Theorem sqpweven 11235

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 11234	. . . . . . 7 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1ocnv 5250	. . . . . . 7 ⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → ◡𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5		f1of 5237	. . . . . . 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	ffvelrni 5417	. . . . 5 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘𝐴) ∈ (𝐽 × ℕ0))
8		xp2nd 5919	. . . . 5 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
9	7, 8	syl 14	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
10	9	nn0zd 8836	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℤ)
11		2nn 8547	. . . . 5 ⊢ 2 ∈ ℕ
12	11	a1i 9	. . . 4 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ)
13	12	nnzd 8837	. . 3 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℤ)
14		dvdsmul2 10901	. . 3 ⊢ (((2nd ‘(◡𝐹‘𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
15	10, 13, 14	syl2anc 403	. 2 ⊢ (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
16		xp1st 5918	. . . . . . . . . 10 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
17	7, 16	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
18		breq2 3841	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
19	18	notbid 627	. . . . . . . . . . 11 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
20	19, 1	elrab2 2772	. . . . . . . . . 10 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 ↔ ((1st ‘(◡𝐹‘𝐴)) ∈ ℕ ∧ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
21	20	simplbi 268	. . . . . . . . 9 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 → (1st ‘(◡𝐹‘𝐴)) ∈ ℕ)
22	17, 21	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℕ)
23	22	nnsqcld 10072	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℕ)
24	20	simprbi 269	. . . . . . . . . 10 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 → ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
25	17, 24	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
26		2prm 11191	. . . . . . . . . 10 ⊢ 2 ∈ ℙ
27	22	nnzd 8837	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℤ)
28		euclemma 11207	. . . . . . . . . . 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 709	. . . . . . . . . . 11 ⊢ ((2 ∥ (1st ‘(◡𝐹‘𝐴)) ∨ 2 ∥ (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
30	28, 29	syl6bb 194	. . . . . . . . . 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 1278	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
32	25, 31	mtbird 633	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
33	22	nncnd 8408	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℂ)
34	33	sqvald 10048	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) = ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
35	34	breq2d 3849	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2) ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴)))))
36	32, 35	mtbird 633	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2))
37		breq2 3841	. . . . . . . . 9 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
38	37	notbid 627	. . . . . . . 8 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
39	38, 1	elrab2 2772	. . . . . . 7 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
40	23, 36, 39	sylanbrc 408	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽)
41	12	nnnn0d 8696	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ0)
42	9, 41	nn0mulcld 8701	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0)
43		opelxp 4457	. . . . . 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 408	. . . . 5 ⊢ (𝐴 ∈ ℕ → ⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0))
45	12	nncnd 8408	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℂ)
46	45, 41, 9	expmuld 10054	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2))
47	46	oveq1d 5649	. . . . . . 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 10073	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) ∈ ℕ)
49	48, 23	nnmulcld 8442	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)) ∈ ℕ)
50		oveq2 5642	. . . . . . . . 9 ⊢ (𝑥 = ((1st ‘(◡𝐹‘𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(◡𝐹‘𝐴))↑2)))
51		oveq2 5642	. . . . . . . . . 10 ⊢ (𝑦 = ((2nd ‘(◡𝐹‘𝐴)) · 2) → (2↑𝑦) = (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)))
52	51	oveq1d 5649	. . . . . . . . 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	ovmpt2g 5761	. . . . . . . 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 1174	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
55		f1ocnvfv2 5539	. . . . . . . . . . . . 13 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
56	3, 55	mpan 415	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
57		1st2nd2 5927	. . . . . . . . . . . . . 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 5293	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
60	56, 59	eqtr3d 2122	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
61		df-ov 5637	. . . . . . . . . . 11 ⊢ ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
62	60, 61	syl6eqr 2138	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))))
63	12, 9	nnexpcld 10073	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℕ)
64	63, 22	nnmulcld 8442	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))) ∈ ℕ)
65		oveq2 5642	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))))
66		oveq2 5642	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → (2↑𝑦) = (2↑(2nd ‘(◡𝐹‘𝐴))))
67	66	oveq1d 5649	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
68	65, 67, 2	ovmpt2g 5761	. . . . . . . . . . 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 1174	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
70	62, 69	eqtrd 2120	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
71	70	oveq1d 5649	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2))
72	63	nncnd 8408	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℂ)
73	72, 33	sqmuld 10063	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
74	71, 73	eqtrd 2120	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
75	47, 54, 74	3eqtr4rd 2131	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)))
76		df-ov 5637	. . . . . 6 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)) = (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩)
77	75, 76	syl6req 2137	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩) = (𝐴↑2))
78		f1ocnvfv 5540	. . . . . 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 415	. . . . 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 61	. . . 4 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘(𝐴↑2)) = ⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩)
81	80	fveq2d 5293	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(𝐴↑2))) = (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩))
82		op2ndg 5904	. . . 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 403	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩) = ((2nd ‘(◡𝐹‘𝐴)) · 2))
84	81, 83	eqtrd 2120	. 2 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(𝐴↑2))) = ((2nd ‘(◡𝐹‘𝐴)) · 2))
85	15, 84	breqtrrd 3863	1 ⊢ (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(◡𝐹‘(𝐴↑2))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103   ∨ wo 664   ∧ w3a 924   = wceq 1289   ∈ wcel 1438  {crab 2363  ⟨cop 3444   class class class wbr 3837   × cxp 4426  ^◡ccnv 4427  ⟶wf 4998  –1-1-onto→wf1o 5001  ‘cfv 5002  (class class class)co 5634   ↦ cmpt2 5636  1^st c1st 5891  2^nd c2nd 5892   · cmul 7334  ℕcn 8394  2c2 8444  ℕ₀cn0 8643  ℤcz 8720  ↑cexp 9919   ∥ cdvds 10878  ℙcprime 11171

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-1o 6163  df-2o 6164  df-er 6272  df-en 6438  df-sup 6658  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10879  df-gcd 11021  df-prm 11172

This theorem is referenced by:  sqne2sq  11237