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Theorem eulerpartlemgvv 30412
Description: Lemma for eulerpart 30418: 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 30409 . . . 4 (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽))))))
1211fveq1d 6180 . . 3 (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴)‘𝐵) = (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵))
1312adantr 481 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → ((𝐺𝐴)‘𝐵) = (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵))
14 nnex 11011 . . . 4 ℕ ∈ V
1514a1i 11 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → ℕ ∈ V)
16 imassrn 5465 . . . . 5 (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ⊆ ran 𝐹
174, 5oddpwdc 30390 . . . . . 6 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
18 f1of 6124 . . . . . 6 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:(𝐽 × ℕ0)⟶ℕ)
19 frn 6040 . . . . . 6 (𝐹:(𝐽 × ℕ0)⟶ℕ → ran 𝐹 ⊆ ℕ)
2017, 18, 19mp2b 10 . . . . 5 ran 𝐹 ⊆ ℕ
2116, 20sstri 3604 . . . 4 (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ⊆ ℕ
2221a1i 11 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ⊆ ℕ)
23 simpr 477 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ)
24 indfval 30052 . . 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 1324 . 2 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))))‘𝐵) = if(𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))), 1, 0))
26 ffn 6032 . . . . . 6 (𝐹:(𝐽 × ℕ0)⟶ℕ → 𝐹 Fn (𝐽 × ℕ0))
2717, 18, 26mp2b 10 . . . . 5 𝐹 Fn (𝐽 × ℕ0)
28 inss1 3825 . . . . . . . 8 (𝒫 (𝐽 × ℕ0) ∩ Fin) ⊆ 𝒫 (𝐽 × ℕ0)
291, 2, 3, 4, 5, 6, 7, 8, 9, 10eulerpartlemmf 30411 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (bits ∘ (𝐴𝐽)) ∈ 𝐻)
301, 2, 3, 4, 5, 6, 7eulerpartlem1 30403 . . . . . . . . . . 11 𝑀:𝐻1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)
31 f1of 6124 . . . . . . . . . . 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 6344 . . . . . . . . 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 3593 . . . . . . 7 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) ∈ 𝒫 (𝐽 × ℕ0))
3635adantr 481 . . . . . 6 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝑀‘(bits ∘ (𝐴𝐽))) ∈ 𝒫 (𝐽 × ℕ0))
3736elpwid 4161 . . . . 5 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝑀‘(bits ∘ (𝐴𝐽))) ⊆ (𝐽 × ℕ0))
38 fvelimab 6240 . . . . 5 ((𝐹 Fn (𝐽 × ℕ0) ∧ (𝑀‘(bits ∘ (𝐴𝐽))) ⊆ (𝐽 × ℕ0)) → (𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ↔ ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵))
3927, 37, 38sylancr 694 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ↔ ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵))
40 ssrab2 3679 . . . . . . . . . 10 {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ⊆ ℕ
414, 40eqsstri 3627 . . . . . . . . 9 𝐽 ⊆ ℕ
427a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑇𝑅) → 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))}))
43 fveq1 6177 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = (bits ∘ (𝐴𝐽)) → (𝑟𝑥) = ((bits ∘ (𝐴𝐽))‘𝑥))
4443eleq2d 2685 . . . . . . . . . . . . . . . . . . 19 (𝑟 = (bits ∘ (𝐴𝐽)) → (𝑦 ∈ (𝑟𝑥) ↔ 𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)))
4544anbi2d 739 . . . . . . . . . . . . . . . . . 18 (𝑟 = (bits ∘ (𝐴𝐽)) → ((𝑥𝐽𝑦 ∈ (𝑟𝑥)) ↔ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))))
4645opabbidv 4707 . . . . . . . . . . . . . . . . 17 (𝑟 = (bits ∘ (𝐴𝐽)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))})
4746adantl 482 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑟 = (bits ∘ (𝐴𝐽))) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))})
4814, 41ssexi 4794 . . . . . . . . . . . . . . . . . 18 𝐽 ∈ V
49 abid2 2743 . . . . . . . . . . . . . . . . . . . 20 {𝑦𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)} = ((bits ∘ (𝐴𝐽))‘𝑥)
50 fvex 6188 . . . . . . . . . . . . . . . . . . . 20 ((bits ∘ (𝐴𝐽))‘𝑥) ∈ V
5149, 50eqeltri 2695 . . . . . . . . . . . . . . . . . . 19 {𝑦𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)} ∈ V
5251a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑥𝐽 → {𝑦𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)} ∈ V)
