ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2sqpwodd GIF version

Theorem 2sqpwodd 11699
Description: The greatest power of two dividing twice the square of an integer is an odd power of two. (Contributed by Jim Kingdon, 17-Nov-2021.)
Hypotheses
Ref Expression
oddpwdc.j 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
oddpwdc.f 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
Assertion
Ref Expression
2sqpwodd (𝐴 ∈ ℕ → ¬ 2 ∥ (2nd ‘(𝐹‘(2 · (𝐴↑2)))))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝐽,𝑦   𝑥,𝐴,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧
Allowed substitution hint:   𝐽(𝑧)

Proof of Theorem 2sqpwodd
StepHypRef Expression
1 oddpwdc.j . . . . . . . . 9 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
2 oddpwdc.f . . . . . . . . 9 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
31, 2oddpwdc 11697 . . . . . . . 8 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4 f1ocnv 5336 . . . . . . . 8 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5 f1of 5323 . . . . . . . 8 (𝐹:ℕ–1-1-onto→(𝐽 × ℕ0) → 𝐹:ℕ⟶(𝐽 × ℕ0))
63, 4, 5mp2b 8 . . . . . . 7 𝐹:ℕ⟶(𝐽 × ℕ0)
76ffvelrni 5508 . . . . . 6 (𝐴 ∈ ℕ → (𝐹𝐴) ∈ (𝐽 × ℕ0))
8 xp2nd 6018 . . . . . 6 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
97, 8syl 14 . . . . 5 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
109nn0zd 9075 . . . 4 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℤ)
11 2nn 8785 . . . . . 6 2 ∈ ℕ
1211a1i 9 . . . . 5 (𝐴 ∈ ℕ → 2 ∈ ℕ)
1312nnzd 9076 . . . 4 (𝐴 ∈ ℕ → 2 ∈ ℤ)
1410, 13zmulcld 9083 . . 3 (𝐴 ∈ ℕ → ((2nd ‘(𝐹𝐴)) · 2) ∈ ℤ)
15 dvdsmul2 11364 . . . 4 (((2nd ‘(𝐹𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
1610, 13, 15syl2anc 406 . . 3 (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
17 oddp1even 11421 . . . . 5 (((2nd ‘(𝐹𝐴)) · 2) ∈ ℤ → (¬ 2 ∥ ((2nd ‘(𝐹𝐴)) · 2) ↔ 2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1)))
1817biimprd 157 . . . 4 (((2nd ‘(𝐹𝐴)) · 2) ∈ ℤ → (2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1) → ¬ 2 ∥ ((2nd ‘(𝐹𝐴)) · 2)))
1918con2d 596 . . 3 (((2nd ‘(𝐹𝐴)) · 2) ∈ ℤ → (2 ∥ ((2nd ‘(𝐹𝐴)) · 2) → ¬ 2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1)))
2014, 16, 19sylc 62 . 2 (𝐴 ∈ ℕ → ¬ 2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1))
21 xp1st 6017 . . . . . . . . . . 11 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(𝐹𝐴)) ∈ 𝐽)
227, 21syl 14 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ 𝐽)
23 breq2 3899 . . . . . . . . . . . . 13 (𝑧 = (1st ‘(𝐹𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(𝐹𝐴))))
2423notbid 639 . . . . . . . . . . . 12 (𝑧 = (1st ‘(𝐹𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2524, 1elrab2 2812 . . . . . . . . . . 11 ((1st ‘(𝐹𝐴)) ∈ 𝐽 ↔ ((1st ‘(𝐹𝐴)) ∈ ℕ ∧ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2625simplbi 270 . . . . . . . . . 10 ((1st ‘(𝐹𝐴)) ∈ 𝐽 → (1st ‘(𝐹𝐴)) ∈ ℕ)
2722, 26syl 14 . . . . . . . . 9 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℕ)
2827nnsqcld 10338 . . . . . . . 8 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ ℕ)
2925simprbi 271 . . . . . . . . . . 11 ((1st ‘(𝐹𝐴)) ∈ 𝐽 → ¬ 2 ∥ (1st ‘(𝐹𝐴)))
3022, 29syl 14 . . . . . . . . . 10 (𝐴 ∈ ℕ → ¬ 2 ∥ (1st ‘(𝐹𝐴)))
31 2prm 11654 . . . . . . . . . . 11 2 ∈ ℙ
3227nnzd 9076 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℤ)
33 euclemma 11670 . . . . . . . . . . . 12 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ (2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴)))))
34 oridm 729 . . . . . . . . . . . 12 ((2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴)))
3533, 34syl6bb 195 . . . . . . . . . . 11 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3631, 32, 32, 35mp3an2i 1303 . . . . . . . . . 10 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3730, 36mtbird 645 . . . . . . . . 9 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
3827nncnd 8644 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℂ)
3938sqvald 10314 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) = ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
4039breq2d 3907 . . . . . . . . 9 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴))↑2) ↔ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴)))))
4137, 40mtbird 645 . . . . . . . 8 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2))
