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Theorem 2sqpwodd 10761
 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 10759 . . . . . . . 8 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4 f1ocnv 5190 . . . . . . . 8 (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → 𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5 f1of 5177 . . . . . . . 8 (𝐹:ℕ–1-1-onto→(𝐽 × ℕ0) → 𝐹:ℕ⟶(𝐽 × ℕ0))
63, 4, 5mp2b 8 . . . . . . 7 𝐹:ℕ⟶(𝐽 × ℕ0)
76ffvelrni 5353 . . . . . 6 (𝐴 ∈ ℕ → (𝐹𝐴) ∈ (𝐽 × ℕ0))
8 xp2nd 5844 . . . . . 6 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
97, 8syl 14 . . . . 5 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℕ0)
109nn0zd 8600 . . . 4 (𝐴 ∈ ℕ → (2nd ‘(𝐹𝐴)) ∈ ℤ)
11 2nn 8312 . . . . . 6 2 ∈ ℕ
1211a1i 9 . . . . 5 (𝐴 ∈ ℕ → 2 ∈ ℕ)
1312nnzd 8601 . . . 4 (𝐴 ∈ ℕ → 2 ∈ ℤ)
1410, 13zmulcld 8608 . . 3 (𝐴 ∈ ℕ → ((2nd ‘(𝐹𝐴)) · 2) ∈ ℤ)
15 dvdsmul2 10426 . . . 4 (((2nd ‘(𝐹𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
1610, 13, 15syl2anc 403 . . 3 (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(𝐹𝐴)) · 2))
17 oddp1even 10483 . . . . 5 (((2nd ‘(𝐹𝐴)) · 2) ∈ ℤ → (¬ 2 ∥ ((2nd ‘(𝐹𝐴)) · 2) ↔ 2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1)))
1817biimprd 156 . . . 4 (((2nd ‘(𝐹𝐴)) · 2) ∈ ℤ → (2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1) → ¬ 2 ∥ ((2nd ‘(𝐹𝐴)) · 2)))
1918con2d 587 . . 3 (((2nd ‘(𝐹𝐴)) · 2) ∈ ℤ → (2 ∥ ((2nd ‘(𝐹𝐴)) · 2) → ¬ 2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1)))
2014, 16, 19sylc 61 . 2 (𝐴 ∈ ℕ → ¬ 2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1))
21 xp1st 5843 . . . . . . . . . . 11 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(𝐹𝐴)) ∈ 𝐽)
227, 21syl 14 . . . . . . . . . 10 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ 𝐽)
23 breq2 3809 . . . . . . . . . . . . 13 (𝑧 = (1st ‘(𝐹𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(𝐹𝐴))))
2423notbid 625 . . . . . . . . . . . 12 (𝑧 = (1st ‘(𝐹𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2524, 1elrab2 2760 . . . . . . . . . . 11 ((1st ‘(𝐹𝐴)) ∈ 𝐽 ↔ ((1st ‘(𝐹𝐴)) ∈ ℕ ∧ ¬ 2 ∥ (1st ‘(𝐹𝐴))))
2625simplbi 268 . . . . . . . . . 10 ((1st ‘(𝐹𝐴)) ∈ 𝐽 → (1st ‘(𝐹𝐴)) ∈ ℕ)
2722, 26syl 14 . . . . . . . . 9 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℕ)
2827nnsqcld 9775 . . . . . . . 8 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ ℕ)
2925simprbi 269 . . . . . . . . . . 11 ((1st ‘(𝐹𝐴)) ∈ 𝐽 → ¬ 2 ∥ (1st ‘(𝐹𝐴)))
3022, 29syl 14 . . . . . . . . . 10 (𝐴 ∈ ℕ → ¬ 2 ∥ (1st ‘(𝐹𝐴)))
31 2prm 10716 . . . . . . . . . . 11 2 ∈ ℙ
3227nnzd 8601 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℤ)
33 euclemma 10732 . . . . . . . . . . . 12 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ (2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴)))))
34 oridm 707 . . . . . . . . . . . 12 ((2 ∥ (1st ‘(𝐹𝐴)) ∨ 2 ∥ (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴)))
3533, 34syl6bb 194 . . . . . . . . . . 11 ((2 ∈ ℙ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ ∧ (1st ‘(𝐹𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3631, 32, 32, 35mp3an2i 1274 . . . . . . . . . 10 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))) ↔ 2 ∥ (1st ‘(𝐹𝐴))))
3730, 36mtbird 631 . . . . . . . . 9 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
3827nncnd 8172 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (1st ‘(𝐹𝐴)) ∈ ℂ)
3938sqvald 9751 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) = ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴))))
4039breq2d 3817 . . . . . . . . 9 (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(𝐹𝐴))↑2) ↔ 2 ∥ ((1st ‘(𝐹𝐴)) · (1st ‘(𝐹𝐴)))))
4137, 40mtbird 631 . . . . . . . 8 (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2))
42 breq2 3809 . . . . . . . . . 10 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
