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Theorem eulerpartlemgvv 30906
Description: Lemma for eulerpart 30912: 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 30903 . . . 4 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽))))))
1211fveq1d 6381 . . 3 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴)‘𝐵) = (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵))
1312adantr 472 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → ((𝐺𝐴)‘𝐵) = (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵))
14 nnex 11285 . . 3 ℕ ∈ V
15 imassrn 5661 . . . 4 (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ⊆ ran 𝐹
164, 5oddpwdc 30884 . . . . 5 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
17 f1of 6324 . . . . 5 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:(𝐽 × ℕ0)⟶ℕ)
18 frn 6231 . . . . 5 (𝐹:(𝐽 × ℕ0)⟶ℕ → ran 𝐹 ⊆ ℕ)
1916, 17, 18mp2b 10 . . . 4 ran 𝐹 ⊆ ℕ
2015, 19sstri 3772 . . 3 (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ⊆ ℕ
21 simpr 477 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ)
22 indfval 30546 . . 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 1589 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵) = if(𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))), 1, 0))
24 ffn 6225 . . . . . 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 30905 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (bits ∘ (𝐴𝐽)) ∈ 𝐻)
271, 2, 3, 4, 5, 6, 7eulerpartlem1 30897 . . . . . . . . . . 11 𝑀:𝐻1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)
28 f1of 6324 . . . . . . . . . . 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)
3029ffvelrni 6552 . . . . . . . . 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 3966 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) ∈ 𝒫 (𝐽 × ℕ0))
3332adantr 472 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝑀‘(bits ∘ (𝐴𝐽))) ∈ 𝒫 (𝐽 × ℕ0))
3433elpwid 4329 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝑀‘(bits ∘ (𝐴𝐽))) ⊆ (𝐽 × ℕ0))
35 fvelimab 6446 . . . . 5 ((𝐹 Fn (𝐽 × ℕ0) ∧ (𝑀‘(bits ∘ (𝐴𝐽))) ⊆ (𝐽 × ℕ0)) → (𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ↔ ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵))
3625, 34, 35sylancr 581 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ↔ ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵))
374ssrab3 3850 . . . . . . . . 9 𝐽 ⊆ ℕ
387a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑇𝑅) → 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))}))
39 fveq1 6378 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = (bits ∘ (𝐴𝐽)) → (𝑟𝑥) = ((bits ∘ (𝐴𝐽))‘𝑥))
4039eleq2d 2830 . . . . . . . . . . . . . . . . . . 19 (𝑟 = (bits ∘ (𝐴𝐽)) → (𝑦 ∈ (𝑟𝑥) ↔ 𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)))
4140anbi2d 622 . . . . . . . . . . . . . . . . . 18 (𝑟 = (bits ∘ (𝐴𝐽)) → ((𝑥𝐽𝑦 ∈ (𝑟𝑥)) ↔ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))))
4241opabbidv 4877 . . . . . . . . . . . . . . . . 17 (𝑟 = (bits ∘ (𝐴𝐽)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))})
4342adantl 473 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑟 = (bits ∘ (𝐴𝐽))) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))})
4414, 37ssexi 4966 . . . . . . . . . . . . . . . . . 18 𝐽 ∈ V
45 abid2 2888 . . . . . . . . . . . . . . . . . . . 20 {𝑦𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)} = ((bits ∘ (𝐴𝐽))‘𝑥)
4645fvexi 6393 . . . . . . . . . . . . . . . . . . 19 {𝑦𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)} ∈ V
4746a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑥𝐽 → {𝑦𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)} ∈ V)
4844, 47opabex3 7348 . . . . . . . . . . . . . . . . 17 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} ∈ V
4948a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑇𝑅) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} ∈ V)
5038, 43, 26, 49fvmptd 6481 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))})
51 simpl 474 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑡𝑦 = 𝑛) → 𝑥 = 𝑡)
5251eleq1d 2829 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑡𝑦 = 𝑛) → (𝑥𝐽𝑡𝐽))
53 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑡𝑦 = 𝑛) → 𝑦 = 𝑛)
5451fveq2d 6383 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑡𝑦 = 𝑛) → ((bits ∘ (𝐴𝐽))‘𝑥) = ((bits ∘ (𝐴𝐽))‘𝑡))
5553, 54eleq12d 2838 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑡𝑦 = 𝑛) → (𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥) ↔ 𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡)))