42 breq2 3899 . . . . . . . . . 10 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
4342notbid 639 . . . . . . . . 9 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
4443, 1elrab2 2812 . . . . . . . 8 (((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(𝐹𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
4528, 41, 44sylanbrc 411 . . . . . . 7 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ 𝐽)
4612nnnn0d 8934 . . . . . . . . 9 (𝐴 ∈ ℕ → 2 ∈ ℕ0)
479, 46nn0mulcld 8939 . . . . . . . 8 (𝐴 ∈ ℕ → ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0)
48 peano2nn0 8921 . . . . . . . 8 (((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0 → (((2nd ‘(𝐹𝐴)) · 2) + 1) ∈ ℕ0)
4947, 48syl 14 . . . . . . 7 (𝐴 ∈ ℕ → (((2nd ‘(𝐹𝐴)) · 2) + 1) ∈ ℕ0)
50 opelxp 4529 . . . . . . 7 (⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0) ↔ (((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ∧ (((2nd ‘(𝐹𝐴)) · 2) + 1) ∈ ℕ0))
5145, 49, 50sylanbrc 411 . . . . . 6 (𝐴 ∈ ℕ → ⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0))
5212nncnd 8644 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 2 ∈ ℂ)
5352, 47expp1d 10318 . . . . . . . . . 10 (𝐴 ∈ ℕ → (2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · 2))
5452, 47expcld 10317 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) ∈ ℂ)
5554, 52mulcomd 7711 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · 2) = (2 · (2↑((2nd ‘(𝐹𝐴)) · 2))))
5652, 46, 9expmuld 10320 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) = ((2↑(2nd ‘(𝐹𝐴)))↑2))
5756oveq2d 5744 . . . . . . . . . 10 (𝐴 ∈ ℕ → (2 · (2↑((2nd ‘(𝐹𝐴)) · 2))) = (2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)))
5853, 55, 573eqtrd 2151 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) = (2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)))
5958oveq1d 5743 . . . . . . . 8 (𝐴 ∈ ℕ → ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)) = ((2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)) · ((1st ‘(𝐹𝐴))↑2)))
6012, 49nnexpcld 10339 . . . . . . . . . 10 (𝐴 ∈ ℕ → (2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) ∈ ℕ)
6160, 28nnmulcld 8679 . . . . . . . . 9 (𝐴 ∈ ℕ → ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)) ∈ ℕ)
62 oveq2 5736 . . . . . . . . . 10 (𝑥 = ((1st ‘(𝐹𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)))
63 oveq2 5736 . . . . . . . . . . 11 (𝑦 = (((2nd ‘(𝐹𝐴)) · 2) + 1) → (2↑𝑦) = (2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)))
6463oveq1d 5743 . . . . . . . . . 10 (𝑦 = (((2nd ‘(𝐹𝐴)) · 2) + 1) → ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)) = ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)))
6562, 64, 2ovmpog 5859 . . . . . . . . 9 ((((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ∧ (((2nd ‘(𝐹𝐴)) · 2) + 1) ∈ ℕ0 ∧ ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)) ∈ ℕ) → (((1st ‘(𝐹𝐴))↑2)𝐹(((2nd ‘(𝐹𝐴)) · 2) + 1)) = ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)))
6645, 49, 61, 65syl3anc 1199 . . . . . . . 8 (𝐴 ∈ ℕ → (((1st ‘(𝐹𝐴))↑2)𝐹(((2nd ‘(𝐹𝐴)) · 2) + 1)) = ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)))
67 f1ocnvfv2 5633 . . . . . . . . . . . . . . . 16 ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(𝐹𝐴)) = 𝐴)
683, 67mpan 418 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = 𝐴)
69 1st2nd2 6027 . . . . . . . . . . . . . . . . 17 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
707, 69syl 14 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℕ → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
7170fveq2d 5379 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
7268, 71eqtr3d 2149 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
73 df-ov 5731 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
7472, 73syl6eqr 2165 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))))
7512, 9nnexpcld 10339 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℕ)
7675, 27nnmulcld 8679 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))) ∈ ℕ)
77 oveq2 5736 . . . . . . . . . . . . . . 15 (𝑥 = (1st ‘(𝐹𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(𝐹𝐴))))
78 oveq2 5736 . . . . . . . . . . . . . . . 16 (𝑦 = (2nd ‘(𝐹𝐴)) → (2↑𝑦) = (2↑(2nd ‘(𝐹𝐴))))