5348, 52opabex3 7131 . . . . . . . . . . . . . . . . 17 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} ∈ V
5453a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑇𝑅) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} ∈ V)
5542, 47, 29, 54fvmptd 6275 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))})
56 simpl 473 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑡𝑦 = 𝑛) → 𝑥 = 𝑡)
5756eleq1d 2684 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑡𝑦 = 𝑛) → (𝑥𝐽𝑡𝐽))
58 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑡𝑦 = 𝑛) → 𝑦 = 𝑛)
5956fveq2d 6182 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑡𝑦 = 𝑛) → ((bits ∘ (𝐴𝐽))‘𝑥) = ((bits ∘ (𝐴𝐽))‘𝑡))
6058, 59eleq12d 2693 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑡𝑦 = 𝑛) → (𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥) ↔ 𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡)))
6157, 60anbi12d 746 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑡𝑦 = 𝑛) → ((𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)) ↔ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))))
6261cbvopabv 4713 . . . . . . . . . . . . . . 15 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} = {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))}
6355, 62syl6eq 2670 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝑇𝑅) → (𝑀‘(bits ∘ (𝐴𝐽))) = {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))})
6463eleq2d 2685 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ 𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))}))
651, 2, 3, 4, 5, 6, 7, 8, 9eulerpartlemt0 30405 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0𝑚 ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
6665simp1bi 1074 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) → 𝐴 ∈ (ℕ0𝑚 ℕ))
67 nn0ex 11283 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ V
6867, 14elmap 7871 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (ℕ0𝑚 ℕ) ↔ 𝐴:ℕ⟶ℕ0)
6966, 68sylib 208 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (𝑇𝑅) → 𝐴:ℕ⟶ℕ0)
70 ffun 6035 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴:ℕ⟶ℕ0 → Fun 𝐴)
71 funres 5917 . . . . . . . . . . . . . . . . . . . . . . 23 (Fun 𝐴 → Fun (𝐴𝐽))
7269, 70, 713syl 18 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (𝑇𝑅) → Fun (𝐴𝐽))
7372adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → Fun (𝐴𝐽))
74 fssres 6057 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴:ℕ⟶ℕ0𝐽 ⊆ ℕ) → (𝐴𝐽):𝐽⟶ℕ0)
7569, 41, 74sylancl 693 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (𝑇𝑅) → (𝐴𝐽):𝐽⟶ℕ0)
76 fdm 6038 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝐽):𝐽⟶ℕ0 → dom (𝐴𝐽) = 𝐽)
7776eleq2d 2685 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝐽):𝐽⟶ℕ0 → (𝑡 ∈ dom (𝐴𝐽) ↔ 𝑡𝐽))
7875, 77syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (𝑇𝑅) → (𝑡 ∈ dom (𝐴𝐽) ↔ 𝑡𝐽))
7978biimpar 502 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → 𝑡 ∈ dom (𝐴𝐽))
80 fvco 6261 . . . . . . . . . . . . . . . . . . . . 21 ((Fun (𝐴𝐽) ∧ 𝑡 ∈ dom (𝐴𝐽)) → ((bits ∘ (𝐴𝐽))‘𝑡) = (bits‘((𝐴𝐽)‘𝑡)))
8173, 79, 80syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → ((bits ∘ (𝐴𝐽))‘𝑡) = (bits‘((𝐴𝐽)‘𝑡)))
82 fvres 6194 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡𝐽 → ((𝐴𝐽)‘𝑡) = (𝐴𝑡))
8382fveq2d 6182 . . . . . . . . . . . . . . . . . . . . 21 (𝑡𝐽 → (bits‘((𝐴𝐽)‘𝑡)) = (bits‘(𝐴𝑡)))
8483adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → (bits‘((𝐴𝐽)‘𝑡)) = (bits‘(𝐴𝑡)))
8581, 84eqtrd 2654 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → ((bits ∘ (𝐴𝐽))‘𝑡) = (bits‘(𝐴𝑡)))
8685eleq2d 2685 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡𝐽) → (𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡) ↔ 𝑛 ∈ (bits‘(𝐴𝑡))))
8786pm5.32da 672 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ (𝑇𝑅) → ((𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡)) ↔ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))))
8887opabbidv 4707 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑇𝑅) → {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} = {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))})