4342notbid 625 . . . . . . . . 9 (𝑧 = ((1st ‘(𝐹𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
4443, 1elrab2 2760 . . . . . . . 8 (((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(𝐹𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(𝐹𝐴))↑2)))
4528, 41, 44sylanbrc 408 . . . . . . 7 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ 𝐽)
4612nnnn0d 8460 . . . . . . . . 9 (𝐴 ∈ ℕ → 2 ∈ ℕ0)
479, 46nn0mulcld 8465 . . . . . . . 8 (𝐴 ∈ ℕ → ((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0)
48 peano2nn0 8447 . . . . . . . 8 (((2nd ‘(𝐹𝐴)) · 2) ∈ ℕ0 → (((2nd ‘(𝐹𝐴)) · 2) + 1) ∈ ℕ0)
4947, 48syl 14 . . . . . . 7 (𝐴 ∈ ℕ → (((2nd ‘(𝐹𝐴)) · 2) + 1) ∈ ℕ0)
50 opelxp 4420 . . . . . . 7 (⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0) ↔ (((1st ‘(𝐹𝐴))↑2) ∈ 𝐽 ∧ (((2nd ‘(𝐹𝐴)) · 2) + 1) ∈ ℕ0))
5145, 49, 50sylanbrc 408 . . . . . 6 (𝐴 ∈ ℕ → ⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0))
5212nncnd 8172 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 2 ∈ ℂ)
5352, 47expp1d 9755 . . . . . . . . . 10 (𝐴 ∈ ℕ → (2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) = ((2↑((2nd ‘(𝐹𝐴)) · 2)) · 2))
5452, 47expcld 9754 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) ∈ ℂ)
5554, 52mulcomd 7254 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((2↑((2nd ‘(𝐹𝐴)) · 2)) · 2) = (2 · (2↑((2nd ‘(𝐹𝐴)) · 2))))
5652, 46, 9expmuld 9757 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (2↑((2nd ‘(𝐹𝐴)) · 2)) = ((2↑(2nd ‘(𝐹𝐴)))↑2))
5756oveq2d 5579 . . . . . . . . . 10 (𝐴 ∈ ℕ → (2 · (2↑((2nd ‘(𝐹𝐴)) · 2))) = (2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)))
5853, 55, 573eqtrd 2119 . . . . . . . . 9 (𝐴 ∈ ℕ → (2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) = (2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)))
5958oveq1d 5578 . . . . . . . 8 (𝐴 ∈ ℕ → ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)) = ((2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)) · ((1st ‘(𝐹𝐴))↑2)))
6012, 49nnexpcld 9776 . . . . . . . . . 10 (𝐴 ∈ ℕ → (2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) ∈ ℕ)
6160, 28nnmulcld 8206 . . . . . . . . 9 (𝐴 ∈ ℕ → ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)) ∈ ℕ)
62 oveq2 5571 . . . . . . . . . 10 (𝑥 = ((1st ‘(𝐹𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)))
63 oveq2 5571 . . . . . . . . . . 11 (𝑦 = (((2nd ‘(𝐹𝐴)) · 2) + 1) → (2↑𝑦) = (2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)))
6463oveq1d 5578 . . . . . . . . . 10 (𝑦 = (((2nd ‘(𝐹𝐴)) · 2) + 1) → ((2↑𝑦) · ((1st ‘(𝐹𝐴))↑2)) = ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)))
6562, 64, 2ovmpt2g 5686 . . . . . . . . 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 1170 . . . . . . . 8 (𝐴 ∈ ℕ → (((1st ‘(𝐹𝐴))↑2)𝐹(((2nd ‘(𝐹𝐴)) · 2) + 1)) = ((2↑(((2nd ‘(𝐹𝐴)) · 2) + 1)) · ((1st ‘(𝐹𝐴))↑2)))
67 f1ocnvfv2 5469 . . . . . . . . . . . . . . . 16 ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(𝐹𝐴)) = 𝐴)
683, 67mpan 415 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = 𝐴)
69 1st2nd2 5852 . . . . . . . . . . . . . . . . 17 ((𝐹𝐴) ∈ (𝐽 × ℕ0) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
707, 69syl 14 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℕ → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
7170fveq2d 5233 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℕ → (𝐹‘(𝐹𝐴)) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
7268, 71eqtr3d 2117 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
73 df-ov 5566 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = (𝐹‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
7472, 73syl6eqr 2133 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))))
7512, 9nnexpcld 9776 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℕ)
7675, 27nnmulcld 8206 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))) ∈ ℕ)
