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Theorem eulerpartlemgvv 29568
Description: Lemma for eulerpart 29574: 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 29565 . . . 4 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽))))))
1211fveq1d 6087 . . 3 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴)‘𝐵) = (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵))
1312adantr 479 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → ((𝐺𝐴)‘𝐵) = (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵))
14 nnex 10870 . . . 4 ℕ ∈ V
1514a1i 11 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → ℕ ∈ V)
16 imassrn 5380 . . . . 5 (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ⊆ ran 𝐹
174, 5oddpwdc 29546 . . . . . 6 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
18 f1of 6032 . . . . . 6 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:(𝐽 × ℕ0)⟶ℕ)
19 frn 5949 . . . . . 6 (𝐹:(𝐽 × ℕ0)⟶ℕ → ran 𝐹 ⊆ ℕ)
2017, 18, 19mp2b 10 . . . . 5 ran 𝐹 ⊆ ℕ
2116, 20sstri 3573 . . . 4 (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ⊆ ℕ
2221a1i 11 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ⊆ ℕ)
23 simpr 475 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ)
24 indfval 29209 . . 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))
2515, 22, 23, 24syl3anc 1317 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵) = if(𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))), 1, 0))
26 ffn 5941 . . . . . 6 (𝐹:(𝐽 × ℕ0)⟶ℕ → 𝐹 Fn (𝐽 × ℕ0))
2717, 18, 26mp2b 10 . . . . 5 𝐹 Fn (𝐽 × ℕ0)
28 inss1 3791 . . . . . . . 8 (𝒫 (𝐽 × ℕ0) ∩ Fin) ⊆ 𝒫 (𝐽 × ℕ0)
291, 2, 3, 4, 5, 6, 7, 8, 9, 10eulerpartlemmf 29567 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (bits ∘ (𝐴𝐽)) ∈ 𝐻)
301, 2, 3, 4, 5, 6, 7eulerpartlem1 29559 . . . . . . . . . . 11 𝑀:𝐻1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)
31 f1of 6032 . . . . . . . . . . 11 (𝑀:𝐻1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) → 𝑀:𝐻⟶(𝒫 (𝐽 × ℕ0) ∩ Fin))
3230, 31ax-mp 5 . . . . . . . . . 10 𝑀:𝐻⟶(𝒫 (𝐽 × ℕ0) ∩ Fin)
3332ffvelrni 6248 . . . . . . . . 9 ((bits ∘ (𝐴𝐽)) ∈ 𝐻 → (𝑀‘(bits ∘ (𝐴𝐽))) ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin))
3429, 33syl 17 . . . . . . . 8 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) ∈ (𝒫 (𝐽 × ℕ0) ∩ Fin))
3528, 34sseldi 3562 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) ∈ 𝒫 (𝐽 × ℕ0))
3635adantr 479 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝑀‘(bits ∘ (𝐴𝐽))) ∈ 𝒫 (𝐽 × ℕ0))
3736elpwid 4114 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝑀‘(bits ∘ (𝐴𝐽))) ⊆ (𝐽 × ℕ0))
38 fvelimab 6145 . . . . 5 ((𝐹 Fn (𝐽 × ℕ0) ∧ (𝑀‘(bits ∘ (𝐴𝐽))) ⊆ (𝐽 × ℕ0)) → (𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ↔ ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵))
3927, 37, 38sylancr 693 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ↔ ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵))
40 ssrab2 3646 . . . . . . . . . 10 {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
414, 40eqsstri 3594 . . . . . . . . 9 𝐽 ⊆ ℕ
427a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑇𝑅) → 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))}))
43 fveq1 6084 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = (bits ∘ (𝐴𝐽)) → (𝑟𝑥) = ((bits ∘ (𝐴𝐽))‘𝑥))
4443eleq2d 2669 . . . . . . . . . . . . . . . . . . 19 (𝑟 = (bits ∘ (𝐴𝐽)) → (𝑦 ∈ (𝑟𝑥) ↔ 𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)))
4544anbi2d 735 . . . . . . . . . . . . . . . . . 18 (𝑟 = (bits ∘ (𝐴𝐽)) → ((𝑥𝐽𝑦 ∈ (𝑟𝑥)) ↔ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))))
4645opabbidv 4639 . . . . . . . . . . . . . . . . 17 (𝑟 = (bits ∘ (𝐴𝐽)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))})
4746adantl 480 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑟 = (bits ∘ (𝐴𝐽))) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))})
4814, 41ssexi 4723 . . . . . . . . . . . . . . . . . 18 𝐽 ∈ V
49 abid2 2728 . . . . . . . . . . . . . . . . . . . 20 {𝑦𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)} = ((bits ∘ (𝐴𝐽))‘𝑥)
50 fvex 6095 . . . . . . . . . . . . . . . . . . . 20 ((bits ∘ (𝐴𝐽))‘𝑥) ∈ V