8988eleq2d 2685 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} ↔ 𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))}))
90 elopab 4973 . . . . . . . . . . . . . . 15 (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))} ↔ ∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))))
9189, 90syl6bb 276 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} ↔ ∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))))))
92 ancom 466 . . . . . . . . . . . . . . . . 17 ((𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ ((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩))
93 anass 680 . . . . . . . . . . . . . . . . 17 (((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) ↔ (𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
9492, 93bitri 264 . . . . . . . . . . . . . . . 16 ((𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ (𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
95942exbii 1773 . . . . . . . . . . . . . . 15 (∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ ∃𝑡𝑛(𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
96 df-rex 2915 . . . . . . . . . . . . . . . . . 18 (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ ↔ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩))
9796anbi2i 729 . . . . . . . . . . . . . . . . 17 ((𝑡𝐽 ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩) ↔ (𝑡𝐽 ∧ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
9897exbii 1772 . . . . . . . . . . . . . . . 16 (∃𝑡(𝑡𝐽 ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩) ↔ ∃𝑡(𝑡𝐽 ∧ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
99 df-rex 2915 . . . . . . . . . . . . . . . 16 (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ ↔ ∃𝑡(𝑡𝐽 ∧ ∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩))
100 exdistr 1917 . . . . . . . . . . . . . . . 16 (∃𝑡𝑛(𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)) ↔ ∃𝑡(𝑡𝐽 ∧ ∃𝑛(𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
10198, 99, 1003bitr4i 292 . . . . . . . . . . . . . . 15 (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ ↔ ∃𝑡𝑛(𝑡𝐽 ∧ (𝑛 ∈ (bits‘(𝐴𝑡)) ∧ 𝑤 = ⟨𝑡, 𝑛⟩)))
10295, 101bitr4i 267 . . . . . . . . . . . . . 14 (∃𝑡𝑛(𝑤 = ⟨𝑡, 𝑛⟩ ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ↔ ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩)
10391, 102syl6bb 276 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ {⟨𝑡, 𝑛⟩ ∣ (𝑡𝐽𝑛 ∈ ((bits ∘ (𝐴𝐽))‘𝑡))} ↔ ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩))
10464, 103bitrd 268 . . . . . . . . . . . 12 (𝐴 ∈ (𝑇𝑅) → (𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩))
105104biimpa 501 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩)
106105adantlr 750 . . . . . . . . . 10 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩)
107 fveq2 6178 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨𝑡, 𝑛⟩ → (𝐹𝑤) = (𝐹‘⟨𝑡, 𝑛⟩))
108107adantl 482 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → (𝐹𝑤) = (𝐹‘⟨𝑡, 𝑛⟩))
109 bitsss 15129 . . . . . . . . . . . . . . . . . . 19 (bits‘(𝐴𝑡)) ⊆ ℕ0
110109sseli 3591 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (bits‘(𝐴𝑡)) → 𝑛 ∈ ℕ0)
111110anim2i 592 . . . . . . . . . . . . . . . . 17 ((𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))) → (𝑡𝐽𝑛 ∈ ℕ0))
112111ad2antlr 762 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → (𝑡𝐽𝑛 ∈ ℕ0))
113 opelxp 5136 . . . . . . . . . . . . . . . . 17 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) ↔ (𝑡𝐽𝑛 ∈ ℕ0))
1144, 5oddpwdcv 30391 . . . . . . . . . . . . . . . . . 18 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)))
115 vex 3198 . . . . . . . . . . . . . . . . . . . . 21 𝑡 ∈ V
116 vex 3198 . . . . . . . . . . . . . . . . . . . . 21 𝑛 ∈ V
117115, 116op2nd 7162 . . . . . . . . . . . . . . . . . . . 20 (2nd ‘⟨𝑡, 𝑛⟩) = 𝑛
118117oveq2i 6646 . . . . . . . . . . . . . . . . . . 19 (2↑(2nd ‘⟨𝑡, 𝑛⟩)) = (2↑𝑛)
119115, 116op1st 7161 . . . . . . . . . . . . . . . . . . 19 (1st ‘⟨𝑡, 𝑛⟩) = 𝑡
120118, 119oveq12i 6647 . . . . . . . . . . . . . . . . . 18 ((2↑(2nd ‘⟨𝑡, 𝑛⟩)) · (1st ‘⟨𝑡, 𝑛⟩)) = ((2↑𝑛) · 𝑡)
