Theorem 2sqpwodd 12041

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 12039	. . . . . . . 8 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1ocnv 5426	. . . . . . . 8 ⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → ◡𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5		f1of 5413	. . . . . . . 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	ffvelrni 5600	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘𝐴) ∈ (𝐽 × ℕ0))
8		xp2nd 6111	. . . . . 6 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
9	7, 8	syl 14	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
10	9	nn0zd 9278	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℤ)
11		2nn 8988	. . . . . 6 ⊢ 2 ∈ ℕ
12	11	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ)
13	12	nnzd 9279	. . . 4 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℤ)
14	10, 13	zmulcld 9286	. . 3 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℤ)
15		dvdsmul2 11702	. . . 4 ⊢ (((2nd ‘(◡𝐹‘𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
16	10, 13, 15	syl2anc 409	. . 3 ⊢ (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
17		oddp1even 11759	. . . . 5 ⊢ (((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℤ → (¬ 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2) ↔ 2 ∥ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
18	17	biimprd 157	. . . 4 ⊢ (((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℤ → (2 ∥ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) → ¬ 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2)))
19	18	con2d 614	. . 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 6110	. . . . . . . . . . 11 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
22	7, 21	syl 14	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
23		breq2 3969	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
24	23	notbid 657	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
25	24, 1	elrab2 2871	. . . . . . . . . . 11 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 ↔ ((1st ‘(◡𝐹‘𝐴)) ∈ ℕ ∧ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
26	25	simplbi 272	. . . . . . . . . 10 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 → (1st ‘(◡𝐹‘𝐴)) ∈ ℕ)
27	22, 26	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℕ)
28	27	nnsqcld 10565	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℕ)
29	25	simprbi 273	. . . . . . . . . . 11 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 → ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
30	22, 29	syl 14	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
31		2prm 11995	. . . . . . . . . . 11 ⊢ 2 ∈ ℙ
32	27	nnzd 9279	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℤ)
33		euclemma 12011	. . . . . . . . . . . 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 747	. . . . . . . . . . . 12 ⊢ ((2 ∥ (1st ‘(◡𝐹‘𝐴)) ∨ 2 ∥ (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
35	33, 34	bitrdi 195	. . . . . . . . . . 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 1324	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
37	30, 36	mtbird 663	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
38	27	nncnd 8841	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℂ)
39	38	sqvald 10541	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) = ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
40	39	breq2d 3977	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2) ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴)))))
41	37, 40	mtbird 663	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2))
42		breq2 3969	. . . . . . . . . 10 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
43	42	notbid 657	. . . . . . . . 9 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
44	43, 1	elrab2 2871	. . . . . . . 8 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
45	28, 41, 44	sylanbrc 414	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽)
46	12	nnnn0d 9137	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ0)
47	9, 46	nn0mulcld 9142	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0)
48		peano2nn0 9124	. . . . . . . 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 4615	. . . . . . 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 414	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩ ∈ (𝐽 × ℕ0))
52	12	nncnd 8841	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℂ)
53	52, 47	expp1d 10545	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · 2))
54	52, 47	expcld 10544	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) ∈ ℂ)
55	54, 52	mulcomd 7893	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · 2) = (2 · (2↑((2nd ‘(◡𝐹‘𝐴)) · 2))))
56	52, 46, 9	expmuld 10547	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2))
57	56	oveq2d 5837	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 · (2↑((2nd ‘(◡𝐹‘𝐴)) · 2))) = (2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)))
58	53, 55, 57	3eqtrd 2194	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = (2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)))
59	58	oveq1d 5836	. . . . . . . 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 10566	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) ∈ ℕ)