5149, 50eqeltri 2680 . . . . . . . . . . . . . . . . . . 19 {𝑦𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)} ∈ V
5251a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑥𝐽 → {𝑦𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)} ∈ V)
5348, 52opabex3 7012 . . . . . . . . . . . . . . . . 17 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} ∈ V
5453a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑇𝑅) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} ∈ V)
5542, 47, 29, 54fvmptd 6179 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))})
56 simpl 471 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑡𝑦 = 𝑛) → 𝑥 = 𝑡)
5756eleq1d 2668 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑡𝑦 = 𝑛) → (𝑥𝐽𝑡𝐽))
58 simpr 475 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑡𝑦 = 𝑛) → 𝑦 = 𝑛)
5956fveq2d 6089 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑡𝑦 = 𝑛) → ((bits ∘ (𝐴𝐽))‘𝑥) = ((bits ∘ (𝐴𝐽))‘𝑡))
6058, 59eleq12d 2678 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑡𝑦 = 𝑛) → (𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥) ↔ 𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡)))
6157, 60anbi12d 742 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑡𝑦 = 𝑛) → ((𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)) ↔ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))))
6261cbvopabv 4645 . . . . . . . . . . . . . . 15 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} = {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))}
6355, 62syl6eq 2656 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) = {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))})
6463eleq2d 2669 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ 𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))}))
651, 2, 3, 4, 5, 6, 7, 8, 9eulerpartlemt0 29561 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
6665simp1bi 1068 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0𝑚 ℕ))
67 nn0ex 11142 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ V
6867, 14elmap 7746 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (ℕ0𝑚 ℕ) ↔ 𝐴:ℕ⟶ℕ0)
6966, 68sylib 206 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
70 ffun 5944 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴:ℕ⟶ℕ0 → Fun 𝐴)
71 funres 5826 . . . . . . . . . . . . . . . . . . . . . . 23 (Fun 𝐴 → Fun (𝐴𝐽))
7269, 70, 713syl 18 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (𝑇𝑅) → Fun (𝐴𝐽))
7372adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → Fun (𝐴𝐽))
74 fssres 5965 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴:ℕ⟶ℕ0𝐽 ⊆ ℕ) → (𝐴𝐽):𝐽⟶ℕ0)
7569, 41, 74sylancl 692 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (𝑇𝑅) → (𝐴𝐽):𝐽⟶ℕ0)
76 fdm 5947 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝐽):𝐽⟶ℕ0 → dom (𝐴𝐽) = 𝐽)
7776eleq2d 2669 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝐽):𝐽⟶ℕ0 → (𝑡 ∈ dom (𝐴𝐽) ↔ 𝑡𝐽))
7875, 77syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (𝑇𝑅) → (𝑡 ∈ dom (𝐴𝐽) ↔ 𝑡𝐽))
7978biimpar 500 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → 𝑡 ∈ dom (𝐴𝐽))
80 fvco 6166 . . . . . . . . . . . . . . . . . . . . 21 ((Fun (𝐴𝐽) ∧ 𝑡 ∈ dom (𝐴𝐽)) → ((bits ∘ (𝐴𝐽))‘𝑡) = (bits‘((𝐴𝐽)‘𝑡)))
8173, 79, 80syl2anc 690 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → ((bits ∘ (𝐴𝐽))‘𝑡) = (bits‘((𝐴𝐽)‘𝑡)))
82 fvres 6099 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡𝐽 → ((𝐴𝐽)‘𝑡) = (𝐴𝑡))
8382fveq2d 6089 . . . . . . . . . . . . . . . . . . . . 21 (𝑡𝐽 → (bits‘((𝐴𝐽)‘𝑡)) = (bits‘(𝐴𝑡)))
8483adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → (bits‘((𝐴𝐽)‘𝑡)) = (bits‘(𝐴𝑡)))
8581, 84eqtrd 2640 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → ((bits ∘ (𝐴𝐽))‘𝑡) = (bits‘(𝐴𝑡)))
8685eleq2d 2669 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → (𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡) ↔ 𝑛 ∈ (bits‘(𝐴𝑡))))
8786pm5.32da 670 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ (𝑇𝑅) → ((𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡)) ↔ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))))