121114, 120syl6eq 2670 . . . . . . . . . . . . . . . . 17 (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡))
122113, 121sylbir 225 . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . 14 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑤 = ⟨𝑡, 𝑛⟩) → ((2↑𝑛) · 𝑡) = (𝐹𝑤))
125124ex 450 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑤 = ⟨𝑡, 𝑛⟩ → ((2↑𝑛) · 𝑡) = (𝐹𝑤)))
126125anassrs 679 . . . . . . . . . . . 12 (((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ 𝑡𝐽) ∧ 𝑛 ∈ (bits‘(𝐴𝑡))) → (𝑤 = ⟨𝑡, 𝑛⟩ → ((2↑𝑛) · 𝑡) = (𝐹𝑤)))
127126reximdva 3014 . . . . . . . . . . 11 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ 𝑡𝐽) → (∃𝑛 ∈ (bits‘(𝐴𝑡))𝑤 = ⟨𝑡, 𝑛⟩ → ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤)))
128127reximdva 3014 . . . . . . . . . 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 3659 . . . . . . . . 9 (𝐽 ⊆ ℕ → (∃𝑡𝐽𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤)))
13141, 129, 130mpsyl 68 . . . . . . . 8 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤))
132131adantr 481 . . . . . . 7 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝐹𝑤) = 𝐵) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤))
133 eqeq2 2631 . . . . . . . . . 10 ((𝐹𝑤) = 𝐵 → (((2↑𝑛) · 𝑡) = (𝐹𝑤) ↔ ((2↑𝑛) · 𝑡) = 𝐵))
134133rexbidv 3048 . . . . . . . . 9 ((𝐹𝑤) = 𝐵 → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = (𝐹𝑤) ↔ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
135134adantl 482 . . . . . . . 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 3048 . . . . . . 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 222 . . . . . 6 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ 𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))) ∧ (𝐹𝑤) = 𝐵) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵)
138137r19.29an 3073 . . . . 5 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵) → ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵)
139 simp-5l 807 . . . . . . . 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 798 . . . . . . . 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 791 . . . . . . . . 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 481 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → Fun (𝐴𝐽))
14376eleq2d 2685 . . . . . . . . . . . . . 14 ((𝐴𝐽):𝐽⟶ℕ0 → (𝑥 ∈ dom (𝐴𝐽) ↔ 𝑥𝐽))
14475, 143syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ (𝑇𝑅) → (𝑥 ∈ dom (𝐴𝐽) ↔ 𝑥𝐽))
145144biimpar 502 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → 𝑥 ∈ dom (𝐴𝐽))
146 fvco 6261 . . . . . . . . . . . 12 ((Fun (𝐴𝐽) ∧ 𝑥 ∈ dom (𝐴𝐽)) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘((𝐴𝐽)‘𝑥)))
147142, 145, 146syl2anc 692 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘((𝐴𝐽)‘𝑥)))
148 fvres 6194 . . . . . . . . . . . . 13 (𝑥𝐽 → ((𝐴𝐽)‘𝑥) = (𝐴𝑥))
149148fveq2d 6182 . . . . . . . . . . . 12 (𝑥𝐽 → (bits‘((𝐴𝐽)‘𝑥)) = (bits‘(𝐴𝑥)))
150149adantl 482 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → (bits‘((𝐴𝐽)‘𝑥)) = (bits‘(𝐴𝑥)))
151147, 150eqtrd 2654 . . . . . . . . . 10 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥𝐽) → ((bits ∘ (𝐴𝐽))‘𝑥) = (bits‘(𝐴𝑥)))
152139, 140, 151syl2anc 692 . . . . . . . . 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 2702 . . . . . . . 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 2685 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → (⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))}))
155 opabid 4972 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))} ↔ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥)))
156154, 155syl6bb 276 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ↔ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))))
157156biimpar 502 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥𝐽𝑦 ∈ ((bits ∘ (𝐴𝐽))‘𝑥))) → ⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))))
158139, 140, 153, 157syl12anc 1322 . . . . . . 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 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)