5652, 55anbi12d 624 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑡𝑦 = 𝑛) → ((𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)) ↔ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))))
5756cbvopabv 4883 . . . . . . . . . . . . . . 15 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} = {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))}
5850, 57syl6eq 2815 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) = {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))})
5958eleq2d 2830 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ 𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))}))
601, 2, 3, 4, 5, 6, 7, 8, 9eulerpartlemt0 30899 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
6160simp1bi 1175 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0𝑚 ℕ))
62 nn0ex 11549 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
6362, 14elmap 8093 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (ℕ0𝑚 ℕ) ↔ 𝐴:ℕ⟶ℕ0)
6461, 63sylib 209 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
65 ffun 6228 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴:ℕ⟶ℕ0 → Fun 𝐴)
66 funres 6112 . . . . . . . . . . . . . . . . . . . . . 22 (Fun 𝐴 → Fun (𝐴𝐽))
6764, 65, 663syl 18 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ (𝑇𝑅) → Fun (𝐴𝐽))
68 fssres 6254 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴:ℕ⟶ℕ0𝐽 ⊆ ℕ) → (𝐴𝐽):𝐽⟶ℕ0)
6964, 37, 68sylancl 580 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (𝑇𝑅) → (𝐴𝐽):𝐽⟶ℕ0)
70 fdm 6233 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝐽):𝐽⟶ℕ0 → dom (𝐴𝐽) = 𝐽)
7170eleq2d 2830 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝐽):𝐽⟶ℕ0 → (𝑡 ∈ dom (𝐴𝐽) ↔ 𝑡𝐽))
7269, 71syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (𝑇𝑅) → (𝑡 ∈ dom (𝐴𝐽) ↔ 𝑡𝐽))
7372biimpar 469 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → 𝑡 ∈ dom (𝐴𝐽))
74 fvco 6467 . . . . . . . . . . . . . . . . . . . . 21 ((Fun (𝐴𝐽) ∧ 𝑡 ∈ dom (𝐴𝐽)) → ((bits ∘ (𝐴𝐽))‘𝑡) = (bits‘((𝐴𝐽)‘𝑡)))
7567, 73, 74syl2an2r 675 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → ((bits ∘ (𝐴𝐽))‘𝑡) = (bits‘((𝐴𝐽)‘𝑡)))
76 fvres 6398 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡𝐽 → ((𝐴𝐽)‘𝑡) = (𝐴𝑡))
7776fveq2d 6383 . . . . . . . . . . . . . . . . . . . . 21 (𝑡𝐽 → (bits‘((𝐴𝐽)‘𝑡)) = (bits‘(𝐴𝑡)))
7877adantl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → (bits‘((𝐴𝐽)‘𝑡)) = (bits‘(𝐴𝑡)))
7975, 78eqtrd 2799 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → ((bits ∘ (𝐴𝐽))‘𝑡) = (bits‘(𝐴𝑡)))
8079eleq2d 2830 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → (𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡) ↔ 𝑛 ∈ (bits‘(𝐴𝑡))))
8180pm5.32da 574 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ (𝑇𝑅) → ((𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡)) ↔ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))))
8281opabbidv 4877 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑇𝑅) → {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} = {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))})
8382eleq2d 2830 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} ↔ 𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))}))
84 elopab 5146 . . . . . . . . . . . . . . 15 (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))} ↔ ∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))))
8583, 84syl6bb 278 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} ↔ ∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))))))
86 ancom 452 . . . . . . . . . . . . . . . . 17 ((𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ ((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩))
87 anass 460 . . . . . . . . . . . . . . . . 17 (((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) ↔ (𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
8886, 87bitri 266 . . . . . . . . . . . . . . . 16 ((𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ (𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
89882exbii 1944 . . . . . . . . . . . . . . 15 (∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ ∃𝑡𝑛(𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
90 df-rex 3061 . . . . . . . . . . . . . . . . . 18 (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ ↔ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩))
9190anbi2i 616 . . . . . . . . . . . . . . . . 17 ((𝑡𝐽 ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩) ↔ (𝑡𝐽 ∧ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
