Theorem 2sqpwodd 12981

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 12979	. . . . . . . 8 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1ocnv 5655	. . . . . . . 8 ⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → ◡𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5		f1of 5642	. . . . . . . 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 5845	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘𝐴) ∈ (𝐽 × ℕ0))
8		xp2nd 6404	. . . . . 6 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
9	7, 8	syl 14	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
10	9	nn0zd 9777	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℤ)
11		2nn 9477	. . . . . 6 ⊢ 2 ∈ ℕ
12	11	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ)
13	12	nnzd 9778	. . . 4 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℤ)
14	10, 13	zmulcld 9785	. . 3 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℤ)
15		dvdsmul2 12606	. . . 4 ⊢ (((2nd ‘(◡𝐹‘𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
16	10, 13, 15	syl2anc 415	. . 3 ⊢ (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
17		oddp1even 12668	. . . . 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 633	. . 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 6403	. . . . . . . . . . 11 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
22	7, 21	syl 14	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
23		breq2 4135	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
24	23	notbid 677	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
25	24, 1	elrab2 2985	. . . . . . . . . . 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 11153	. . . . . . . 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 12930	. . . . . . . . . . 11 ⊢ 2 ∈ ℙ
32	27	nnzd 9778	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℤ)
33		euclemma 12950	. . . . . . . . . . . 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 769	. . . . . . . . . . . 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 1383	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
37	30, 36	mtbird 684	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
38	27	nncnd 9327	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℂ)
39	38	sqvald 11129	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) = ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
40	39	breq2d 4143	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2) ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴)))))
41	37, 40	mtbird 684	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2))
42		breq2 4135	. . . . . . . . . 10 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
43	42	notbid 677	. . . . . . . . 9 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
44	43, 1	elrab2 2985	. . . . . . . 8 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
45	28, 41, 44	sylanbrc 421	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽)
46	12	nnnn0d 9631	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ0)
47	9, 46	nn0mulcld 9636	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0)
48		peano2nn0 9614	. . . . . . . 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 4806	. . . . . . 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 421	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0))
52	12	nncnd 9327	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℂ)
53	52, 47	expp1d 11133	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · 2))
54	52, 47	expcld 11132	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) ∈ ℂ)
55	54, 52	mulcomd 8353	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · 2) = (2 · (2↑((2nd ‘(◡𝐹‘𝐴)) · 2))))
56	52, 46, 9	expmuld 11135	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2))
57	56	oveq2d 6104	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 · (2↑((2nd ‘(◡𝐹‘𝐴)) · 2))) = (2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)))
58	53, 55, 57	3eqtrd 2275	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = (2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)))
59	58	oveq1d 6103	. . . . . . . 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 11154	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) ∈ ℕ)
61	60, 28	nnmulcld 9362	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) · ((1st ‘(◡𝐹‘𝐴))↑2)) ∈ ℕ)
62		oveq2 6096	. . . . . . . . . 10 ⊢ (𝑥 = ((1st ‘(◡𝐹‘𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(◡𝐹‘𝐴))↑2)))
63		oveq2 6096	. . . . . . . . . . 11 ⊢ (𝑦 = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) → (2↑𝑦) = (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
64	63	oveq1d 6103	. . . . . . . . . 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 6226	. . . . . . . . 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 1278	. . . . . . . 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 5987	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
68	3, 67	mpan 428	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
69		1st2nd2 6413	. . . . . . . . . . . . . . . . 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 5702	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
72	68, 71	eqtr3d 2273	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
73		df-ov 6091	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
74	72, 73	eqtr4di 2289	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))))
75	12, 9	nnexpcld 11154	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℕ)
76	75, 27	nnmulcld 9362	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))) ∈ ℕ)
77		oveq2 6096	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (1st ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))))
78		oveq2 6096	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → (2↑𝑦) = (2↑(2nd ‘(◡𝐹‘𝐴))))
79	78	oveq1d 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
80	77, 79, 2	ovmpog 6226	. . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
82	74, 81	eqtrd 2271	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
83	82	oveq1d 6103	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2))
84	75	nncnd 9327	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℂ)
85	84, 38	sqmuld 11144	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
86	83, 85	eqtrd 2271	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
87	86	oveq2d 6104	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (2 · (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2))))
88	56, 54	eqeltrrd 2316	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) ∈ ℂ)
89	28	nncnd 9327	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℂ)
90	52, 88, 89	mulassd 8355	. . . . . . . . 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 2274	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = ((2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
92	59, 66, 91	3eqtr4rd 2282	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (((1st ‘(◡𝐹‘𝐴))↑2)𝐹(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
93		df-ov 6091	. . . . . . 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 2288	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩) = (2 · (𝐴↑2)))
95		f1ocnvfv 5988	. . . . . . 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 428	. . . . . 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 5702	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) = (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩))
99		op2ndg 6389	. . . . 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 415	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩) = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1))
101	98, 100	eqtrd 2271	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1))
102	101	breq2d 4143	. 2 ⊢ (𝐴 ∈ ℕ → (2 ∥ (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) ↔ 2 ∥ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
103	20, 102	mtbird 684	1 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 720   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {crab 2532  ⟨cop 3712   class class class wbr 4131   × cxp 4774  ^◡ccnv 4775  ⟶wf 5376  –1-1-onto→wf1o 5379  ‘cfv 5380  (class class class)co 6088   ∈ cmpo 6090  1^st c1st 6376  2^nd c2nd 6377  ℂcc 8183  1c1 8186   + caddc 8188   · cmul 8190  ℕcn 9313  2c2 9364  ℕ₀cn0 9574  ℤcz 9655  ↑cexp 10996   ∥ cdvds 12579  ℙcprime 12910

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-2o 6692  df-er 6811  df-en 7027  df-sup 7330  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-dvds 12580  df-gcd 12756  df-prm 12911

This theorem is used by:  sqne2sq  12982