16037ad4antr 767 . . . . . . . . 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 3596 . . . . . . . 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 4393 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ⟨𝑡, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
163162eleq1d 2684 . . . . . . . . . . 11 (𝑡 = 𝑥 → (⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0)))
164162fveq2d 6182 . . . . . . . . . . . 12 (𝑡 = 𝑥 → (𝐹‘⟨𝑡, 𝑦⟩) = (𝐹‘⟨𝑥, 𝑦⟩))
165 oveq2 6643 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ((2↑𝑦) · 𝑡) = ((2↑𝑦) · 𝑥))
166164, 165eqeq12d 2635 . . . . . . . . . . 11 (𝑡 = 𝑥 → ((𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥)))
167163, 166imbi12d 334 . . . . . . . . . 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 4394 . . . . . . . . . . . . 13 (𝑛 = 𝑦 → ⟨𝑡, 𝑛⟩ = ⟨𝑡, 𝑦⟩)
169168eleq1d 2684 . . . . . . . . . . . 12 (𝑛 = 𝑦 → (⟨𝑡, 𝑛⟩ ∈ (𝐽 × ℕ0) ↔ ⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0)))
170168fveq2d 6182 . . . . . . . . . . . . 13 (𝑛 = 𝑦 → (𝐹‘⟨𝑡, 𝑛⟩) = (𝐹‘⟨𝑡, 𝑦⟩))
171 oveq2 6643 . . . . . . . . . . . . . 14 (𝑛 = 𝑦 → (2↑𝑛) = (2↑𝑦))
172171oveq1d 6650 . . . . . . . . . . . . 13 (𝑛 = 𝑦 → ((2↑𝑛) · 𝑡) = ((2↑𝑦) · 𝑡))
173170, 172eqeq12d 2635 . . . . . . . . . . . 12 (𝑛 = 𝑦 → ((𝐹‘⟨𝑡, 𝑛⟩) = ((2↑𝑛) · 𝑡) ↔ (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡)))
174169, 173imbi12d 334 . . . . . . . . . . 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 2261 . . . . . . . . . 10 (⟨𝑡, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑡, 𝑦⟩) = ((2↑𝑦) · 𝑡))
176167, 175chvarv 2261 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0) → (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥))
177 eqeq2 2631 . . . . . . . . . 10 (((2↑𝑦) · 𝑥) = 𝐵 → ((𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵))
178177biimpa 501 . . . . . . . . 9 ((((2↑𝑦) · 𝑥) = 𝐵 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = ((2↑𝑦) · 𝑥)) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵)
179176, 178sylan2 491 . . . . . . . 8 ((((2↑𝑦) · 𝑥) = 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐽 × ℕ0)) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵)
180159, 161, 179syl2anc 692 . . . . . . 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 6178 . . . . . . . . 9 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
182181eqeq1d 2622 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑤) = 𝐵 ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵))
183182rspcev 3304 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (𝑀‘(bits ∘ (𝐴𝐽))) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝐵) → ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵)
184158, 180, 183syl2anc 692 . . . . . 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 6643 . . . . . . . . . . 11 (𝑡 = 𝑥 → ((2↑𝑛) · 𝑡) = ((2↑𝑛) · 𝑥))
186185eqeq1d 2622 . . . . . . . . . 10 (𝑡 = 𝑥 → (((2↑𝑛) · 𝑡) = 𝐵 ↔ ((2↑𝑛) · 𝑥) = 𝐵))
187171oveq1d 6650 . . . . . . . . . . 11 (𝑛 = 𝑦 → ((2↑𝑛) · 𝑥) = ((2↑𝑦) · 𝑥))
188187eqeq1d 2622 . . . . . . . . . 10 (𝑛 = 𝑦 → (((2↑𝑛) · 𝑥) = 𝐵 ↔ ((2↑𝑦) · 𝑥) = 𝐵))
189186, 188sylan9bb 735 . . . . . . . . 9 ((𝑡 = 𝑥𝑛 = 𝑦) → (((2↑𝑛) · 𝑡) = 𝐵 ↔ ((2↑𝑦) · 𝑥) = 𝐵))
190 simpl 473 . . . . . . . . . . 11 ((𝑡 = 𝑥𝑛 = 𝑦) → 𝑡 = 𝑥)
191190fveq2d 6182 . . . . . . . . . 10 ((𝑡 = 𝑥𝑛 = 𝑦) → (𝐴𝑡) = (𝐴𝑥))
192191fveq2d 6182 . . . . . . . . 9 ((𝑡 = 𝑥𝑛 = 𝑦) → (bits‘(𝐴𝑡)) = (bits‘(𝐴𝑥)))
193189, 192cbvrexdva2 3171 . . . . . . . 8 (𝑡 = 𝑥 → (∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵 ↔ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
194193cbvrexv 3167 . . . . . . 7 (∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵 ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
195 nfv 1841 . . . . . . . . . . . . . 14 𝑦 𝐴 ∈ (𝑇𝑅)
196 nfv 1841 . . . . . . . . . . . . . . 15 𝑦 𝑥 ∈ ℕ
197 nfre1 3002 . . . . . . . . . . . . . . 15 𝑦𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵
198196, 197nfan 1826 . . . . . . . . . . . . . 14 𝑦(𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
199195, 198nfan 1826 . . . . . . . . . . . . 13 𝑦(𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