61	60, 28	nnmulcld 8876	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) · ((1st ‘(◡𝐹‘𝐴))↑2)) ∈ ℕ)
62		oveq2 5829	. . . . . . . . . 10 ⊢ (𝑥 = ((1st ‘(◡𝐹‘𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(◡𝐹‘𝐴))↑2)))
63		oveq2 5829	. . . . . . . . . . 11 ⊢ (𝑦 = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) → (2↑𝑦) = (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
64	63	oveq1d 5836	. . . . . . . . . 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 5952	. . . . . . . . 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 1220	. . . . . . . 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 5725	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
68	3, 67	mpan 421	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
69		1st2nd2 6120	. . . . . . . . . . . . . . . . 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 5471	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
72	68, 71	eqtr3d 2192	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
73		df-ov 5824	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
74	72, 73	eqtr4di 2208	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))))
75	12, 9	nnexpcld 10566	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℕ)
76	75, 27	nnmulcld 8876	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))) ∈ ℕ)
77		oveq2 5829	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (1st ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))))
78		oveq2 5829	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → (2↑𝑦) = (2↑(2nd ‘(◡𝐹‘𝐴))))
79	78	oveq1d 5836	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
80	77, 79, 2	ovmpog 5952	. . . . . . . . . . . . . 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 1220	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
82	74, 81	eqtrd 2190	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
83	82	oveq1d 5836	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2))
84	75	nncnd 8841	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℂ)
85	84, 38	sqmuld 10556	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
86	83, 85	eqtrd 2190	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
87	86	oveq2d 5837	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (2 · (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2))))
88	56, 54	eqeltrrd 2235	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) ∈ ℂ)
89	28	nncnd 8841	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℂ)
90	52, 88, 89	mulassd 7895	. . . . . . . . 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 2193	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = ((2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
92	59, 66, 91	3eqtr4rd 2201	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (((1st ‘(◡𝐹‘𝐴))↑2)𝐹(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
93		df-ov 5824	. . . . . . 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 2207	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩) = (2 · (𝐴↑2)))
95		f1ocnvfv 5726	. . . . . . 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 421	. . . . . 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 5471	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) = (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩))
99		op2ndg 6096	. . . . 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 409	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩) = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1))
101	98, 100	eqtrd 2190	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1))
102	101	breq2d 3977	. 2 ⊢ (𝐴 ∈ ℕ → (2 ∥ (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) ↔ 2 ∥ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
103	20, 102	mtbird 663	1 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 698   ∧ w3a 963   = wceq 1335   ∈ wcel 2128  {crab 2439  ⟨cop 3563   class class class wbr 3965   × cxp 4583  ^◡ccnv 4584  ⟶wf 5165  –1-1-onto→wf1o 5168  ‘cfv 5169  (class class class)co 5821   ∈ cmpo 5823  1^st c1st 6083  2^nd c2nd 6084  ℂcc 7724  1c1 7727   + caddc 7729   · cmul 7731  ℕcn 8827  2c2 8878  ℕ₀cn0 9084  ℤcz 9161  ↑cexp 10411   ∥ cdvds 11676  ℙcprime 11975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-mulrcl 7825  ax-addcom 7826  ax-mulcom 7827  ax-addass 7828  ax-mulass 7829  ax-distr 7830  ax-i2m1 7831  ax-0lt1 7832  ax-1rid 7833  ax-0id 7834  ax-rnegex 7835  ax-precex 7836  ax-cnre 7837  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840  ax-pre-apti 7841  ax-pre-ltadd 7842  ax-pre-mulgt0 7843  ax-pre-mulext 7844  ax-arch 7845  ax-caucvg 7846

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-xor 1358  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-frec 6335  df-1o 6360  df-2o 6361  df-er 6477  df-en 6683  df-sup 6924  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-sub 8042  df-neg 8043  df-reap 8444  df-ap 8451  df-div 8540  df-inn 8828  df-2 8886  df-3 8887  df-4 8888  df-n0 9085  df-z 9162  df-uz 9434  df-q 9522  df-rp 9554  df-fz 9906  df-fzo 10035  df-fl 10162  df-mod 10215  df-seqfrec 10338  df-exp 10412  df-cj 10735  df-re 10736  df-im 10737  df-rsqrt 10891  df-abs 10892  df-dvds 11677  df-gcd 11822  df-prm 11976

This theorem is referenced by:  sqne2sq  12042