Theorem 2sqpwodd 12811

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 = (𝑥 ∈ 𝐽, 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

Step	Hyp	Ref	Expression
1		oddpwdc.j	. . . . . . . . 9 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
2		oddpwdc.f	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
3	1, 2	oddpwdc 12809	. . . . . . . 8 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1ocnv 5605	. . . . . . . 8 ⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → ◡𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5		f1of 5592	. . . . . . . 8 ⊢ (◡𝐹:ℕ–1-1-onto→(𝐽 × ℕ0) → ◡𝐹:ℕ⟶(𝐽 × ℕ0))
6	3, 4, 5	mp2b 8	. . . . . . 7 ⊢ ◡𝐹:ℕ⟶(𝐽 × ℕ0)
7	6	ffvelcdmi 5789	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘𝐴) ∈ (𝐽 × ℕ0))
8		xp2nd 6338	. . . . . 6 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
9	7, 8	syl 14	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
10	9	nn0zd 9644	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℤ)
11		2nn 9347	. . . . . 6 ⊢ 2 ∈ ℕ
12	11	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ)
13	12	nnzd 9645	. . . 4 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℤ)
14	10, 13	zmulcld 9652	. . 3 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℤ)
15		dvdsmul2 12438	. . . 4 ⊢ (((2nd ‘(◡𝐹‘𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
16	10, 13, 15	syl2anc 411	. . 3 ⊢ (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
17		oddp1even 12500	. . . . 5 ⊢ (((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℤ → (¬ 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2) ↔ 2 ∥ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
18	17	biimprd 158	. . . 4 ⊢ (((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℤ → (2 ∥ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) → ¬ 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2)))
19	18	con2d 629	. . 3 ⊢ (((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℤ → (2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2) → ¬ 2 ∥ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
20	14, 16, 19	sylc 62	. 2 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1))
21		xp1st 6337	. . . . . . . . . . 11 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
22	7, 21	syl 14	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
23		breq2 4097	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
24	23	notbid 673	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
25	24, 1	elrab2 2966	. . . . . . . . . . 11 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 ↔ ((1st ‘(◡𝐹‘𝐴)) ∈ ℕ ∧ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
26	25	simplbi 274	. . . . . . . . . 10 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 → (1st ‘(◡𝐹‘𝐴)) ∈ ℕ)
27	22, 26	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℕ)
28	27	nnsqcld 11002	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℕ)
29	25	simprbi 275	. . . . . . . . . . 11 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 → ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
30	22, 29	syl 14	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
31		2prm 12762	. . . . . . . . . . 11 ⊢ 2 ∈ ℙ
32	27	nnzd 9645	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℤ)
33		euclemma 12781	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℙ ∧ (1st ‘(◡𝐹‘𝐴)) ∈ ℤ ∧ (1st ‘(◡𝐹‘𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ (2 ∥ (1st ‘(◡𝐹‘𝐴)) ∨ 2 ∥ (1st ‘(◡𝐹‘𝐴)))))
34		oridm 765	. . . . . . . . . . . 12 ⊢ ((2 ∥ (1st ‘(◡𝐹‘𝐴)) ∨ 2 ∥ (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
35	33, 34	bitrdi 196	. . . . . . . . . . 11 ⊢ ((2 ∈ ℙ ∧ (1st ‘(◡𝐹‘𝐴)) ∈ ℤ ∧ (1st ‘(◡𝐹‘𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
36	31, 32, 32, 35	mp3an2i 1379	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
37	30, 36	mtbird 680	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
38	27	nncnd 9199	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℂ)
39	38	sqvald 10978	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) = ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
40	39	breq2d 4105	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2) ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴)))))
41	37, 40	mtbird 680	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2))
42		breq2 4097	. . . . . . . . . 10 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
43	42	notbid 673	. . . . . . . . 9 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
44	43, 1	elrab2 2966	. . . . . . . 8 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
45	28, 41, 44	sylanbrc 417	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽)
46	12	nnnn0d 9499	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ0)
47	9, 46	nn0mulcld 9504	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0)
48		peano2nn0 9484	. . . . . . . 8 ⊢ (((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0 → (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) ∈ ℕ0)
49	47, 48	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) ∈ ℕ0)
50		opelxp 4761	. . . . . . 7 ⊢ (⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0) ↔ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ∧ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) ∈ ℕ0))
51	45, 49, 50	sylanbrc 417	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0))
52	12	nncnd 9199	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℂ)
53	52, 47	expp1d 10982	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · 2))
54	52, 47	expcld 10981	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) ∈ ℂ)
55	54, 52	mulcomd 8243	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · 2) = (2 · (2↑((2nd ‘(◡𝐹‘𝐴)) · 2))))
56	52, 46, 9	expmuld 10984	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2))
57	56	oveq2d 6044	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 · (2↑((2nd ‘(◡𝐹‘𝐴)) · 2))) = (2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)))
58	53, 55, 57	3eqtrd 2268	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = (2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)))
59	58	oveq1d 6043	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) · ((1st ‘(◡𝐹‘𝐴))↑2)) = ((2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
60	12, 49	nnexpcld 11003	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) ∈ ℕ)
