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Theorem 2sqpwodd 12317
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 12315 . . . . . . . 8 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4 f1ocnv 5514 . . . . . . . 8 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5 f1of 5501 . . . . . . . 8 (𝐹:ℕ–1-1-onto→(𝐽 × ℕ0) → 𝐹:ℕ⟶(𝐽 × ℕ0))
63, 4, 5mp2b 8 . . . . . . 7 𝐹:ℕ⟶(𝐽 × ℕ0)
76ffvelcdmi 5693 . . . . . 6 (𝐴 ∈ ℕ → (𝐹𝐴) ∈ (𝐽 × ℕ0))
8 xp2nd 6221 . . . . . 6 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
97, 8syl 14 . . . . 5 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
109nn0zd 9440 . . . 4 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℤ)
11 2nn 9146 . . . . . 6 2 ∈ ℕ
1211a1i 9 . . . . 5 (𝐴 ∈ ℕ → 2 ∈ ℕ)
1312nnzd 9441 . . . 4 (𝐴 ∈ ℕ → 2 ∈ ℤ)
1410, 13zmulcld 9448 . . 3 (𝐴 ∈ ℕ → ((2nd ‘(𝐹𝐴)) · 2) ∈ ℤ)
15 dvdsmul2 11960 . . . 4 (((2nd ‘(𝐹𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
1610, 13, 15syl2anc 411 . . 3 (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
17 oddp1even 12020 . . . . 5 (((2nd ‘(𝐹𝐴)) · 2) ∈ ℤ → (¬ 2 ∥ ((2nd ‘(𝐹𝐴)) · 2) ↔ 2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1)))
1817biimprd 158 . . . 4 (((2nd ‘(𝐹𝐴)) · 2) ∈ ℤ → (2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1) → ¬ 2 ∥ ((2nd ‘(𝐹𝐴)) · 2)))
1918con2d 625 . . 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 6220 . . . . . . . . . . 11 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(𝐹𝐴)) ∈ 𝐽)
227, 21syl 14 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ 𝐽)
23 breq2 4034 . . . . . . . . . . . . 13 (𝑧 = (1st ‘(𝐹𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(𝐹𝐴))))
2423notbid 668 . . . . . . . . . . . 12 (𝑧 = (1st ‘(𝐹𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2524, 1elrab2 2920 . . . . . . . . . . 11 ((1st ‘(𝐹𝐴)) ∈ 𝐽 ↔ ((1st ‘(𝐹𝐴)) ∈ ℕ ∧ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2625simplbi 274 . . . . . . . . . 10 ((1st ‘(𝐹𝐴)) ∈ 𝐽 → (1st ‘(𝐹𝐴)) ∈ ℕ)
2722, 26syl 14 . . . . . . . . 9 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℕ)
2827nnsqcld 10768 . . . . . . . 8 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ ℕ)
2925simprbi 275 . . . . . . . . . . 11 ((1st ‘(𝐹𝐴)) ∈ 𝐽 → ¬ 2 ∥ (1st ‘(𝐹𝐴)))
3022, 29syl 14 . . . . . . . . . 10 (𝐴 ∈ ℕ → ¬ 2 ∥ (1st ‘(𝐹𝐴)))
31 2prm 12268 . . . . . . . . . . 11 2 ∈ ℙ
3227nnzd 9441 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℤ)
33 euclemma 12287 . . . . . . . . . . . 12 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ (2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴)))))
34 oridm 758 . . . . . . . . . . . 12 ((2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴)))
3533, 34bitrdi 196 . . . . . . . . . . 11 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3631, 32, 32, 35mp3an2i 1353 . . . . . . . . . 10 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3730, 36mtbird 674 . . . . . . . . 9 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
3827nncnd 8998 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℂ)
3938sqvald 10744 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) = ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
4039breq2d 4042 . . . . . . . . 9 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴))↑2) ↔ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴)))))
4137, 40mtbird 674 . . . . . . . 8 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2))
42 breq2 4034 . . . . . . . . . 10 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
4342notbid 668 . . . . . . . . 9 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
4443, 1elrab2 2920 . . . . . . . 8 (((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(𝐹𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
4528, 41, 44sylanbrc 417 . . . . . . 7 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ 𝐽)
4612nnnn0d 9296 . . . . . . . . 9 (𝐴 ∈ ℕ → 2 ∈ ℕ0)
479, 46nn0mulcld 9301 . . . . . . . 8 (𝐴 ∈ ℕ → ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0)
48 peano2nn0 9283 . . . . . . . 8 (((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0 → (((2nd ‘(𝐹𝐴)) · 2) + 1) ∈ ℕ0)
4947, 48syl 14 . . . . . . 7 (𝐴 ∈ ℕ → (((2nd ‘(𝐹𝐴)) · 2) + 1) ∈ ℕ0)
50 opelxp 4690 . . . . . . 7 (⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0) ↔ (((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ∧ (((2nd ‘(𝐹𝐴)) · 2) + 1) ∈ ℕ0))
5145, 49, 50sylanbrc 417 . . . . . 6 (𝐴 ∈ ℕ → ⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0))
5212nncnd 8998 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 2 ∈ ℂ)
5352, 47expp1d 10748 . . . . . . . . . 10 (𝐴 ∈ ℕ → (2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · 2))
5452, 47expcld 10747 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) ∈ ℂ)
5554, 52mulcomd 8043 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · 2) = (2 · (2↑((2nd ‘(𝐹𝐴)) · 2))))
5652, 46, 9expmuld 10750 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) = ((2↑(2nd ‘(𝐹𝐴)))↑2))
5756oveq2d 5935 . . . . . . . . . 10 (𝐴 ∈ ℕ → (2 · (2↑((2nd ‘(𝐹𝐴)) · 2))) = (2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)))
5853, 55, 573eqtrd 2230 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) = (2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)))