8887opabbidv 4639 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑇𝑅) → {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} = {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))})
8988eleq2d 2669 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} ↔ 𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))}))
90 elopab 4895 . . . . . . . . . . . . . . 15 (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))} ↔ ∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))))
9189, 90syl6bb 274 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} ↔ ∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))))))
92 ancom 464 . . . . . . . . . . . . . . . . 17 ((𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ ((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩))
93 anass 678 . . . . . . . . . . . . . . . . 17 (((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) ↔ (𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
9492, 93bitri 262 . . . . . . . . . . . . . . . 16 ((𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ (𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
95942exbii 1764 . . . . . . . . . . . . . . 15 (∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ ∃𝑡𝑛(𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
96 df-rex 2898 . . . . . . . . . . . . . . . . . 18 (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ ↔ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩))
9796anbi2i 725 . . . . . . . . . . . . . . . . 17 ((𝑡𝐽 ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩) ↔ (𝑡𝐽 ∧ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
9897exbii 1763 . . . . . . . . . . . . . . . 16 (∃𝑡(𝑡𝐽 ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩) ↔ ∃𝑡(𝑡𝐽 ∧ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
99 df-rex 2898 . . . . . . . . . . . . . . . 16 (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ ↔ ∃𝑡(𝑡𝐽 ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩))
100 exdistr 1905 . . . . . . . . . . . . . . . 16 (∃𝑡𝑛(𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)) ↔ ∃𝑡(𝑡𝐽 ∧ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
10198, 99, 1003bitr4i 290 . . . . . . . . . . . . . . 15 (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ ↔ ∃𝑡𝑛(𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
10295, 101bitr4i 265 . . . . . . . . . . . . . 14 (∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩)
10391, 102syl6bb 274 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} ↔ ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩))
10464, 103bitrd 266 . . . . . . . . . . . 12 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩))
105104biimpa 499 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩)
106105adantlr 746 . . . . . . . . . 10 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩)
107 fveq2 6085 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨𝑡, 𝑛⟩ → (𝐹𝑤) = (𝐹‘⟨𝑡, 𝑛⟩))
108107adantl 480 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → (𝐹𝑤) = (𝐹‘⟨𝑡, 𝑛⟩))
109 bitsss 14929 . . . . . . . . . . . . . . . . . . 19 (bits‘(𝐴𝑡)) ⊆ ℕ0
110109sseli 3560 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (bits‘(𝐴𝑡)) → 𝑛 ∈ ℕ0)
111110anim2i 590 . . . . . . . . . . . . . . . . 17 ((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) → (𝑡𝐽𝑛 ∈ ℕ0))
112111ad2antlr 758 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → (𝑡𝐽𝑛 ∈ ℕ0))
113 opelxp 5057 . . . . . . . . . . . . . . . . 17 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) ↔ (𝑡𝐽𝑛 ∈ ℕ0))
1144, 5oddpwdcv 29547 . . . . . . . . . . . . . . . . . 18 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)))
115 vex 3172 . . . . . . . . . . . . . . . . . . . . 21 𝑡 ∈ V
116 vex 3172 . . . . . . . . . . . . . . . . . . . . 21 𝑛 ∈ V
117115, 116op2nd 7042 . . . . . . . . . . . . . . . . . . . 20 (2nd ‘⟨𝑡, 𝑛⟩) = 𝑛
118117oveq2i 6535 . . . . . . . . . . . . . . . . . . 19 (2↑(2nd ‘⟨𝑡, 𝑛⟩)) = (2↑𝑛)
119115, 116op1st 7041 . . . . . . . . . . . . . . . . . . 19 (1st ‘⟨𝑡, 𝑛⟩) = 𝑡
120118, 119oveq12i 6536 . . . . . . . . . . . . . . . . . 18 ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)) = ((2↑𝑛) · 𝑡)
