Theorem sqpweven 12868

Description: The greatest power of two dividing the square of an integer is an even 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
sqpweven	⊢ (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(◡𝐹‘(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 sqpweven

Step	Hyp	Ref	Expression
1		oddpwdc.j	. . . . . . . 8 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
2		oddpwdc.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
3	1, 2	oddpwdc 12867	. . . . . . 7 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1ocnv 5626	. . . . . . 7 ⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → ◡𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5		f1of 5613	. . . . . . 7 ⊢ (◡𝐹:ℕ–1-1-onto→(𝐽 × ℕ0) → ◡𝐹:ℕ⟶(𝐽 × ℕ0))
6	3, 4, 5	mp2b 8	. . . . . 6 ⊢ ◡𝐹:ℕ⟶(𝐽 × ℕ0)
7	6	ffvelcdmi 5810	. . . . 5 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘𝐴) ∈ (𝐽 × ℕ0))
8		xp2nd 6359	. . . . 5 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
9	7, 8	syl 14	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
10	9	nn0zd 9697	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℤ)
11		2nn 9398	. . . . 5 ⊢ 2 ∈ ℕ
12	11	a1i 9	. . . 4 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ)
13	12	nnzd 9698	. . 3 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℤ)
14		dvdsmul2 12496	. . 3 ⊢ (((2nd ‘(◡𝐹‘𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
15	10, 13, 14	syl2anc 411	. 2 ⊢ (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
16		xp1st 6358	. . . . . . . . . 10 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
17	7, 16	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
18		breq2 4112	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
19	18	notbid 673	. . . . . . . . . . 11 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
20	19, 1	elrab2 2975	. . . . . . . . . 10 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 ↔ ((1st ‘(◡𝐹‘𝐴)) ∈ ℕ ∧ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
21	20	simplbi 274	. . . . . . . . 9 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 → (1st ‘(◡𝐹‘𝐴)) ∈ ℕ)
22	17, 21	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℕ)
23	22	nnsqcld 11055	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℕ)
24	20	simprbi 275	. . . . . . . . . 10 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 → ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
25	17, 24	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
26		2prm 12820	. . . . . . . . . 10 ⊢ 2 ∈ ℙ
27	22	nnzd 9698	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℤ)
28		euclemma 12839	. . . . . . . . . . 11 ⊢ ((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)))))
29		oridm 765	. . . . . . . . . . 11 ⊢ ((2 ∥ (1st ‘(◡𝐹‘𝐴)) ∨ 2 ∥ (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
30	28, 29	bitrdi 196	. . . . . . . . . 10 ⊢ ((2 ∈ ℙ ∧ (1st ‘(◡𝐹‘𝐴)) ∈ ℤ ∧ (1st ‘(◡𝐹‘𝐴)) ∈ ℤ) → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
31	26, 27, 27, 30	mp3an2i 1379	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
32	25, 31	mtbird 680	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
33	22	nncnd 9250	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℂ)
34	33	sqvald 11031	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) = ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
35	34	breq2d 4120	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2) ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴)))))
36	32, 35	mtbird 680	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2))
37		breq2 4112	. . . . . . . . 9 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
38	37	notbid 673	. . . . . . . 8 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
39	38, 1	elrab2 2975	. . . . . . 7 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
40	23, 36, 39	sylanbrc 417	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽)
41	12	nnnn0d 9552	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ0)
42	9, 41	nn0mulcld 9557	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0)
43		opelxp 4778	. . . . . 6 ⊢ (⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0) ↔ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ∧ ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0))
44	40, 42, 43	sylanbrc 417	. . . . 5 ⊢ (𝐴 ∈ ℕ → ⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0))
45	12	nncnd 9250	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℂ)
46	45, 41, 9	expmuld 11037	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2))
47	46	oveq1d 6064	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
48	12, 42	nnexpcld 11056	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) ∈ ℕ)
49	48, 23	nnmulcld 9285	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)) ∈ ℕ)
50		oveq2 6057	. . . . . . . . 9 ⊢ (𝑥 = ((1st ‘(◡𝐹‘𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(◡𝐹‘𝐴))↑2)))