5958oveq1d 5934 . . . . . . . 8 (𝐴 ∈ ℕ → ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)) = ((2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)) · ((1st ‘(𝐹𝐴))↑2)))
6012, 49nnexpcld 10769 . . . . . . . . . 10 (𝐴 ∈ ℕ → (2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) ∈ ℕ)
6160, 28nnmulcld 9033 . . . . . . . . 9 (𝐴 ∈ ℕ → ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)) ∈ ℕ)
62 oveq2 5927 . . . . . . . . . 10 (𝑥 = ((1st ‘(𝐹𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)))
63 oveq2 5927 . . . . . . . . . . 11 (𝑦 = (((2nd ‘(𝐹𝐴)) · 2) + 1) → (2↑𝑦) = (2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)))
6463oveq1d 5934 . . . . . . . . . 10 (𝑦 = (((2nd ‘(𝐹𝐴)) · 2) + 1) → ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)) = ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)))
6562, 64, 2ovmpog 6054 . . . . . . . . 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 1249 . . . . . . . 8 (𝐴 ∈ ℕ → (((1st ‘(𝐹𝐴))↑2)𝐹(((2nd ‘(𝐹𝐴)) · 2) + 1)) = ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)))
67 f1ocnvfv2 5822 . . . . . . . . . . . . . . . 16 ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(𝐹𝐴)) = 𝐴)
683, 67mpan 424 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = 𝐴)
69 1st2nd2 6230 . . . . . . . . . . . . . . . . 17 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
707, 69syl 14 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℕ → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
7170fveq2d 5559 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
7268, 71eqtr3d 2228 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
73 df-ov 5922 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
7472, 73eqtr4di 2244 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))))
7512, 9nnexpcld 10769 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℕ)
7675, 27nnmulcld 9033 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))) ∈ ℕ)
77 oveq2 5927 . . . . . . . . . . . . . . 15 (𝑥 = (1st ‘(𝐹𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(𝐹𝐴))))
78 oveq2 5927 . . . . . . . . . . . . . . . 16 (𝑦 = (2nd ‘(𝐹𝐴)) → (2↑𝑦) = (2↑(2nd ‘(𝐹𝐴))))
7978oveq1d 5934 . . . . . . . . . . . . . . 15 (𝑦 = (2nd ‘(𝐹𝐴)) → ((2↑𝑦) · (1st ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
8077, 79, 2ovmpog 6054 . . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
8274, 81eqtrd 2226 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
8382oveq1d 5934 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2))
8475nncnd 8998 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℂ)
8584, 38sqmuld 10759 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
8683, 85eqtrd 2226 . . . . . . . . . 10 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
8786oveq2d 5935 . . . . . . . . 9 (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (2 · (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2))))
8856, 54eqeltrrd 2271 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((2↑(2nd ‘(𝐹𝐴)))↑2) ∈ ℂ)
8928nncnd 8998 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ ℂ)
9052, 88, 89mulassd 8045 . . . . . . . . 9 (𝐴 ∈ ℕ → ((2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)) · ((1st ‘(𝐹𝐴))↑2)) = (2 · (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2))))
9187, 90eqtr4d 2229 . . . . . . . 8 (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = ((2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)) · ((1st ‘(𝐹𝐴))↑2)))
9259, 66, 913eqtr4rd 2237 . . . . . . 7 (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (((1st ‘(𝐹𝐴))↑2)𝐹(((2nd ‘(𝐹𝐴)) · 2) + 1)))
93 df-ov 5922 . . . . . . 7 (((1st ‘(𝐹𝐴))↑2)𝐹(((2nd ‘(𝐹𝐴)) · 2) + 1)) = (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩)
9492, 93eqtr2di 2243 . . . . . 6 (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩) = (2 · (𝐴↑2)))
95 f1ocnvfv 5823 . . . . . . 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 424 . . . . . 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 5559 . . . 4 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(2 · (𝐴↑2)))) = (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩))
99 op2ndg 6206 . . . . 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 411 . . . 4 (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩) = (((2nd ‘(𝐹𝐴)) · 2) + 1))
10198, 100eqtrd 2226 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(2 · (𝐴↑2)))) = (((2nd ‘(𝐹𝐴)) · 2) + 1))
102101breq2d 4042 . 2 (𝐴 ∈ ℕ → (2 ∥ (2nd ‘(𝐹‘(2 · (𝐴↑2)))) ↔ 2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1)))
10320, 102mtbird 674 1 (𝐴 ∈ ℕ → ¬ 2 ∥ (2nd ‘(𝐹‘(2 · (𝐴↑2)))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wo 709  w3a 980   = wceq 1364  wcel 2164  {crab 2476  cop 3622   class class class wbr 4030   × cxp 4658  ccnv 4659  wf 5251  1-1-ontowf1o 5254  cfv 5255  (class class class)co 5919  cmpo 5921  1st c1st 6193  2nd c2nd 6194  cc 7872  1c1 7875   + caddc 7877   · cmul 7879  cn 8984  2c2 9035  0cn0 9243  cz 9320  cexp 10612  cdvds 11933  cprime 12248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-1o 6471  df-2o 6472  df-er 6589  df-en 6797  df-sup 7045  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-fl 10342  df-mod 10397  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-dvds 11934  df-gcd 12083  df-prm 12249
This theorem is referenced by:  sqne2sq  12318
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