Theorem 2sqpwodd 12693

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 12691	. . . . . . . 8 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1ocnv 5584	. . . . . . . 8 ⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → ◡𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5		f1of 5571	. . . . . . . 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 5768	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘𝐴) ∈ (𝐽 × ℕ0))
8		xp2nd 6310	. . . . . 6 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
9	7, 8	syl 14	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
10	9	nn0zd 9563	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℤ)
11		2nn 9268	. . . . . 6 ⊢ 2 ∈ ℕ
12	11	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ)
13	12	nnzd 9564	. . . 4 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℤ)
14	10, 13	zmulcld 9571	. . 3 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℤ)
15		dvdsmul2 12320	. . . 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 12382	. . . . 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 627	. . 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 6309	. . . . . . . . . . 11 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
22	7, 21	syl 14	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
23		breq2 4086	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
24	23	notbid 671	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
25	24, 1	elrab2 2962	. . . . . . . . . . 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 10911	. . . . . . . 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 12644	. . . . . . . . . . 11 ⊢ 2 ∈ ℙ
32	27	nnzd 9564	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℤ)
33		euclemma 12663	. . . . . . . . . . . 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 762	. . . . . . . . . . . 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 1376	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
37	30, 36	mtbird 677	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
38	27	nncnd 9120	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℂ)
39	38	sqvald 10887	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) = ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
40	39	breq2d 4094	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2) ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴)))))
41	37, 40	mtbird 677	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2))
42		breq2 4086	. . . . . . . . . 10 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
43	42	notbid 671	. . . . . . . . 9 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
44	43, 1	elrab2 2962	. . . . . . . 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 9418	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ0)
47	9, 46	nn0mulcld 9423	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0)
48		peano2nn0 9405	. . . . . . . 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 4748	. . . . . . 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 9120	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℂ)
53	52, 47	expp1d 10891	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · 2))
54	52, 47	expcld 10890	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) ∈ ℂ)
55	54, 52	mulcomd 8164	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · 2) = (2 · (2↑((2nd ‘(◡𝐹‘𝐴)) · 2))))
56	52, 46, 9	expmuld 10893	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2))
57	56	oveq2d 6016	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 · (2↑((2nd ‘(◡𝐹‘𝐴)) · 2))) = (2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)))
58	53, 55, 57	3eqtrd 2266	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = (2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)))
59	58	oveq1d 6015	. . . . . . . 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 10912	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) ∈ ℕ)
61	60, 28	nnmulcld 9155	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) · ((1st ‘(◡𝐹‘𝐴))↑2)) ∈ ℕ)
62		oveq2 6008	. . . . . . . . . 10 ⊢ (𝑥 = ((1st ‘(◡𝐹‘𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(◡𝐹‘𝐴))↑2)))
63		oveq2 6008	. . . . . . . . . . 11 ⊢ (𝑦 = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) → (2↑𝑦) = (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
64	63	oveq1d 6015	. . . . . . . . . 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 6138	. . . . . . . . 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 1271	. . . . . . . 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 5901	. . . . . . . . . . . . . . . 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 6319	. . . . . . . . . . . . . . . . 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 5630	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
72	68, 71	eqtr3d 2264	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
73		df-ov 6003	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
74	72, 73	eqtr4di 2280	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))))
75	12, 9	nnexpcld 10912	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℕ)
76	75, 27	nnmulcld 9155	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))) ∈ ℕ)
77		oveq2 6008	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (1st ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))))
78		oveq2 6008	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → (2↑𝑦) = (2↑(2nd ‘(◡𝐹‘𝐴))))
79	78	oveq1d 6015	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
80	77, 79, 2	ovmpog 6138	. . . . . . . . . . . . . 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 1271	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
82	74, 81	eqtrd 2262	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
83	82	oveq1d 6015	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2))
84	75	nncnd 9120	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℂ)
85	84, 38	sqmuld 10902	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
86	83, 85	eqtrd 2262	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
87	86	oveq2d 6016	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (2 · (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2))))
88	56, 54	eqeltrrd 2307	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) ∈ ℂ)
89	28	nncnd 9120	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℂ)
90	52, 88, 89	mulassd 8166	. . . . . . . . 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 2265	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = ((2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
92	59, 66, 91	3eqtr4rd 2273	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (((1st ‘(◡𝐹‘𝐴))↑2)𝐹(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
93		df-ov 6003	. . . . . . 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 2279	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩) = (2 · (𝐴↑2)))
95		f1ocnvfv 5902	. . . . . . 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 5630	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) = (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩))
99		op2ndg 6295	. . . . 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 2262	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1))
102	101	breq2d 4094	. 2 ⊢ (𝐴 ∈ ℕ → (2 ∥ (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) ↔ 2 ∥ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
103	20, 102	mtbird 677	1 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 713   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  {crab 2512  ⟨cop 3669   class class class wbr 4082   × cxp 4716  ^◡ccnv 4717  ⟶wf 5313  –1-1-onto→wf1o 5316  ‘cfv 5317  (class class class)co 6000   ∈ cmpo 6002  1^st c1st 6282  2^nd c2nd 6283  ℂcc 7993  1c1 7996   + caddc 7998   · cmul 8000  ℕcn 9106  2c2 9157  ℕ₀cn0 9365  ℤcz 9442  ↑cexp 10755   ∥ cdvds 12293  ℙcprime 12624

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-2o 6561  df-er 6678  df-en 6886  df-sup 7147  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-fl 10485  df-mod 10540  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-dvds 12294  df-gcd 12470  df-prm 12625

This theorem is referenced by:  sqne2sq  12694