvres 6910 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡𝐽 → ((𝐴𝐽)‘𝑡) = (𝐴𝑡))
7574fveq2d 6895 . . . . . . . . . . . . . . . . . . . . 21 (𝑡𝐽 → (bits‘((𝐴𝐽)‘𝑡)) = (bits‘(𝐴𝑡)))
7675adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → (bits‘((𝐴𝐽)‘𝑡)) = (bits‘(𝐴𝑡)))
7773, 76eqtrd 2767 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → ((bits ∘ (𝐴𝐽))‘𝑡) = (bits‘(𝐴𝑡)))
7877eleq2d 2814 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → (𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡) ↔ 𝑛 ∈ (bits‘(𝐴𝑡))))
7978pm5.32da 578 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ (𝑇𝑅) → ((𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡)) ↔ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))))
8079opabbidv 5208 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑇𝑅) → {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} = {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))})
8180eleq2d 2814 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} ↔ 𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))}))
82 elopab 5523 . . . . . . . . . . . . . . 15 (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))} ↔ ∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))))
8381, 82bitrdi 287 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} ↔ ∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))))))
84 ancom 460 . . . . . . . . . . . . . . . . 17 ((𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ ((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩))
85 anass 468 . . . . . . . . . . . . . . . . 17 (((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) ↔ (𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
8684, 85bitri 275 . . . . . . . . . . . . . . . 16 ((𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ (𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
87862exbii 1844 . . . . . . . . . . . . . . 15 (∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ ∃𝑡𝑛(𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
88 df-rex 3066 . . . . . . . . . . . . . . . . . 18 (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ ↔ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩))
8988anbi2i 622 . . . . . . . . . . . . . . . . 17 ((𝑡𝐽 ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩) ↔ (𝑡𝐽 ∧ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
9089exbii 1843 . . . . . . . . . . . . . . . 16 (∃𝑡(𝑡𝐽 ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩) ↔ ∃𝑡(𝑡𝐽 ∧ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
91 df-rex 3066 . . . . . . . . . . . . . . . 16 (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ ↔ ∃𝑡(𝑡𝐽 ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩))
92 exdistr 1951 . . . . . . . . . . . . . . . 16 (∃𝑡𝑛(𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)) ↔ ∃𝑡(𝑡𝐽 ∧ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
9390, 91, 923bitr4i 303 . . . . . . . . . . . . . . 15 (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ ↔ ∃𝑡𝑛(𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
9487, 93bitr4i 278 . . . . . . . . . . . . . 14 (∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩)
9583, 94bitrdi 287 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} ↔ ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩))
9657, 95bitrd 279 . . . . . . . . . . . 12 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩))
9796biimpa 476 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩)
9897adantlr 714 . . . . . . . . . 10 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩)
99 fveq2 6891 . . . . . . . . . . . . . 14 (𝑤 = ⟨𝑡, 𝑛⟩ → (𝐹𝑤) = (𝐹‘⟨𝑡, 𝑛⟩))
10099adantl 481 . . . . . . . . . . . . 13 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → (𝐹𝑤) = (𝐹‘⟨𝑡, 𝑛⟩))
101 bitsss 16392 . . . . . . . . . . . . . . . . 17 (bits‘(𝐴𝑡)) ⊆ ℕ0
102101sseli 3974 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (bits‘(𝐴𝑡)) → 𝑛 ∈ ℕ0)
103102anim2i 616 . . . . . . . . . . . . . . 15 ((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) → (𝑡𝐽𝑛 ∈ ℕ0))
104103ad2antlr 726 . . . . . . . . . . . . . 14 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → (𝑡𝐽𝑛 ∈ ℕ0))