200 simplr 791 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → 𝑥 ∈ ℕ)
201 n0i 3912 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (bits‘(𝐴𝑥)) → ¬ (bits‘(𝐴𝑥)) = ∅)
202201adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ¬ (bits‘(𝐴𝑥)) = ∅)
203 fveq2 6178 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑥) = 0 → (bits‘(𝐴𝑥)) = (bits‘0))
204 0bits 15142 . . . . . . . . . . . . . . . . . . . 20 (bits‘0) = ∅
205203, 204syl6eq 2670 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑥) = 0 → (bits‘(𝐴𝑥)) = ∅)
206202, 205nsyl 135 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ¬ (𝐴𝑥) = 0)
20769ffvelrnda 6345 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) → (𝐴𝑥) ∈ ℕ0)
208207adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → (𝐴𝑥) ∈ ℕ0)
209 elnn0 11279 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑥) ∈ ℕ0 ↔ ((𝐴𝑥) ∈ ℕ ∨ (𝐴𝑥) = 0))
210208, 209sylib 208 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ((𝐴𝑥) ∈ ℕ ∨ (𝐴𝑥) = 0))
211210orcomd 403 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ((𝐴𝑥) = 0 ∨ (𝐴𝑥) ∈ ℕ))
212211orcanai 951 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ¬ (𝐴𝑥) = 0) → (𝐴𝑥) ∈ ℕ)
213206, 212mpdan 701 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → (𝐴𝑥) ∈ ℕ)
21465simp3bi 1076 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ (𝑇𝑅) → (𝐴 “ ℕ) ⊆ 𝐽)
215214sselda 3595 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑛 ∈ (𝐴 “ ℕ)) → 𝑛𝐽)
216 breq2 4648 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑛 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑛))
217216notbid 308 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑛 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑛))
218217, 4elrab2 3360 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛𝐽 ↔ (𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛))
219218simprbi 480 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝐽 → ¬ 2 ∥ 𝑛)
220215, 219syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑛 ∈ (𝐴 “ ℕ)) → ¬ 2 ∥ 𝑛)
221220ralrimiva 2963 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ (𝑇𝑅) → ∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛)
222 ffn 6032 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴:ℕ⟶ℕ0𝐴 Fn ℕ)
223 elpreima 6323 . . . . . . . . . . . . . . . . . . . . . . . . 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 331 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ (𝑇𝑅) → ((𝑛 ∈ (𝐴 “ ℕ) → ¬ 2 ∥ 𝑛) ↔ ((𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ) → ¬ 2 ∥ 𝑛)))
226 impexp 462 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ) → ¬ 2 ∥ 𝑛) ↔ (𝑛 ∈ ℕ → ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛)))
227225, 226syl6bb 276 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (𝑇𝑅) → ((𝑛 ∈ (𝐴 “ ℕ) → ¬ 2 ∥ 𝑛) ↔ (𝑛 ∈ ℕ → ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛))))
228227ralbidv2 2981 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ (𝑇𝑅) → (∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛 ↔ ∀𝑛 ∈ ℕ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛)))
229221, 228mpbid 222 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ (𝑇𝑅) → ∀𝑛 ∈ ℕ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛))
230 fveq2 6178 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑛 → (𝐴𝑥) = (𝐴𝑛))
231230eleq1d 2684 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑛 → ((𝐴𝑥) ∈ ℕ ↔ (𝐴𝑛) ∈ ℕ))
232 breq2 4648 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑛 → (2 ∥ 𝑥 ↔ 2 ∥ 𝑛))
233232notbid 308 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑛 → (¬ 2 ∥ 𝑥 ↔ ¬ 2 ∥ 𝑛))
234231, 233imbi12d 334 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑛 → (((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥) ↔ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛)))
235234cbvralv 3166 . . . . . . . . . . . . . . . . . . . 20 (∀𝑥 ∈ ℕ ((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥) ↔ ∀𝑛 ∈ ℕ ((𝐴𝑛) ∈ ℕ → ¬ 2 ∥ 𝑛))
236229, 235sylibr 224 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ (𝑇𝑅) → ∀𝑥 ∈ ℕ ((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥))