9291exbii 1943 . . . . . . . . . . . . . . . 16 (∃𝑡(𝑡𝐽 ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩) ↔ ∃𝑡(𝑡𝐽 ∧ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
93 df-rex 3061 . . . . . . . . . . . . . . . 16 (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ ↔ ∃𝑡(𝑡𝐽 ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩))
94 exdistr 2049 . . . . . . . . . . . . . . . 16 (∃𝑡𝑛(𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)) ↔ ∃𝑡(𝑡𝐽 ∧ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
9592, 93, 943bitr4i 294 . . . . . . . . . . . . . . 15 (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ ↔ ∃𝑡𝑛(𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
9689, 95bitr4i 269 . . . . . . . . . . . . . 14 (∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩)
9785, 96syl6bb 278 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} ↔ ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩))
9859, 97bitrd 270 . . . . . . . . . . . 12 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩))
9998biimpa 468 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩)
10099adantlr 706 . . . . . . . . . 10 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩)
101 fveq2 6379 . . . . . . . . . . . . . 14 (𝑤 = ⟨𝑡, 𝑛⟩ → (𝐹𝑤) = (𝐹‘⟨𝑡, 𝑛⟩))
102101adantl 473 . . . . . . . . . . . . 13 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → (𝐹𝑤) = (𝐹‘⟨𝑡, 𝑛⟩))
103 bitsss 15443 . . . . . . . . . . . . . . . . 17 (bits‘(𝐴𝑡)) ⊆ ℕ0
104103sseli 3759 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (bits‘(𝐴𝑡)) → 𝑛 ∈ ℕ0)
105104anim2i 610 . . . . . . . . . . . . . . 15 ((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) → (𝑡𝐽𝑛 ∈ ℕ0))
106105ad2antlr 718 . . . . . . . . . . . . . 14 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → (𝑡𝐽𝑛 ∈ ℕ0))
107 opelxp 5315 . . . . . . . . . . . . . . 15 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) ↔ (𝑡𝐽𝑛 ∈ ℕ0))
1084, 5oddpwdcv 30885 . . . . . . . . . . . . . . . 16 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)))
109 vex 3353 . . . . . . . . . . . . . . . . . . 19 𝑡 ∈ V
110 vex 3353 . . . . . . . . . . . . . . . . . . 19 𝑛 ∈ V
111109, 110op2nd 7379 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨𝑡, 𝑛⟩) = 𝑛
112111oveq2i 6857 . . . . . . . . . . . . . . . . 17 (2↑(2nd ‘⟨𝑡, 𝑛⟩)) = (2↑𝑛)
113109, 110op1st 7378 . . . . . . . . . . . . . . . . 17 (1st ‘⟨𝑡, 𝑛⟩) = 𝑡
114112, 113oveq12i 6858 . . . . . . . . . . . . . . . 16 ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)) = ((2↑𝑛) · 𝑡)
115108, 114syl6eq 2815 . . . . . . . . . . . . . . 15 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
116107, 115sylbir 226 . . . . . . . . . . . . . 14 ((𝑡𝐽𝑛 ∈ ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
117106, 116syl 17 . . . . . . . . . . . . 13 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
118102, 117eqtr2d 2800 . . . . . . . . . . . 12 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → ((2↑𝑛) · 𝑡) = (𝐹𝑤))
119118ex 401 . . . . . . . . . . 11 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑤 = ⟨𝑡, 𝑛⟩ → ((2↑𝑛) · 𝑡) = (𝐹𝑤)))
120119reximdvva 3166 . . . . . . . . . 10 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤)))
121100, 120mpd 15 . . . . . . . . 9 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤))
122 ssrexv 3829 . . . . . . . . 9 (𝐽 ⊆ ℕ → (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤)))
12337, 121, 122mpsyl 68 . . . . . . . 8 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤))
124123adantr 472 . . . . . . 7 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝐹𝑤) = 𝐵) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤))
125 eqeq2 2776 . . . . . . . . . 10 ((𝐹𝑤) = 𝐵 → (((2↑𝑛) · 𝑡) = (𝐹𝑤) ↔ ((2↑𝑛) · 𝑡) = 𝐵))
126125rexbidv 3199 . . . . . . . . 9 ((𝐹𝑤) = 𝐵 → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
127126adantl 473 . . . . . . . 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))
128127rexbidv 3199 . . . . . . 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))
129124, 128mpbid 223 . . . . . 6 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝐹𝑤) = 𝐵) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵)
130129r19.29an 3224 . . . . 5 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵)
131 simp-5l 805 . . . . . . . 8 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → 𝐴 ∈ (𝑇𝑅))
132 simpllr 793 . . . . . . . 8 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → 𝑥𝐽)
133 simplr 785 . . . . . . . . 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)))
13470eleq2d 2830 . . . . . . . . . . . . . 14 ((𝐴𝐽):𝐽⟶ℕ0 → (𝑥 ∈ dom (𝐴𝐽) ↔ 𝑥𝐽))