61	60, 28	nnmulcld 9234	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) · ((1st ‘(◡𝐹‘𝐴))↑2)) ∈ ℕ)
62		oveq2 6036	. . . . . . . . . 10 ⊢ (𝑥 = ((1st ‘(◡𝐹‘𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(◡𝐹‘𝐴))↑2)))
63		oveq2 6036	. . . . . . . . . . 11 ⊢ (𝑦 = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) → (2↑𝑦) = (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
64	63	oveq1d 6043	. . . . . . . . . 10 ⊢ (𝑦 = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) → ((2↑𝑦) · ((1st ‘(◡𝐹‘𝐴))↑2)) = ((2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
65	62, 64, 2	ovmpog 6166	. . . . . . . . 9 ⊢ ((((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ∧ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) ∈ ℕ0 ∧ ((2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) · ((1st ‘(◡𝐹‘𝐴))↑2)) ∈ ℕ) → (((1st ‘(◡𝐹‘𝐴))↑2)𝐹(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = ((2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
66	45, 49, 61, 65	syl3anc 1274	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (((1st ‘(◡𝐹‘𝐴))↑2)𝐹(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = ((2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
67		f1ocnvfv2 5929	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
68	3, 67	mpan 424	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
69		1st2nd2 6347	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (◡𝐹‘𝐴) = ⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
70	7, 69	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘𝐴) = ⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
71	70	fveq2d 5652	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
72	68, 71	eqtr3d 2266	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
73		df-ov 6031	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
74	72, 73	eqtr4di 2282	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))))
75	12, 9	nnexpcld 11003	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℕ)
76	75, 27	nnmulcld 9234	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))) ∈ ℕ)
77		oveq2 6036	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (1st ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))))
78		oveq2 6036	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → (2↑𝑦) = (2↑(2nd ‘(◡𝐹‘𝐴))))
79	78	oveq1d 6043	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
80	77, 79, 2	ovmpog 6166	. . . . . . . . . . . . . 14 ⊢ (((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 ∧ (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0 ∧ ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))) ∈ ℕ) → ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
81	22, 9, 76, 80	syl3anc 1274	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
82	74, 81	eqtrd 2264	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
83	82	oveq1d 6043	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2))
84	75	nncnd 9199	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℂ)
85	84, 38	sqmuld 10993	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
86	83, 85	eqtrd 2264	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
87	86	oveq2d 6044	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (2 · (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2))))
88	56, 54	eqeltrrd 2309	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) ∈ ℂ)
89	28	nncnd 9199	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℂ)
90	52, 88, 89	mulassd 8245	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)) · ((1st ‘(◡𝐹‘𝐴))↑2)) = (2 · (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2))))
91	87, 90	eqtr4d 2267	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = ((2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
92	59, 66, 91	3eqtr4rd 2275	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (((1st ‘(◡𝐹‘𝐴))↑2)𝐹(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
93		df-ov 6031	. . . . . . 7 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2)𝐹(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩)
94	92, 93	eqtr2di 2281	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩) = (2 · (𝐴↑2)))
95		f1ocnvfv 5930	. . . . . . 7 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ ⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0)) → ((𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩) = (2 · (𝐴↑2)) → (◡𝐹‘(2 · (𝐴↑2))) = ⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩))
96	3, 95	mpan 424	. . . . . 6 ⊢ (⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0) → ((𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩) = (2 · (𝐴↑2)) → (◡𝐹‘(2 · (𝐴↑2))) = ⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩))
97	51, 94, 96	sylc 62	. . . . 5 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘(2 · (𝐴↑2))) = ⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩)
98	97	fveq2d 5652	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) = (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩))
99		op2ndg 6323	. . . . 5 ⊢ ((((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ∧ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) ∈ ℕ0) → (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩) = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1))
100	45, 49, 99	syl2anc 411	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩) = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1))
101	98, 100	eqtrd 2264	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1))
102	101	breq2d 4105	. 2 ⊢ (𝐴 ∈ ℕ → (2 ∥ (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) ↔ 2 ∥ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
103	20, 102	mtbird 680	1 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  {crab 2515  ⟨cop 3676   class class class wbr 4093   × cxp 4729  ^◡ccnv 4730  ⟶wf 5329  –1-1-onto→wf1o 5332  ‘cfv 5333  (class class class)co 6028   ∈ cmpo 6030  1^st c1st 6310  2^nd c2nd 6311  ℂcc 8073  1c1 8076   + caddc 8078   · cmul 8080  ℕcn 9185  2c2 9236  ℕ₀cn0 9444  ℤcz 9523  ↑cexp 10846   ∥ cdvds 12411  ℙcprime 12742

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-sup 7226  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fz 10289  df-fzo 10423  df-fl 10576  df-mod 10631  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-dvds 12412  df-gcd 12588  df-prm 12743

This theorem is referenced by:  sqne2sq  12812