237236r19.21bi 2929 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) → ((𝐴𝑥) ∈ ℕ → ¬ 2 ∥ 𝑥))
238237imp 445 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ (𝐴𝑥) ∈ ℕ) → ¬ 2 ∥ 𝑥)
239213, 238syldan 487 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → ¬ 2 ∥ 𝑥)
240 breq2 4648 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑥))
241240notbid 308 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑥))
242241, 4elrab2 3360 . . . . . . . . . . . . . . . 16 (𝑥𝐽 ↔ (𝑥 ∈ ℕ ∧ ¬ 2 ∥ 𝑥))
243200, 239, 242sylanbrc 697 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (𝑇𝑅) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → 𝑥𝐽)
244243adantlrr 756 . . . . . . . . . . . . . 14 (((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) → 𝑥𝐽)
245244adantr 481 . . . . . . . . . . . . 13 ((((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) ∧ 𝑦 ∈ (bits‘(𝐴𝑥))) ∧ ((2↑𝑦) · 𝑥) = 𝐵) → 𝑥𝐽)
246 simprr 795 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) → ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
247199, 245, 246r19.29af 3072 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) → 𝑥𝐽)
248247, 246jca 554 . . . . . . . . . . 11 ((𝐴 ∈ (𝑇𝑅) ∧ (𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)) → (𝑥𝐽 ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
249248ex 450 . . . . . . . . . 10 (𝐴 ∈ (𝑇𝑅) → ((𝑥 ∈ ℕ ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵) → (𝑥𝐽 ∧ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)))
250249reximdv2 3011 . . . . . . . . 9 (𝐴 ∈ (𝑇𝑅) → (∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵 → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵))
251250imp 445 . . . . . . . 8 ((𝐴 ∈ (𝑇𝑅) ∧ ∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵) → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
252251adantlr 750 . . . . . . 7 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑥 ∈ ℕ ∃𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵) → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
253194, 252sylan2b 492 . . . . . 6 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) → ∃𝑥𝐽𝑦 ∈ (bits‘(𝐴𝑥))((2↑𝑦) · 𝑥) = 𝐵)
254184, 253r19.29vva 3076 . . . . 5 (((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) ∧ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵) → ∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵)
255138, 254impbida 876 . . . 4 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (∃𝑤 ∈ (𝑀‘(bits ∘ (𝐴𝐽)))(𝐹𝑤) = 𝐵 ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
25639, 255bitrd 268 . . 3 ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → (𝐵 ∈ (𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽)))) ↔ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵))
257256ifbid 4099 . 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 2658 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 196  wo 383  wa 384   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wral 2909  wrex 2910  {crab 2913  Vcvv 3195  cin 3566  wss 3567  c0 3907  ifcif 4077  𝒫 cpw 4149  cop 4174   class class class wbr 4644  {copab 4703  cmpt 4720   × cxp 5102  ccnv 5103  dom cdm 5104  ran crn 5105  cres 5106  cima 5107  ccom 5108  Fun wfun 5870   Fn wfn 5871  wf 5872  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-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-ac2 9270  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-fal 1487  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-int 4467  df-iun 4513  df-disj 4612  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-se 5064  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-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-acn 8753  df-ac 8924  df-cda 8975  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-3 11065  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-rp 11818  df-fz 12312  df-fzo 12450  df-fl 12576  df-mod 12652  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-sum 14398  df-dvds 14965  df-bits 15125  df-ind 30047
This theorem is referenced by:  eulerpartlemgs2  30416
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