13569, 134syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑥 ∈ dom (𝐴𝐽) ↔ 𝑥𝐽))
136135biimpar 469 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → 𝑥 ∈ dom (𝐴𝐽))
137 fvco 6467 . . . . . . . . . . . 12 ((Fun (𝐴𝐽) ∧ 𝑥 ∈ dom (𝐴𝐽)) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘((𝐴𝐽)‘𝑥)))
13867, 136, 137syl2an2r 675 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘((𝐴𝐽)‘𝑥)))
139 fvres 6398 . . . . . . . . . . . . 13 (𝑥𝐽 → ((𝐴𝐽)‘𝑥) = (𝐴𝑥))
140139fveq2d 6383 . . . . . . . . . . . 12 (𝑥𝐽 → (bits‘((𝐴𝐽)‘𝑥)) = (bits‘(𝐴𝑥)))
141140adantl 473 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → (bits‘((𝐴𝐽)‘𝑥)) = (bits‘(𝐴𝑥)))
142138, 141eqtrd 2799 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘(𝐴𝑥)))
143131, 132, 142syl2anc 579 . . . . . . . . 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)))
144133, 143eleqtrrd 2847 . . . . . . . 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))
14550eleq2d 2830 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → (⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))}))
146 opabid 5145 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} ↔ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)))
147145, 146syl6bb 278 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))))
148147biimpar 469 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))) → ⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))))
149131, 132, 144, 148syl12anc 865 . . . . . . 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))))
150 simpr 477 . . . . . . . 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)
15134ad4antr 724 . . . . . . . . 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))
152151, 149sseldd 3764 . . . . . . . 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))
153 opeq1 4561 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ⟨𝑡, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
154153eleq1d 2829 . . . . . . . . . . 11 (𝑡 = 𝑥 → (⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0)))
155153fveq2d 6383 . . . . . . . . . . . 12 (𝑡 = 𝑥 → (𝐹‘⟨𝑡, 𝑦⟩) = (𝐹‘⟨𝑥, 𝑦⟩))
156 oveq2 6854 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ((2↑𝑦) · 𝑡) = ((2↑𝑦) · 𝑥))
157155, 156eqeq12d 2780 . . . . . . . . . . 11 (𝑡 = 𝑥 → ((𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥)))
158154, 157imbi12d 335 . . . . . . . . . 10 (𝑡 = 𝑥 → ((⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥))))
159 opeq2 4562 . . . . . . . . . . . . 13 (𝑛 = 𝑦 → ⟨𝑡, 𝑛⟩ = ⟨𝑡, 𝑦⟩)
160159eleq1d 2829 . . . . . . . . . . . 12 (𝑛 = 𝑦 → (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) ↔ ⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0)))
161159fveq2d 6383 . . . . . . . . . . . . 13 (𝑛 = 𝑦 → (𝐹‘⟨𝑡, 𝑛⟩) = (𝐹‘⟨𝑡, 𝑦⟩))
162 oveq2 6854 . . . . . . . . . . . . . 14 (𝑛 = 𝑦 → (2↑𝑛) = (2↑𝑦))
163162oveq1d 6861 . . . . . . . . . . . . 13 (𝑛 = 𝑦 → ((2↑𝑛) · 𝑡) = ((2↑𝑦) · 𝑡))
164161, 163eqeq12d 2780 . . . . . . . . . . . 12 (𝑛 = 𝑦 → ((𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡) ↔ (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡)))
165160, 164imbi12d 335 . . . . . . . . . . 11 (𝑛 = 𝑦 → ((⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡)) ↔ (⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡))))
166165, 115chvarv 2369 . . . . . . . . . 10 (⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡))
167158, 166chvarv 2369 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥))
168 eqeq2 2776 . . . . . . . . . 10 (((2↑𝑦) · 𝑥) = 𝐵 → ((𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵))
169168biimpa 468 . . . . . . . . 9 ((((2↑𝑦) · 𝑥) = 𝐵 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥)) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵)
170167, 169sylan2 586 . . . . . . . 8 ((((2↑𝑦) · 𝑥) = 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0)) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵)
171150, 152, 170syl2anc 579 . . . . . . 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)
172 fveqeq2 6388 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑤) = 𝐵 ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵))
173172rspcev 3462 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵) → ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵)
174149, 171, 173syl2anc 579 . . . . . 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)
175 oveq2 6854 . . . . . . . . . . 11 (𝑡 = 𝑥 → ((2↑𝑛) · 𝑡) = ((2↑𝑛) · 𝑥))
176175eqeq1d 2767 . . . . . . . . . 10 (𝑡 = 𝑥 → (((2↑𝑛) · 𝑡) = 𝐵 ↔ ((2↑𝑛) · 𝑥) = 𝐵))