105 opelxp 5708 . . . . . . . . . . . . . . 15 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) ↔ (𝑡𝐽𝑛 ∈ ℕ0))
1064, 5oddpwdcv 33911 . . . . . . . . . . . . . . . 16 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)))
107 vex 3473 . . . . . . . . . . . . . . . . . . 19 𝑡 ∈ V
108 vex 3473 . . . . . . . . . . . . . . . . . . 19 𝑛 ∈ V
109107, 108op2nd 7996 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨𝑡, 𝑛⟩) = 𝑛
110109oveq2i 7425 . . . . . . . . . . . . . . . . 17 (2↑(2nd ‘⟨𝑡, 𝑛⟩)) = (2↑𝑛)
111107, 108op1st 7995 . . . . . . . . . . . . . . . . 17 (1st ‘⟨𝑡, 𝑛⟩) = 𝑡
112110, 111oveq12i 7426 . . . . . . . . . . . . . . . 16 ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)) = ((2↑𝑛) · 𝑡)
113106, 112eqtrdi 2783 . . . . . . . . . . . . . . 15 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
114105, 113sylbir 234 . . . . . . . . . . . . . 14 ((𝑡𝐽𝑛 ∈ ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
115104, 114syl 17 . . . . . . . . . . . . 13 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
116100, 115eqtr2d 2768 . . . . . . . . . . . 12 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → ((2↑𝑛) · 𝑡) = (𝐹𝑤))
117116ex 412 . . . . . . . . . . 11 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑤 = ⟨𝑡, 𝑛⟩ → ((2↑𝑛) · 𝑡) = (𝐹𝑤)))
118117reximdvva 3200 . . . . . . . . . 10 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤)))
11998, 118mpd 15 . . . . . . . . 9 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤))
120 ssrexv 4047 . . . . . . . . 9 (𝐽 ⊆ ℕ → (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤)))
12137, 119, 120mpsyl 68 . . . . . . . 8 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤))
122121adantr 480 . . . . . . 7 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝐹𝑤) = 𝐵) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤))
123 eqeq2 2739 . . . . . . . . . 10 ((𝐹𝑤) = 𝐵 → (((2↑𝑛) · 𝑡) = (𝐹𝑤) ↔ ((2↑𝑛) · 𝑡) = 𝐵))
124123rexbidv 3173 . . . . . . . . 9 ((𝐹𝑤) = 𝐵 → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
125124adantl 481 . . . . . . . 8 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝐹𝑤) = 𝐵) → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
126125rexbidv 3173 . . . . . . 7 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝐹𝑤) = 𝐵) → (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤) ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
127122, 126mpbid 231 . . . . . 6 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝐹𝑤) = 𝐵) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵)
128127r19.29an 3153 . . . . 5 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵)
129 simp-5l 784 . . . . . . . 8 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → 𝐴 ∈ (𝑇𝑅))
130 simpllr 775 . . . . . . . 8 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → 𝑥𝐽)
131 simplr 768 . . . . . . . . 9 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → 𝑦 ∈ (bits‘(𝐴𝑥)))
13268eleq2d 2814 . . . . . . . . . . . . . 14 ((𝐴𝐽):𝐽⟶ℕ0 → (𝑥 ∈ dom (𝐴𝐽) ↔ 𝑥𝐽))
13367, 132syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑥 ∈ dom (𝐴𝐽) ↔ 𝑥𝐽))
134133biimpar 477 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → 𝑥 ∈ dom (𝐴𝐽))
135 fvco 6990 . . . . . . . . . . . 12 ((Fun (𝐴𝐽) ∧ 𝑥 ∈ dom (𝐴𝐽)) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘((𝐴𝐽)‘𝑥)))
13665, 134, 135syl2an2r 684 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘((𝐴𝐽)‘𝑥)))
137 fvres 6910 . . . . . . . . . . . . 13 (𝑥𝐽 → ((𝐴𝐽)‘𝑥) = (𝐴𝑥))
138137fveq2d 6895 . . . . . . . . . . . 12 (𝑥𝐽 → (bits‘((𝐴𝐽)‘𝑥)) = (bits‘(𝐴𝑥)))
139138adantl 481 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → (bits‘((𝐴𝐽)‘𝑥)) = (bits‘(𝐴𝑥)))
140136, 139eqtrd 2767 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘(𝐴𝑥)))
141129, 130, 140syl2anc 583 . . . . . . . . 9 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘(𝐴𝑥)))
142131, 141eleqtrrd 2831 . . . . . . . 8 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → 𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))
14348eleq2d 2814 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → (⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))}))