7978oveq1d 5743 . . . . . . . . . . . . . . 15 (𝑦 = (2nd ‘(𝐹𝐴)) → ((2↑𝑦) · (1st ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
8077, 79, 2ovmpog 5859 . . . . . . . . . . . . . 14 (((1st ‘(𝐹𝐴)) ∈ 𝐽 ∧ (2nd ‘(𝐹𝐴)) ∈ ℕ0 ∧ ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))) ∈ ℕ) → ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
8122, 9, 76, 80syl3anc 1199 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
8274, 81eqtrd 2147 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
8382oveq1d 5743 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2))
8475nncnd 8644 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℂ)
8584, 38sqmuld 10329 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
8683, 85eqtrd 2147 . . . . . . . . . 10 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
8786oveq2d 5744 . . . . . . . . 9 (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (2 · (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2))))
8856, 54eqeltrrd 2192 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((2↑(2nd ‘(𝐹𝐴)))↑2) ∈ ℂ)
8928nncnd 8644 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ ℂ)
9052, 88, 89mulassd 7713 . . . . . . . . 9 (𝐴 ∈ ℕ → ((2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)) · ((1st ‘(𝐹𝐴))↑2)) = (2 · (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2))))
9187, 90eqtr4d 2150 . . . . . . . 8 (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = ((2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)) · ((1st ‘(𝐹𝐴))↑2)))
9259, 66, 913eqtr4rd 2158 . . . . . . 7 (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (((1st ‘(𝐹𝐴))↑2)𝐹(((2nd ‘(𝐹𝐴)) · 2) + 1)))
93 df-ov 5731 . . . . . . 7 (((1st ‘(𝐹𝐴))↑2)𝐹(((2nd ‘(𝐹𝐴)) · 2) + 1)) = (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩)
9492, 93syl6req 2164 . . . . . 6 (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩) = (2 · (𝐴↑2)))
95 f1ocnvfv 5634 . . . . . . 7 ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ ⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0)) → ((𝐹‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩) = (2 · (𝐴↑2)) → (𝐹‘(2 · (𝐴↑2))) = ⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩))
963, 95mpan 418 . . . . . 6 (⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0) → ((𝐹‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩) = (2 · (𝐴↑2)) → (𝐹‘(2 · (𝐴↑2))) = ⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩))
9751, 94, 96sylc 62 . . . . 5 (𝐴 ∈ ℕ → (𝐹‘(2 · (𝐴↑2))) = ⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩)
9897fveq2d 5379 . . . 4 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(2 · (𝐴↑2)))) = (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩))
99 op2ndg 6003 . . . . 5 ((((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ∧ (((2nd ‘(𝐹𝐴)) · 2) + 1) ∈ ℕ0) → (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩) = (((2nd ‘(𝐹𝐴)) · 2) + 1))
10045, 49, 99syl2anc 406 . . . 4 (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩) = (((2nd ‘(𝐹𝐴)) · 2) + 1))
10198, 100eqtrd 2147 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(2 · (𝐴↑2)))) = (((2nd ‘(𝐹𝐴)) · 2) + 1))
102101breq2d 3907 . 2 (𝐴 ∈ ℕ → (2 ∥ (2nd ‘(𝐹‘(2 · (𝐴↑2)))) ↔ 2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1)))
10320, 102mtbird 645 1 (𝐴 ∈ ℕ → ¬ 2 ∥ (2nd ‘(𝐹‘(2 · (𝐴↑2)))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wo 680  w3a 945   = wceq 1314  wcel 1463  {crab 2394  cop 3496   class class class wbr 3895   × cxp 4497  ccnv 4498  wf 5077  1-1-ontowf1o 5080  cfv 5081  (class class class)co 5728  cmpo 5730  1st c1st 5990  2nd c2nd 5991  cc 7545  1c1 7548   + caddc 7550   · cmul 7552  cn 8630  2c2 8681  0cn0 8881  cz 8958  cexp 10185  cdvds 11341  cprime 11634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664  ax-caucvg 7665
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-xor 1337  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-1o 6267  df-2o 6268  df-er 6383  df-en 6589  df-sup 6823  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-3 8690  df-4 8691  df-n0 8882  df-z 8959  df-uz 9229  df-q 9314  df-rp 9344  df-fz 9684  df-fzo 9813  df-fl 9936  df-mod 9989  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663  df-dvds 11342  df-gcd 11484  df-prm 11635
This theorem is referenced by:  sqne2sq  11700
  Copyright terms: Public domain W3C validator