121114, 120syl6eq 2656 . . . . . . . . . . . . . . . . 17 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
122113, 121sylbir 223 . . . . . . . . . . . . . . . 16 ((𝑡𝐽𝑛 ∈ ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
123112, 122syl 17 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
124108, 123eqtr2d 2641 . . . . . . . . . . . . . 14 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → ((2↑𝑛) · 𝑡) = (𝐹𝑤))
125124ex 448 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑤 = ⟨𝑡, 𝑛⟩ → ((2↑𝑛) · 𝑡) = (𝐹𝑤)))
126125anassrs 677 . . . . . . . . . . . 12 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ 𝑡𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → (𝑤 = ⟨𝑡, 𝑛⟩ → ((2↑𝑛) · 𝑡) = (𝐹𝑤)))
127126reximdva 2996 . . . . . . . . . . 11 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ 𝑡𝐽) → (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ → ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤)))
128127reximdva 2996 . . . . . . . . . 10 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤)))
129106, 128mpd 15 . . . . . . . . 9 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤))
130 ssrexv 3626 . . . . . . . . 9 (𝐽 ⊆ ℕ → (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤)))
13141, 129, 130mpsyl 65 . . . . . . . 8 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤))
132131adantr 479 . . . . . . 7 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝐹𝑤) = 𝐵) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤))
133 eqeq2 2617 . . . . . . . . . 10 ((𝐹𝑤) = 𝐵 → (((2↑𝑛) · 𝑡) = (𝐹𝑤) ↔ ((2↑𝑛) · 𝑡) = 𝐵))
134133rexbidv 3030 . . . . . . . . 9 ((𝐹𝑤) = 𝐵 → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
135134adantl 480 . . . . . . . 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))
136135rexbidv 3030 . . . . . . 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))
137132, 136mpbid 220 . . . . . 6 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝐹𝑤) = 𝐵) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵)
138137r19.29an 3055 . . . . 5 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵)
139 simp-5l 803 . . . . . . . 8 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → 𝐴 ∈ (𝑇𝑅))
140 simpllr 794 . . . . . . . 8 ((((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) ∧ 𝑥𝐽) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → 𝑥𝐽)
141 simplr 787 . . . . . . . . 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)))
14272adantr 479 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → Fun (𝐴𝐽))
14376eleq2d 2669 . . . . . . . . . . . . . 14 ((𝐴𝐽):𝐽⟶ℕ0 → (𝑥 ∈ dom (𝐴𝐽) ↔ 𝑥𝐽))
14475, 143syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑥 ∈ dom (𝐴𝐽) ↔ 𝑥𝐽))
145144biimpar 500 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → 𝑥 ∈ dom (𝐴𝐽))
146 fvco 6166 . . . . . . . . . . . 12 ((Fun (𝐴𝐽) ∧ 𝑥 ∈ dom (𝐴𝐽)) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘((𝐴𝐽)‘𝑥)))
147142, 145, 146syl2anc 690 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘((𝐴𝐽)‘𝑥)))
148 fvres 6099 . . . . . . . . . . . . 13 (𝑥𝐽 → ((𝐴𝐽)‘𝑥) = (𝐴𝑥))
149148fveq2d 6089 . . . . . . . . . . . 12 (𝑥𝐽 → (bits‘((𝐴𝐽)‘𝑥)) = (bits‘(𝐴𝑥)))
150149adantl 480 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → (bits‘((𝐴𝐽)‘𝑥)) = (bits‘(𝐴𝑥)))
151147, 150eqtrd 2640 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘(𝐴𝑥)))
152139, 140, 151syl2anc 690 . . . . . . . . 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)))
153141, 152eleqtrrd 2687 . . . . . . . 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))
15455eleq2d 2669 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → (⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))}))
155 opabid 4894 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} ↔ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)))
156154, 155syl6bb 274 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))))
157156biimpar 500 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))) → ⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))))
158139, 140, 153, 157syl12anc 1315 . . . . . . 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))))