144 opabidw 5520 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} ↔ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)))
145143, 144bitrdi 287 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))))
146145biimpar 477 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))) → ⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))))
147129, 130, 142, 146syl12anc 836 . . . . . . 7 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))))
148 simpr 484 . . . . . . . 8 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → ((2↑𝑦) · 𝑥) = 𝐵)
14934ad4antr 731 . . . . . . . . 9 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → (𝑀‘(bits ∘ (𝐴𝐽))) ⊆ (𝐽 × ℕ0))
150149, 147sseldd 3979 . . . . . . . 8 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0))
151 opeq1 4869 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ⟨𝑡, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
152151eleq1d 2813 . . . . . . . . . . 11 (𝑡 = 𝑥 → (⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0)))
153151fveq2d 6895 . . . . . . . . . . . 12 (𝑡 = 𝑥 → (𝐹‘⟨𝑡, 𝑦⟩) = (𝐹‘⟨𝑥, 𝑦⟩))
154 oveq2 7422 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ((2↑𝑦) · 𝑡) = ((2↑𝑦) · 𝑥))
155153, 154eqeq12d 2743 . . . . . . . . . . 11 (𝑡 = 𝑥 → ((𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥)))
156152, 155imbi12d 344 . . . . . . . . . 10 (𝑡 = 𝑥 → ((⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥))))
157 opeq2 4870 . . . . . . . . . . . . 13 (𝑛 = 𝑦 → ⟨𝑡, 𝑛⟩ = ⟨𝑡, 𝑦⟩)
158157eleq1d 2813 . . . . . . . . . . . 12 (𝑛 = 𝑦 → (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) ↔ ⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0)))
159157fveq2d 6895 . . . . . . . . . . . . 13 (𝑛 = 𝑦 → (𝐹‘⟨𝑡, 𝑛⟩) = (𝐹‘⟨𝑡, 𝑦⟩))
160 oveq2 7422 . . . . . . . . . . . . . 14 (𝑛 = 𝑦 → (2↑𝑛) = (2↑𝑦))
161160oveq1d 7429 . . . . . . . . . . . . 13 (𝑛 = 𝑦 → ((2↑𝑛) · 𝑡) = ((2↑𝑦) · 𝑡))
162159, 161eqeq12d 2743 . . . . . . . . . . . 12 (𝑛 = 𝑦 → ((𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡) ↔ (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡)))
163158, 162imbi12d 344 . . . . . . . . . . 11 (𝑛 = 𝑦 → ((⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡)) ↔ (⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡))))
164163, 113chvarvv 1995 . . . . . . . . . 10 (⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡))
165156, 164chvarvv 1995 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥))
166 eqeq2 2739 . . . . . . . . . 10 (((2↑𝑦) · 𝑥) = 𝐵 → ((𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵))
167166biimpa 476 . . . . . . . . 9 ((((2↑𝑦) · 𝑥) = 𝐵 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥)) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵)
168165, 167sylan2 592 . . . . . . . 8 ((((2↑𝑦) · 𝑥) = 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0)) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵)
169148, 150, 168syl2anc 583 . . . . . . 7 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵)
170 fveqeq2 6900 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑤) = 𝐵 ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵))
171170rspcev 3607 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵) → ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵)
172147, 169, 171syl2anc 583 . . . . . 6 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵)
173 oveq2 7422 . . . . . . . . . . 11 (𝑡 = 𝑥 → ((2↑𝑛) · 𝑡) = ((2↑𝑛) · 𝑥))
174173eqeq1d 2729 . . . . . . . . . 10 (𝑡 = 𝑥 → (((2↑𝑛) · 𝑡) = 𝐵 ↔ ((2↑𝑛) · 𝑥) = 𝐵))
175160oveq1d 7429 . . . . . . . . . . 11 (𝑛 = 𝑦 → ((2↑𝑛) · 𝑥) = ((2↑𝑦) · 𝑥))
176175eqeq1d 2729 . . . . . . . . . 10 (𝑛 = 𝑦 → (((2↑𝑛) · 𝑥) = 𝐵 ↔ ((2↑𝑦) · 𝑥) = 𝐵))
177174, 176sylan9bb 509 . . . . . . . . 9 ((𝑡 = 𝑥𝑛 = 𝑦) → (((2↑𝑛) · 𝑡) = 𝐵 ↔ ((2↑𝑦) · 𝑥) = 𝐵))
178 simpl 482 . . . . . . . . . . 11 ((𝑡 = 𝑥𝑛 = 𝑦) → 𝑡 = 𝑥)
179178fveq2d 6895 . . . . . . . . . 10 ((𝑡 = 𝑥𝑛 = 𝑦) → (𝐴𝑡) = (𝐴𝑥))
180179fveq2d 6895 . . . . . . . . 9 ((𝑡 = 𝑥𝑛 = 𝑦) → (bits‘(𝐴𝑡)) = (bits‘(𝐴𝑥)))
181177, 180cbvrexdva2 3340 . . . . . . . 8 (𝑡 = 𝑥 → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵 ↔ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
