Theorem 2sqpwodd 12314

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 12312	. . . . . . . 8 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1ocnv 5513	. . . . . . . 8 ⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → ◡𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5		f1of 5500	. . . . . . . 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 5692	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘𝐴) ∈ (𝐽 × ℕ0))
8		xp2nd 6219	. . . . . 6 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
9	7, 8	syl 14	. . . . 5 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
10	9	nn0zd 9437	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℤ)
11		2nn 9143	. . . . . 6 ⊢ 2 ∈ ℕ
12	11	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ)
13	12	nnzd 9438	. . . 4 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℤ)
14	10, 13	zmulcld 9445	. . 3 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℤ)
15		dvdsmul2 11957	. . . 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 12017	. . . . 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 625	. . 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 6218	. . . . . . . . . . 11 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
22	7, 21	syl 14	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
23		breq2 4033	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
24	23	notbid 668	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
25	24, 1	elrab2 2919	. . . . . . . . . . 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 10765	. . . . . . . 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 12265	. . . . . . . . . . 11 ⊢ 2 ∈ ℙ
32	27	nnzd 9438	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℤ)
33		euclemma 12284	. . . . . . . . . . . 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 758	. . . . . . . . . . . 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 1353	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
37	30, 36	mtbird 674	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
38	27	nncnd 8996	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℂ)
39	38	sqvald 10741	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) = ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
40	39	breq2d 4041	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2) ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴)))))
41	37, 40	mtbird 674	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2))
42		breq2 4033	. . . . . . . . . 10 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
43	42	notbid 668	. . . . . . . . 9 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
44	43, 1	elrab2 2919	. . . . . . . 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 9293	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ0)
47	9, 46	nn0mulcld 9298	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0)
48		peano2nn0 9280	. . . . . . . 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 4689	. . . . . . 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 8996	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℂ)
53	52, 47	expp1d 10745	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · 2))
54	52, 47	expcld 10744	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) ∈ ℂ)
55	54, 52	mulcomd 8041	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · 2) = (2 · (2↑((2nd ‘(◡𝐹‘𝐴)) · 2))))
56	52, 46, 9	expmuld 10747	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2))
57	56	oveq2d 5934	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2 · (2↑((2nd ‘(◡𝐹‘𝐴)) · 2))) = (2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)))
58	53, 55, 57	3eqtrd 2230	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) = (2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)))
59	58	oveq1d 5933	. . . . . . . 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 10766	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) ∈ ℕ)
61	60, 28	nnmulcld 9031	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)) · ((1st ‘(◡𝐹‘𝐴))↑2)) ∈ ℕ)
62		oveq2 5926	. . . . . . . . . 10 ⊢ (𝑥 = ((1st ‘(◡𝐹‘𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(◡𝐹‘𝐴))↑2)))
63		oveq2 5926	. . . . . . . . . . 11 ⊢ (𝑦 = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1) → (2↑𝑦) = (2↑(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
64	63	oveq1d 5933	. . . . . . . . . 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 6053	. . . . . . . . 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 1249	. . . . . . . 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 5821	. . . . . . . . . . . . . . . 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 6228	. . . . . . . . . . . . . . . . 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 5558	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
72	68, 71	eqtr3d 2228	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
73		df-ov 5921	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
74	72, 73	eqtr4di 2244	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))))
75	12, 9	nnexpcld 10766	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℕ)
76	75, 27	nnmulcld 9031	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))) ∈ ℕ)
77		oveq2 5926	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (1st ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))))
78		oveq2 5926	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → (2↑𝑦) = (2↑(2nd ‘(◡𝐹‘𝐴))))
79	78	oveq1d 5933	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
80	77, 79, 2	ovmpog 6053	. . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
82	74, 81	eqtrd 2226	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
83	82	oveq1d 5933	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2))
84	75	nncnd 8996	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℂ)
85	84, 38	sqmuld 10756	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
86	83, 85	eqtrd 2226	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
87	86	oveq2d 5934	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (2 · (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2))))
88	56, 54	eqeltrrd 2271	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) ∈ ℂ)
89	28	nncnd 8996	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℂ)
90	52, 88, 89	mulassd 8043	. . . . . . . . 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 2229	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = ((2 · ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
92	59, 66, 91	3eqtr4rd 2237	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (2 · (𝐴↑2)) = (((1st ‘(◡𝐹‘𝐴))↑2)𝐹(((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
93		df-ov 5921	. . . . . . 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 2243	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩) = (2 · (𝐴↑2)))
95		f1ocnvfv 5822	. . . . . . 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 5558	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) = (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)⟩))
99		op2ndg 6204	. . . . 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 2226	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) = (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1))
102	101	breq2d 4041	. 2 ⊢ (𝐴 ∈ ℕ → (2 ∥ (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))) ↔ 2 ∥ (((2nd ‘(◡𝐹‘𝐴)) · 2) + 1)))
103	20, 102	mtbird 674	1 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (2nd ‘(◡𝐹‘(2 · (𝐴↑2)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  {crab 2476  ⟨cop 3621   class class class wbr 4029   × cxp 4657  ^◡ccnv 4658  ⟶wf 5250  –1-1-onto→wf1o 5253  ‘cfv 5254  (class class class)co 5918   ∈ cmpo 5920  1^st c1st 6191  2^nd c2nd 6192  ℂcc 7870  1c1 7873   + caddc 7875   · cmul 7877  ℕcn 8982  2c2 9033  ℕ₀cn0 9240  ℤcz 9317  ↑cexp 10609   ∥ cdvds 11930  ℙcprime 12245

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-en 6795  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080  df-prm 12246

This theorem is referenced by:  sqne2sq  12315