159 simpr 475 . . . . . . . 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)
16037ad4antr 763 . . . . . . . . 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))
161160, 158sseldd 3565 . . . . . . . 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))
162 opeq1 4331 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ⟨𝑡, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
163162eleq1d 2668 . . . . . . . . . . 11 (𝑡 = 𝑥 → (⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0)))
164162fveq2d 6089 . . . . . . . . . . . 12 (𝑡 = 𝑥 → (𝐹‘⟨𝑡, 𝑦⟩) = (𝐹‘⟨𝑥, 𝑦⟩))
165 oveq2 6532 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ((2↑𝑦) · 𝑡) = ((2↑𝑦) · 𝑥))
166164, 165eqeq12d 2621 . . . . . . . . . . 11 (𝑡 = 𝑥 → ((𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥)))
167163, 166imbi12d 332 . . . . . . . . . 10 (𝑡 = 𝑥 → ((⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥))))
168 opeq2 4332 . . . . . . . . . . . . 13 (𝑛 = 𝑦 → ⟨𝑡, 𝑛⟩ = ⟨𝑡, 𝑦⟩)
169168eleq1d 2668 . . . . . . . . . . . 12 (𝑛 = 𝑦 → (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) ↔ ⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0)))
170168fveq2d 6089 . . . . . . . . . . . . 13 (𝑛 = 𝑦 → (𝐹‘⟨𝑡, 𝑛⟩) = (𝐹‘⟨𝑡, 𝑦⟩))
171 oveq2 6532 . . . . . . . . . . . . . 14 (𝑛 = 𝑦 → (2↑𝑛) = (2↑𝑦))
172171oveq1d 6539 . . . . . . . . . . . . 13 (𝑛 = 𝑦 → ((2↑𝑛) · 𝑡) = ((2↑𝑦) · 𝑡))
173170, 172eqeq12d 2621 . . . . . . . . . . . 12 (𝑛 = 𝑦 → ((𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡) ↔ (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡)))
174169, 173imbi12d 332 . . . . . . . . . . 11 (𝑛 = 𝑦 → ((⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡)) ↔ (⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡))))
175174, 121chvarv 2246 . . . . . . . . . 10 (⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡))
176167, 175chvarv 2246 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥))
177 eqeq2 2617 . . . . . . . . . 10 (((2↑𝑦) · 𝑥) = 𝐵 → ((𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵))
178177biimpa 499 . . . . . . . . 9 ((((2↑𝑦) · 𝑥) = 𝐵 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥)) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵)
179176, 178sylan2 489 . . . . . . . 8 ((((2↑𝑦) · 𝑥) = 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0)) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵)
180159, 161, 179syl2anc 690 . . . . . . 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)
181 fveq2 6085 . . . . . . . . 9 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
182181eqeq1d 2608 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑤) = 𝐵 ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵))
183182rspcev 3278 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵) → ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵)
184158, 180, 183syl2anc 690 . . . . . 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)
185 oveq2 6532 . . . . . . . . . . 11 (𝑡 = 𝑥 → ((2↑𝑛) · 𝑡) = ((2↑𝑛) · 𝑥))
186185eqeq1d 2608 . . . . . . . . . 10 (𝑡 = 𝑥 → (((2↑𝑛) · 𝑡) = 𝐵 ↔ ((2↑𝑛) · 𝑥) = 𝐵))
187171oveq1d 6539 . . . . . . . . . . 11 (𝑛 = 𝑦 → ((2↑𝑛) · 𝑥) = ((2↑𝑦) · 𝑥))
188187eqeq1d 2608 . . . . . . . . . 10 (𝑛 = 𝑦 → (((2↑𝑛) · 𝑥) = 𝐵 ↔ ((2↑𝑦) · 𝑥) = 𝐵))
189186, 188sylan9bb 731 . . . . . . . . 9 ((𝑡 = 𝑥𝑛 = 𝑦) → (((2↑𝑛) · 𝑡) = 𝐵 ↔ ((2↑𝑦) · 𝑥) = 𝐵))
190 simpl 471 . . . . . . . . . . 11 ((𝑡 = 𝑥𝑛 = 𝑦) → 𝑡 = 𝑥)
191190fveq2d 6089 . . . . . . . . . 10 ((𝑡 = 𝑥𝑛 = 𝑦) → (𝐴𝑡) = (𝐴𝑥))
192191fveq2d 6089 . . . . . . . . 9 ((𝑡 = 𝑥𝑛 = 𝑦) → (bits‘(𝐴𝑡)) = (bits‘(𝐴𝑥)))
193189, 192cbvrexdva2 3148 . . . . . . . 8 (𝑡 = 𝑥 → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵 ↔ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
194193cbvrexv 3144 . . . . . . 7 (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵 ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
195 nfv 1829 . . . . . . . . . . . . . 14 𝑦 𝐴 ∈ (𝑇𝑅)
196 nfv 1829 . . . . . . . . . . . . . . 15 𝑦 𝑥 ∈ ℕ
197 nfre1 2984 . . . . . . . . . . . . . . 15 𝑦𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵
198196, 197nfan 1815 . . . . . . . . . . . . . 14 𝑦(𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
199195, 198nfan 1815 . . . . . . . . . . . . 13 𝑦(𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
200 simplr 787 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → 𝑥 ∈ ℕ)
201 n0i 3875 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (bits‘(𝐴𝑥)) → ¬ (bits‘(𝐴𝑥)) = ∅)
202201adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ¬ (bits‘(𝐴𝑥)) = ∅)
203 fveq2 6085 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑥) = 0 → (bits‘(𝐴𝑥)) = (bits‘0))
204 0bits 14942 . . . . . . . . . . . . . . . . . . . 20 (bits‘0) = ∅
205203, 204syl6eq 2656 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑥) = 0 → (bits‘(𝐴𝑥)) = ∅)
206202, 205nsyl 133 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ¬ (𝐴𝑥) = 0)
20769ffvelrnda 6249 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) → (𝐴𝑥) ∈ ℕ0)
208207adantr 479 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → (𝐴𝑥) ∈ ℕ0)
209 elnn0 11138 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑥) ∈ ℕ0 ↔ ((𝐴𝑥) ∈ ℕ ∨ (𝐴𝑥) = 0))
210208, 209sylib 206 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ((𝐴𝑥) ∈ ℕ ∨ (𝐴𝑥) = 0))
211210orcomd 401 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ((𝐴𝑥) = 0 ∨ (𝐴𝑥) ∈ ℕ))
212211orcanai 949 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ¬ (𝐴𝑥) = 0) → (𝐴𝑥) ∈ ℕ)
213206, 212mpdan 698 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → (𝐴𝑥) ∈ ℕ)
21465simp3bi 1070 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ 𝐽)
215214sselda 3564 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑛 ∈ (𝐴 “ ℕ)) → 𝑛𝐽)
216 breq2 4578 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
217216notbid 306 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
218217, 4elrab2 3329 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
219218simprbi 478 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝐽 → ¬ 2 ∥ 𝑛)
220215, 219syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑛 ∈ (𝐴 “ ℕ)) → ¬ 2 ∥ 𝑛)
221220ralrimiva 2945 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ (𝑇𝑅) → ∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛)
222 ffn 5941 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴:ℕ⟶ℕ0𝐴 Fn ℕ)
223 elpreima 6227 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 Fn ℕ → (𝑛 ∈ (𝐴 “ ℕ) ↔ (𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ)))
22469, 222, 2233syl 18 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) → (𝑛 ∈ (𝐴 “ ℕ) ↔ (𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ)))
225224imbi1d 329 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (𝑇𝑅) → ((𝑛 ∈ (𝐴 “ ℕ) → ¬ 2 ∥ 𝑛) ↔ ((𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ) → ¬ 2 ∥ 𝑛)))
226 impexp 460 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ) → ¬ 2 ∥ 𝑛) ↔ (𝑛 ∈ ℕ → ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛)))
227225, 226syl6bb 274 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (𝑇𝑅) → ((𝑛 ∈ (𝐴 “ ℕ) → ¬ 2 ∥ 𝑛) ↔ (𝑛 ∈ ℕ → ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛))))
228227ralbidv2 2963 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ (𝑇𝑅) → (∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ ℕ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛)))
229221, 228mpbid 220 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ (𝑇𝑅) → ∀𝑛 ∈ ℕ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛))
230 fveq2 6085 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑛 → (𝐴𝑥) = (𝐴𝑛))
231230eleq1d 2668 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑛 → ((𝐴𝑥) ∈ ℕ ↔ (𝐴𝑛) ∈ ℕ))
232 breq2 4578 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑛 → (2 ∥ 𝑥 ↔ 2 ∥ 𝑛))
233232notbid 306 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑛 → (¬ 2 ∥ 𝑥 ↔ ¬ 2 ∥ 𝑛))
234231, 233imbi12d 332 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑛 → (((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥) ↔ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛)))
235234cbvralv 3143 . . . . . . . . . . . . . . . . . . . 20 (∀𝑥 ∈ ℕ ((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥) ↔ ∀𝑛 ∈ ℕ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛))