182181cbvrexvw 3230 . . . . . . 7 (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵 ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
183 nfv 1910 . . . . . . . . . . . . . 14 𝑦 𝐴 ∈ (𝑇𝑅)
184 nfv 1910 . . . . . . . . . . . . . . 15 𝑦 𝑥 ∈ ℕ
185 nfre1 3277 . . . . . . . . . . . . . . 15 𝑦𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵
186184, 185nfan 1895 . . . . . . . . . . . . . 14 𝑦(𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
187183, 186nfan 1895 . . . . . . . . . . . . 13 𝑦(𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
188 simplr 768 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → 𝑥 ∈ ℕ)
18962ffvelcdmda 7088 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) → (𝐴𝑥) ∈ ℕ0)
190189adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → (𝐴𝑥) ∈ ℕ0)
191 elnn0 12496 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑥) ∈ ℕ0 ↔ ((𝐴𝑥) ∈ ℕ ∨ (𝐴𝑥) = 0))
192190, 191sylib 217 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ((𝐴𝑥) ∈ ℕ ∨ (𝐴𝑥) = 0))
193 n0i 4329 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (bits‘(𝐴𝑥)) → ¬ (bits‘(𝐴𝑥)) = ∅)
194193adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ¬ (bits‘(𝐴𝑥)) = ∅)
195 fveq2 6891 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑥) = 0 → (bits‘(𝐴𝑥)) = (bits‘0))
196 0bits 16405 . . . . . . . . . . . . . . . . . . . 20 (bits‘0) = ∅
197195, 196eqtrdi 2783 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑥) = 0 → (bits‘(𝐴𝑥)) = ∅)
198194, 197nsyl 140 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ¬ (𝐴𝑥) = 0)
199192, 198olcnd 876 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → (𝐴𝑥) ∈ ℕ)
20058simp3bi 1145 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ 𝐽)
201200sselda 3978 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑛 ∈ (𝐴 “ ℕ)) → 𝑛𝐽)
202 breq2 5146 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
203202notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
204203, 4elrab2 3683 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
205204simprbi 496 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝐽 → ¬ 2 ∥ 𝑛)
206201, 205syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑛 ∈ (𝐴 “ ℕ)) → ¬ 2 ∥ 𝑛)
207206ralrimiva 3141 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ (𝑇𝑅) → ∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛)
208 ffn 6716 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴:ℕ⟶ℕ0𝐴 Fn ℕ)
209 elpreima 7061 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 Fn ℕ → (𝑛 ∈ (𝐴 “ ℕ) ↔ (𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ)))
21062, 208, 2093syl 18 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) → (𝑛 ∈ (𝐴 “ ℕ) ↔ (𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ)))
211210imbi1d 341 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (𝑇𝑅) → ((𝑛 ∈ (𝐴 “ ℕ) → ¬ 2 ∥ 𝑛) ↔ ((𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ) → ¬ 2 ∥ 𝑛)))
212 impexp 450 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ) → ¬ 2 ∥ 𝑛) ↔ (𝑛 ∈ ℕ → ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛)))
213211, 212bitrdi 287 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (𝑇𝑅) → ((𝑛 ∈ (𝐴 “ ℕ) → ¬ 2 ∥ 𝑛) ↔ (𝑛 ∈ ℕ → ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛))))
214213ralbidv2 3168 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ (𝑇𝑅) → (∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ ℕ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛)))
215207, 214mpbid 231 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ (𝑇𝑅) → ∀𝑛 ∈ ℕ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛))
216 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑛 → (𝐴𝑥) = (𝐴𝑛))
217216eleq1d 2813 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑛 → ((𝐴𝑥) ∈ ℕ ↔ (𝐴𝑛) ∈ ℕ))
218 breq2 5146 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑛 → (2 ∥ 𝑥 ↔ 2 ∥ 𝑛))
219218notbid 318 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑛 → (¬ 2 ∥ 𝑥 ↔ ¬ 2 ∥ 𝑛))
220217, 219imbi12d 344 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑛 → (((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥) ↔ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛)))
