Theorem sqpweven 12612

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 12611	. . . . . . 7 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1ocnv 5557	. . . . . . 7 ⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → ◡𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5		f1of 5544	. . . . . . 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 5737	. . . . 5 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘𝐴) ∈ (𝐽 × ℕ0))
8		xp2nd 6275	. . . . 5 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
9	7, 8	syl 14	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
10	9	nn0zd 9528	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℤ)
11		2nn 9233	. . . . 5 ⊢ 2 ∈ ℕ
12	11	a1i 9	. . . 4 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ)
13	12	nnzd 9529	. . 3 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℤ)
14		dvdsmul2 12240	. . 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 6274	. . . . . . . . . 10 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
17	7, 16	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
18		breq2 4063	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
19	18	notbid 669	. . . . . . . . . . 11 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
20	19, 1	elrab2 2939	. . . . . . . . . 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 10876	. . . . . . 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 12564	. . . . . . . . . 10 ⊢ 2 ∈ ℙ
27	22	nnzd 9529	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℤ)
28		euclemma 12583	. . . . . . . . . . 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 759	. . . . . . . . . . 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 1355	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
32	25, 31	mtbird 675	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
33	22	nncnd 9085	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℂ)
34	33	sqvald 10852	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) = ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
35	34	breq2d 4071	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2) ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴)))))
36	32, 35	mtbird 675	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2))
37		breq2 4063	. . . . . . . . 9 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
38	37	notbid 669	. . . . . . . 8 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
39	38, 1	elrab2 2939	. . . . . . 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 9383	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ0)
42	9, 41	nn0mulcld 9388	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0)
43		opelxp 4723	. . . . . 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 9085	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℂ)
46	45, 41, 9	expmuld 10858	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2))
47	46	oveq1d 5982	. . . . . . 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 10877	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) ∈ ℕ)
49	48, 23	nnmulcld 9120	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)) ∈ ℕ)
50		oveq2 5975	. . . . . . . . 9 ⊢ (𝑥 = ((1st ‘(◡𝐹‘𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(◡𝐹‘𝐴))↑2)))
51		oveq2 5975	. . . . . . . . . 10 ⊢ (𝑦 = ((2nd ‘(◡𝐹‘𝐴)) · 2) → (2↑𝑦) = (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)))
52	51	oveq1d 5982	. . . . . . . . 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 6103	. . . . . . . 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 1250	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
55		f1ocnvfv2 5870	. . . . . . . . . . . . 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 6284	. . . . . . . . . . . . . 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 5603	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
60	56, 59	eqtr3d 2242	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
61		df-ov 5970	. . . . . . . . . . 11 ⊢ ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
62	60, 61	eqtr4di 2258	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))))
63	12, 9	nnexpcld 10877	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℕ)
64	63, 22	nnmulcld 9120	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))) ∈ ℕ)
65		oveq2 5975	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))))
66		oveq2 5975	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → (2↑𝑦) = (2↑(2nd ‘(◡𝐹‘𝐴))))
67	66	oveq1d 5982	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
68	65, 67, 2	ovmpog 6103	. . . . . . . . . . 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 1250	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
70	62, 69	eqtrd 2240	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
71	70	oveq1d 5982	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2))
72	63	nncnd 9085	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℂ)
73	72, 33	sqmuld 10867	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
74	71, 73	eqtrd 2240	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
75	47, 54, 74	3eqtr4rd 2251	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)))
76		df-ov 5970	. . . . . 6 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)) = (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩)
77	75, 76	eqtr2di 2257	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩) = (𝐴↑2))
78		f1ocnvfv 5871	. . . . . 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 5603	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(𝐴↑2))) = (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩))
82		op2ndg 6260	. . . 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 2240	. 2 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(𝐴↑2))) = ((2nd ‘(◡𝐹‘𝐴)) · 2))
85	15, 84	breqtrrd 4087	1 ⊢ (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(◡𝐹‘(𝐴↑2))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 710   ∧ w3a 981   = wceq 1373   ∈ wcel 2178  {crab 2490  ⟨cop 3646   class class class wbr 4059   × cxp 4691  ^◡ccnv 4692  ⟶wf 5286  –1-1-onto→wf1o 5289  ‘cfv 5290  (class class class)co 5967   ∈ cmpo 5969  1^st c1st 6247  2^nd c2nd 6248   · cmul 7965  ℕcn 9071  2c2 9122  ℕ₀cn0 9330  ℤcz 9407  ↑cexp 10720   ∥ cdvds 12213  ℙcprime 12544

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-2o 6526  df-er 6643  df-en 6851  df-sup 7112  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-gcd 12390  df-prm 12545

This theorem is referenced by:  sqne2sq  12614