177162oveq1d 6861 . . . . . . . . . . 11 (𝑛 = 𝑦 → ((2↑𝑛) · 𝑥) = ((2↑𝑦) · 𝑥))
178177eqeq1d 2767 . . . . . . . . . 10 (𝑛 = 𝑦 → (((2↑𝑛) · 𝑥) = 𝐵 ↔ ((2↑𝑦) · 𝑥) = 𝐵))
179176, 178sylan9bb 505 . . . . . . . . 9 ((𝑡 = 𝑥𝑛 = 𝑦) → (((2↑𝑛) · 𝑡) = 𝐵 ↔ ((2↑𝑦) · 𝑥) = 𝐵))
180 simpl 474 . . . . . . . . . . 11 ((𝑡 = 𝑥𝑛 = 𝑦) → 𝑡 = 𝑥)
181180fveq2d 6383 . . . . . . . . . 10 ((𝑡 = 𝑥𝑛 = 𝑦) → (𝐴𝑡) = (𝐴𝑥))
182181fveq2d 6383 . . . . . . . . 9 ((𝑡 = 𝑥𝑛 = 𝑦) → (bits‘(𝐴𝑡)) = (bits‘(𝐴𝑥)))
183179, 182cbvrexdva2 3324 . . . . . . . 8 (𝑡 = 𝑥 → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵 ↔ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
184183cbvrexv 3320 . . . . . . 7 (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵 ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
185 nfv 2009 . . . . . . . . . . . . . 14 𝑦 𝐴 ∈ (𝑇𝑅)
186 nfv 2009 . . . . . . . . . . . . . . 15 𝑦 𝑥 ∈ ℕ
187 nfre1 3151 . . . . . . . . . . . . . . 15 𝑦𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵
188186, 187nfan 1998 . . . . . . . . . . . . . 14 𝑦(𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
189185, 188nfan 1998 . . . . . . . . . . . . 13 𝑦(𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
190 simplr 785 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → 𝑥 ∈ ℕ)
191 n0i 4086 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (bits‘(𝐴𝑥)) → ¬ (bits‘(𝐴𝑥)) = ∅)
192191adantl 473 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ¬ (bits‘(𝐴𝑥)) = ∅)
193 fveq2 6379 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑥) = 0 → (bits‘(𝐴𝑥)) = (bits‘0))
194 0bits 15456 . . . . . . . . . . . . . . . . . . . 20 (bits‘0) = ∅
195193, 194syl6eq 2815 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑥) = 0 → (bits‘(𝐴𝑥)) = ∅)
196192, 195nsyl 137 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ¬ (𝐴𝑥) = 0)
19764ffvelrnda 6553 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) → (𝐴𝑥) ∈ ℕ0)
198197adantr 472 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → (𝐴𝑥) ∈ ℕ0)
199 elnn0 11544 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑥) ∈ ℕ0 ↔ ((𝐴𝑥) ∈ ℕ ∨ (𝐴𝑥) = 0))
200198, 199sylib 209 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ((𝐴𝑥) ∈ ℕ ∨ (𝐴𝑥) = 0))
201200orcomd 897 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ((𝐴𝑥) = 0 ∨ (𝐴𝑥) ∈ ℕ))
202201orcanai 1025 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ¬ (𝐴𝑥) = 0) → (𝐴𝑥) ∈ ℕ)
203196, 202mpdan 678 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → (𝐴𝑥) ∈ ℕ)
20460simp3bi 1177 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ 𝐽)
205204sselda 3763 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑛 ∈ (𝐴 “ ℕ)) → 𝑛𝐽)
206 breq2 4815 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
207206notbid 309 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
208207, 4elrab2 3525 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
209208simprbi 490 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝐽 → ¬ 2 ∥ 𝑛)
210205, 209syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑛 ∈ (𝐴 “ ℕ)) → ¬ 2 ∥ 𝑛)
211210ralrimiva 3113 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ (𝑇𝑅) → ∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛)
212 ffn 6225 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴:ℕ⟶ℕ0𝐴 Fn ℕ)
213 elpreima 6531 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 Fn ℕ → (𝑛 ∈ (𝐴 “ ℕ) ↔ (𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ)))
21464, 212, 2133syl 18 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) → (𝑛 ∈ (𝐴 “ ℕ) ↔ (𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ)))
215214imbi1d 332 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (𝑇𝑅) → ((𝑛 ∈ (𝐴 “ ℕ) → ¬ 2 ∥ 𝑛) ↔ ((𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ) → ¬ 2 ∥ 𝑛)))
216 impexp 441 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ) → ¬ 2 ∥ 𝑛) ↔ (𝑛 ∈ ℕ → ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛)))
217215, 216syl6bb 278 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (𝑇𝑅) → ((𝑛 ∈ (𝐴 “ ℕ) → ¬ 2 ∥ 𝑛) ↔ (𝑛 ∈ ℕ → ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛))))
218217ralbidv2 3131 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ (𝑇𝑅) → (∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ ℕ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛)))
219211, 218mpbid 223 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ (𝑇𝑅) → ∀𝑛 ∈ ℕ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛))
220 fveq2 6379 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑛 → (𝐴𝑥) = (𝐴𝑛))