221220cbvralvw 3229 . . . . . . . . . . . . . . . . . . . 20 (∀𝑥 ∈ ℕ ((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥) ↔ ∀𝑛 ∈ ℕ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛))
222215, 221sylibr 233 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ (𝑇𝑅) → ∀𝑥 ∈ ℕ ((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥))
223222r19.21bi 3243 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) → ((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥))
224223imp 406 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ (𝐴𝑥) ∈ ℕ) → ¬ 2 ∥ 𝑥)
225199, 224syldan 590 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ¬ 2 ∥ 𝑥)
226 breq2 5146 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑥))
227226notbid 318 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑥))
228227, 4elrab2 3683 . . . . . . . . . . . . . . . 16 (𝑥𝐽 ↔ (𝑥 ∈ ℕ ∧ ¬ 2 ∥ 𝑥))
229188, 225, 228sylanbrc 582 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → 𝑥𝐽)
230229adantlrr 720 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → 𝑥𝐽)
231230adantr 480 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → 𝑥𝐽)
232 simprr 772 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) → ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
233187, 231, 232r19.29af 3260 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) → 𝑥𝐽)
234233, 232jca 511 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) → (𝑥𝐽 ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
235234ex 412 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → ((𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵) → (𝑥𝐽 ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)))
236235reximdv2 3159 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵 → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
237236imp 406 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ ∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵) → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
238237adantlr 714 . . . . . . 7 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵) → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
239182, 238sylan2b 593 . . . . . 6 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
240172, 239r19.29vva 3208 . . . . 5 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) → ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵)
241128, 240impbida 800 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
24236, 241bitrd 279 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
243242ifbid 4547 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → if(𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))), 1, 0) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵, 1, 0))
24413, 23, 2433eqtrd 2771 1 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → ((𝐺𝐴)‘𝐵) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵, 1, 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846   = wceq 1534  wex 1774  wcel 2099  {cab 2704  wral 3056  wrex 3065  {crab 3427  Vcvv 3469  cin 3943  wss 3944  c0 4318  ifcif 4524  𝒫 cpw 4598  cop 4630   class class class wbr 5142  {copab 5204  cmpt 5225   × cxp 5670  ccnv 5671  dom cdm 5672  ran crn 5673  cres 5674  cima 5675  ccom 5676  Fun wfun 6536   Fn wfn 6537  wf 6538  1-1-ontowf1o 6541  cfv 6542  (class class class)co 7414  cmpo 7416  1st c1st 7985  2nd c2nd 7986   supp csupp 8159  m cmap 8836  Fincfn 8955  0cc0 11130  1c1 11131   · cmul 11135  cle 11271  cn 12234  2c2 12289  0cn0 12494  cexp 14050  Σcsu 15656  cdvds 16222  bitscbits 16385  𝟭cind 33565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-ac2 10478  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-disj 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-oi 9525  df-dju 9916  df-card 9954  df-acn 9957  df-ac 10131  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-rp 12999  df-fz 13509  df-fzo 13652  df-fl 13781  df-mod 13859  df-seq 13991  df-exp 14051  df-hash 14314  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-sum 15657  df-dvds 16223  df-bits 16388  df-ind 33566
This theorem is referenced by:  eulerpartlemgs2  33936
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