Theorem sqpweven 12981

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 12980	. . . . . . 7 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1ocnv 5655	. . . . . . 7 ⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → ◡𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5		f1of 5642	. . . . . . 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 5845	. . . . 5 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘𝐴) ∈ (𝐽 × ℕ0))
8		xp2nd 6404	. . . . 5 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
9	7, 8	syl 14	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
10	9	nn0zd 9778	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℤ)
11		2nn 9478	. . . . 5 ⊢ 2 ∈ ℕ
12	11	a1i 9	. . . 4 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ)
13	12	nnzd 9779	. . 3 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℤ)
14		dvdsmul2 12607	. . 3 ⊢ (((2nd ‘(◡𝐹‘𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
15	10, 13, 14	syl2anc 415	. 2 ⊢ (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
16		xp1st 6403	. . . . . . . . . 10 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
17	7, 16	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
18		breq2 4135	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
19	18	notbid 677	. . . . . . . . . . 11 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
20	19, 1	elrab2 2985	. . . . . . . . . 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 11154	. . . . . . 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 12931	. . . . . . . . . 10 ⊢ 2 ∈ ℙ
27	22	nnzd 9779	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℤ)
28		euclemma 12951	. . . . . . . . . . 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 769	. . . . . . . . . . 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 1383	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
32	25, 31	mtbird 684	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
33	22	nncnd 9328	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℂ)
34	33	sqvald 11130	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) = ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
35	34	breq2d 4143	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2) ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴)))))
36	32, 35	mtbird 684	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2))
37		breq2 4135	. . . . . . . . 9 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
38	37	notbid 677	. . . . . . . 8 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
39	38, 1	elrab2 2985	. . . . . . 7 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
40	23, 36, 39	sylanbrc 421	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽)
41	12	nnnn0d 9632	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ0)
42	9, 41	nn0mulcld 9637	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0)
43		opelxp 4806	. . . . . 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 421	. . . . 5 ⊢ (𝐴 ∈ ℕ → ⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0))
45	12	nncnd 9328	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℂ)
46	45, 41, 9	expmuld 11136	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2))
47	46	oveq1d 6103	. . . . . . 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 11155	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) ∈ ℕ)
49	48, 23	nnmulcld 9363	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)) ∈ ℕ)
50		oveq2 6096	. . . . . . . . 9 ⊢ (𝑥 = ((1st ‘(◡𝐹‘𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(◡𝐹‘𝐴))↑2)))
51		oveq2 6096	. . . . . . . . . 10 ⊢ (𝑦 = ((2nd ‘(◡𝐹‘𝐴)) · 2) → (2↑𝑦) = (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)))
52	51	oveq1d 6103	. . . . . . . . 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 6226	. . . . . . . 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 1278	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
55		f1ocnvfv2 5987	. . . . . . . . . . . . 13 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
56	3, 55	mpan 428	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
57		1st2nd2 6413	. . . . . . . . . . . . . 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 5702	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
60	56, 59	eqtr3d 2273	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
61		df-ov 6091	. . . . . . . . . . 11 ⊢ ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
62	60, 61	eqtr4di 2289	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))))
63	12, 9	nnexpcld 11155	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℕ)
64	63, 22	nnmulcld 9363	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))) ∈ ℕ)
65		oveq2 6096	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))))
66		oveq2 6096	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → (2↑𝑦) = (2↑(2nd ‘(◡𝐹‘𝐴))))
67	66	oveq1d 6103	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
68	65, 67, 2	ovmpog 6226	. . . . . . . . . . 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 1278	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
70	62, 69	eqtrd 2271	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
71	70	oveq1d 6103	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2))
72	63	nncnd 9328	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℂ)
73	72, 33	sqmuld 11145	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
74	71, 73	eqtrd 2271	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
75	47, 54, 74	3eqtr4rd 2282	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)))
76		df-ov 6091	. . . . . 6 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)) = (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩)
77	75, 76	eqtr2di 2288	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩) = (𝐴↑2))
78		f1ocnvfv 5988	. . . . . 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 428	. . . . 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 5702	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(𝐴↑2))) = (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩))
82		op2ndg 6389	. . . 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 415	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩) = ((2nd ‘(◡𝐹‘𝐴)) · 2))
84	81, 83	eqtrd 2271	. 2 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(𝐴↑2))) = ((2nd ‘(◡𝐹‘𝐴)) · 2))
85	15, 84	breqtrrd 4159	1 ⊢ (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(◡𝐹‘(𝐴↑2))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 720   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {crab 2532  ⟨cop 3712   class class class wbr 4131   × cxp 4774  ^◡ccnv 4775  ⟶wf 5376  –1-1-onto→wf1o 5379  ‘cfv 5380  (class class class)co 6088   ∈ cmpo 6090  1^st c1st 6376  2^nd c2nd 6377   · cmul 8191  ℕcn 9314  2c2 9365  ℕ₀cn0 9575  ℤcz 9656  ↑cexp 10997   ∥ cdvds 12580  ℙcprime 12911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-2o 6692  df-er 6811  df-en 7028  df-sup 7331  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-fz 10430  df-fzo 10568  df-fl 10723  df-mod 10782  df-seqfrec 10907  df-exp 10998  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-dvds 12581  df-gcd 12757  df-prm 12912

This theorem is used by:  sqne2sq  12983