221220eleq1d 2829 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑛 → ((𝐴𝑥) ∈ ℕ ↔ (𝐴𝑛) ∈ ℕ))
222 breq2 4815 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑛 → (2 ∥ 𝑥 ↔ 2 ∥ 𝑛))
223222notbid 309 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑛 → (¬ 2 ∥ 𝑥 ↔ ¬ 2 ∥ 𝑛))
224221, 223imbi12d 335 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑛 → (((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥) ↔ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛)))
225224cbvralv 3319 . . . . . . . . . . . . . . . . . . . 20 (∀𝑥 ∈ ℕ ((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥) ↔ ∀𝑛 ∈ ℕ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛))
226219, 225sylibr 225 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ (𝑇𝑅) → ∀𝑥 ∈ ℕ ((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥))
227226r19.21bi 3079 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) → ((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥))
228227imp 395 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ (𝐴𝑥) ∈ ℕ) → ¬ 2 ∥ 𝑥)
229203, 228syldan 585 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ¬ 2 ∥ 𝑥)
230 breq2 4815 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑥))
231230notbid 309 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑥))
232231, 4elrab2 3525 . . . . . . . . . . . . . . . 16 (𝑥𝐽 ↔ (𝑥 ∈ ℕ ∧ ¬ 2 ∥ 𝑥))
233190, 229, 232sylanbrc 578 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → 𝑥𝐽)
234233adantlrr 712 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → 𝑥𝐽)
235234adantr 472 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → 𝑥𝐽)
236 simprr 789 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) → ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
237189, 235, 236r19.29af 3223 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) → 𝑥𝐽)
238237, 236jca 507 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) → (𝑥𝐽 ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
239238ex 401 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → ((𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵) → (𝑥𝐽 ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)))
240239reximdv2 3160 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵 → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
241240imp 395 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ ∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵) → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
242241adantlr 706 . . . . . . 7 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵) → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
243184, 242sylan2b 587 . . . . . 6 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
244174, 243r19.29vva 3228 . . . . 5 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) → ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵)
245130, 244impbida 835 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
24636, 245bitrd 270 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
247246ifbid 4267 . 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))
24813, 23, 2473eqtrd 2803 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 197  wa 384  wo 873   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cin 3733  wss 3734  c0 4081  ifcif 4245  𝒫 cpw 4317  cop 4342   class class class wbr 4811  {copab 4873  cmpt 4890   × cxp 5277  ccnv 5278  dom cdm 5279  ran crn 5280  cres 5281  cima 5282  ccom 5283  Fun wfun 6064   Fn wfn 6065  wf 6066  1-1-ontowf1o 6069  cfv 6070  (class class class)co 6846  cmpt2 6848  1st c1st 7368  2nd c2nd 7369   supp csupp 7501  𝑚 cmap 8064  Fincfn 8164  0cc0 10193  1c1 10194   · cmul 10198  cle 10333  cn 11278  2c2 11331  0cn0 11542  cexp 13072  Σcsu 14715  cdvds 15279  bitscbits 15436  𝟭cind 30540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-ac2 9542  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-disj 4780  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-supp 7502  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-er 7951  df-map 8066  df-pm 8067  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-acn 9023  df-ac 9194  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-n0 11543  df-xnn0 11615  df-z 11629  df-uz 11892  df-rp 12034  df-fz 12539  df-fzo 12679  df-fl 12806  df-mod 12882  df-seq 13014  df-exp 13073  df-hash 13327  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-clim 14518  df-sum 14716  df-dvds 15280  df-bits 15439  df-ind 30541
This theorem is referenced by:  eulerpartlemgs2  30910
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