77 oveq2 5571 . . . . . . . . . . . . . . 15 (𝑥 = (1st ‘(𝐹𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(𝐹𝐴))))
78 oveq2 5571 . . . . . . . . . . . . . . . 16 (𝑦 = (2nd ‘(𝐹𝐴)) → (2↑𝑦) = (2↑(2nd ‘(𝐹𝐴))))
7978oveq1d 5578 . . . . . . . . . . . . . . 15 (𝑦 = (2nd ‘(𝐹𝐴)) → ((2↑𝑦) · (1st ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
8077, 79, 2ovmpt2g 5686 . . . . . . . . . . . . . 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 1170 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))𝐹(2nd ‘(𝐹𝐴))) = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
8274, 81eqtrd 2115 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴))))
8382oveq1d 5578 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2))
8475nncnd 8172 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → (2↑(2nd ‘(𝐹𝐴))) ∈ ℂ)
8584, 38sqmuld 9766 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (((2↑(2nd ‘(𝐹𝐴))) · (1st ‘(𝐹𝐴)))↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
8683, 85eqtrd 2115 . . . . . . . . . 10 (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2)))
8786oveq2d 5579 . . . . . . . . 9 (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (2 · (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2))))
8856, 54eqeltrrd 2160 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((2↑(2nd ‘(𝐹𝐴)))↑2) ∈ ℂ)
8928nncnd 8172 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((1st ‘(𝐹𝐴))↑2) ∈ ℂ)
9052, 88, 89mulassd 7256 . . . . . . . . 9 (𝐴 ∈ ℕ → ((2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)) · ((1st ‘(𝐹𝐴))↑2)) = (2 · (((2↑(2nd ‘(𝐹𝐴)))↑2) · ((1st ‘(𝐹𝐴))↑2))))
9187, 90eqtr4d 2118 . . . . . . . 8 (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = ((2 · ((2↑(2nd ‘(𝐹𝐴)))↑2)) · ((1st ‘(𝐹𝐴))↑2)))
9259, 66, 913eqtr4rd 2126 . . . . . . 7 (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (((1st ‘(𝐹𝐴))↑2)𝐹(((2nd ‘(𝐹𝐴)) · 2) + 1)))
93 df-ov 5566 . . . . . . 7 (((1st ‘(𝐹𝐴))↑2)𝐹(((2nd ‘(𝐹𝐴)) · 2) + 1)) = (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩)
9492, 93syl6req 2132 . . . . . 6 (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩) = (2 · (𝐴↑2)))
95 f1ocnvfv 5470 . . . . . . 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 415 . . . . . 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 61 . . . . 5 (𝐴 ∈ ℕ → (𝐹‘(2 · (𝐴↑2))) = ⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩)
9897fveq2d 5233 . . . 4 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(2 · (𝐴↑2)))) = (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩))
99 op2ndg 5829 . . . . 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 403 . . . 4 (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(𝐹𝐴))↑2), (((2nd ‘(𝐹𝐴)) · 2) + 1)⟩) = (((2nd ‘(𝐹𝐴)) · 2) + 1))
10198, 100eqtrd 2115 . . 3 (𝐴 ∈ ℕ → (2nd ‘(𝐹‘(2 · (𝐴↑2)))) = (((2nd ‘(𝐹𝐴)) · 2) + 1))
102101breq2d 3817 . 2 (𝐴 ∈ ℕ → (2 ∥ (2nd ‘(𝐹‘(2 · (𝐴↑2)))) ↔ 2 ∥ (((2nd ‘(𝐹𝐴)) · 2) + 1)))
10320, 102mtbird 631 1 (𝐴 ∈ ℕ → ¬ 2 ∥ (2nd ‘(𝐹‘(2 · (𝐴↑2)))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103   ∨ wo 662   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  {crab 2357  ⟨cop 3419   class class class wbr 3805   × cxp 4389  ◡ccnv 4390  ⟶wf 4948  –1-1-onto→wf1o 4951  ‘cfv 4952  (class class class)co 5563   ↦ cmpt2 5565  1st c1st 5816  2nd c2nd 5817  ℂcc 7093  1c1 7096   + caddc 7098   · cmul 7100  ℕcn 8158  2c2 8208  ℕ0cn0 8407  ℤcz 8484  ↑cexp 9624   ∥ cdvds 10403  ℙcprime 10696 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210 This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-xor 1308  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-1o 6085  df-2o 6086  df-er 6193  df-en 6309  df-sup 6491  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-gcd 10546  df-prm 10697 This theorem is referenced by:  sqne2sq  10762
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