236229, 235sylibr 222 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ (𝑇𝑅) → ∀𝑥 ∈ ℕ ((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥))
237236r19.21bi 2912 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) → ((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥))
238237imp 443 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ (𝐴𝑥) ∈ ℕ) → ¬ 2 ∥ 𝑥)
239213, 238syldan 485 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ¬ 2 ∥ 𝑥)
240 breq2 4578 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑥))
241240notbid 306 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑥))
242241, 4elrab2 3329 . . . . . . . . . . . . . . . 16 (𝑥𝐽 ↔ (𝑥 ∈ ℕ ∧ ¬ 2 ∥ 𝑥))
243200, 239, 242sylanbrc 694 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → 𝑥𝐽)
244243adantlrr 752 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → 𝑥𝐽)
245244adantr 479 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → 𝑥𝐽)
246 simprr 791 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) → ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
247199, 245, 246r19.29af 3054 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) → 𝑥𝐽)
248247, 246jca 552 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) → (𝑥𝐽 ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
249248ex 448 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → ((𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵) → (𝑥𝐽 ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)))
250249reximdv2 2993 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵 → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
251250imp 443 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ ∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵) → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
252251adantlr 746 . . . . . . 7 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵) → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
253194, 252sylan2b 490 . . . . . 6 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
254184, 253r19.29vva 3058 . . . . 5 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) → ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵)
255138, 254impbida 872 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
25639, 255bitrd 266 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
257256ifbid 4054 . 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))
25813, 25, 2573eqtrd 2644 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 194  wo 381  wa 382   = wceq 1474  wex 1694  wcel 1976  {cab 2592  wral 2892  wrex 2893  {crab 2896  Vcvv 3169  cin 3535  wss 3536  c0 3870  ifcif 4032  𝒫 cpw 4104  cop 4127   class class class wbr 4574  {copab 4633  cmpt 4634   × cxp 5023  ccnv 5024  dom cdm 5025  ran crn 5026  cres 5027  cima 5028  ccom 5029  Fun wfun 5781   Fn wfn 5782  wf 5783  1-1-ontowf1o 5786  cfv 5787  (class class class)co 6524  cmpt2 6526  1st c1st 7031  2nd c2nd 7032   supp csupp 7156  𝑚 cmap 7718  Fincfn 7815  0cc0 9789  1c1 9790   · cmul 9794  cle 9928  cn 10864  2c2 10914  0cn0 11136  cexp 12674  Σcsu 14207  cdvds 14764  bitscbits 14922  𝟭cind 29203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395  ax-ac2 9142  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-disj 4545  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-supp 7157  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-2o 7422  df-oadd 7425  df-er 7603  df-map 7720  df-pm 7721  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-sup 8205  df-inf 8206  df-oi 8272  df-card 8622  df-acn 8625  df-ac 8796  df-cda 8847  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-3 10924  df-n0 11137  df-z 11208  df-uz 11517  df-rp 11662  df-fz 12150  df-fzo 12287  df-fl 12407  df-mod 12483  df-seq 12616  df-exp 12675  df-hash 12932  df-cj 13630  df-re 13631  df-im 13632  df-sqrt 13766  df-abs 13767  df-clim 14010  df-sum 14208  df-dvds 14765  df-bits 14925  df-ind 29204
This theorem is referenced by:  eulerpartlemgs2  29572
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