51		oveq2 6057	. . . . . . . . . 10 ⊢ (𝑦 = ((2nd ‘(◡𝐹‘𝐴)) · 2) → (2↑𝑦) = (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)))
52	51	oveq1d 6064	. . . . . . . . 9 ⊢ (𝑦 = ((2nd ‘(◡𝐹‘𝐴)) · 2) → ((2↑𝑦) · ((1st ‘(◡𝐹‘𝐴))↑2)) = ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
53	50, 52, 2	ovmpog 6187	. . . . . . . 8 ⊢ ((((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ∧ ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0 ∧ ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)) ∈ ℕ) → (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
54	40, 42, 49, 53	syl3anc 1274	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
55		f1ocnvfv2 5950	. . . . . . . . . . . . 13 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
56	3, 55	mpan 424	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
57		1st2nd2 6368	. . . . . . . . . . . . . 14 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (◡𝐹‘𝐴) = ⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
58	7, 57	syl 14	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘𝐴) = ⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
59	58	fveq2d 5673	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
60	56, 59	eqtr3d 2267	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
61		df-ov 6052	. . . . . . . . . . 11 ⊢ ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
62	60, 61	eqtr4di 2283	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))))
63	12, 9	nnexpcld 11056	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℕ)
64	63, 22	nnmulcld 9285	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))) ∈ ℕ)
65		oveq2 6057	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))))
66		oveq2 6057	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → (2↑𝑦) = (2↑(2nd ‘(◡𝐹‘𝐴))))
67	66	oveq1d 6064	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
68	65, 67, 2	ovmpog 6187	. . . . . . . . . . 11 ⊢ (((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))))
69	17, 9, 64, 68	syl3anc 1274	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
70	62, 69	eqtrd 2265	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
71	70	oveq1d 6064	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2))
72	63	nncnd 9250	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℂ)
73	72, 33	sqmuld 11046	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
74	71, 73	eqtrd 2265	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
75	47, 54, 74	3eqtr4rd 2276	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)))
76		df-ov 6052	. . . . . 6 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)) = (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩)
77	75, 76	eqtr2di 2282	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩) = (𝐴↑2))
78		f1ocnvfv 5951	. . . . . 6 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ ⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0)) → ((𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩) = (𝐴↑2) → (◡𝐹‘(𝐴↑2)) = ⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩))
79	3, 78	mpan 424	. . . . 5 ⊢ (⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0) → ((𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩) = (𝐴↑2) → (◡𝐹‘(𝐴↑2)) = ⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩))
80	44, 77, 79	sylc 62	. . . 4 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘(𝐴↑2)) = ⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩)
81	80	fveq2d 5673	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(𝐴↑2))) = (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩))
82		op2ndg 6344	. . . 4 ⊢ ((((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ∧ ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0) → (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩) = ((2nd ‘(◡𝐹‘𝐴)) · 2))
83	40, 42, 82	syl2anc 411	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩) = ((2nd ‘(◡𝐹‘𝐴)) · 2))
84	81, 83	eqtrd 2265	. 2 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(𝐴↑2))) = ((2nd ‘(◡𝐹‘𝐴)) · 2))
85	15, 84	breqtrrd 4136	1 ⊢ (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(◡𝐹‘(𝐴↑2))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  {crab 2524  ⟨cop 3691   class class class wbr 4108   × cxp 4746  ^◡ccnv 4747  ⟶wf 5347  –1-1-onto→wf1o 5350  ‘cfv 5351  (class class class)co 6049   ∈ cmpo 6051  1^st c1st 6331  2^nd c2nd 6332   · cmul 8131  ℕcn 9236  2c2 9287  ℕ₀cn0 9495  ℤcz 9576  ↑cexp 10899   ∥ cdvds 12469  ℙcprime 12800

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

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-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-1o 6646  df-2o 6647  df-er 6766  df-en 6975  df-sup 7274  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-fz 10342  df-fzo 10476  df-fl 10629  df-mod 10684  df-seqfrec 10809  df-exp 10900  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-dvds 12470  df-gcd 12646  df-prm 12801

This theorem is referenced by:  sqne2sq  12870