Theorem sqpweven 11842

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 11841	. . . . . . 7 ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
4		f1ocnv 5373	. . . . . . 7 ⊢ (𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ → ◡𝐹:ℕ–1-1-onto→(𝐽 × ℕ0))
5		f1of 5360	. . . . . . 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 5547	. . . . 5 ⊢ (𝐴 ∈ ℕ → (◡𝐹‘𝐴) ∈ (𝐽 × ℕ0))
8		xp2nd 6057	. . . . 5 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
9	7, 8	syl 14	. . . 4 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℕ0)
10	9	nn0zd 9164	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘𝐴)) ∈ ℤ)
11		2nn 8874	. . . . 5 ⊢ 2 ∈ ℕ
12	11	a1i 9	. . . 4 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ)
13	12	nnzd 9165	. . 3 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℤ)
14		dvdsmul2 11505	. . 3 ⊢ (((2nd ‘(◡𝐹‘𝐴)) ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
15	10, 13, 14	syl2anc 408	. 2 ⊢ (𝐴 ∈ ℕ → 2 ∥ ((2nd ‘(◡𝐹‘𝐴)) · 2))
16		xp1st 6056	. . . . . . . . . 10 ⊢ ((◡𝐹‘𝐴) ∈ (𝐽 × ℕ0) → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
17	7, 16	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ 𝐽)
18		breq2 3928	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (2 ∥ 𝑧 ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
19	18	notbid 656	. . . . . . . . . . 11 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝐴)) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
20	19, 1	elrab2 2838	. . . . . . . . . 10 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 ↔ ((1st ‘(◡𝐹‘𝐴)) ∈ ℕ ∧ ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
21	20	simplbi 272	. . . . . . . . 9 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 → (1st ‘(◡𝐹‘𝐴)) ∈ ℕ)
22	17, 21	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℕ)
23	22	nnsqcld 10438	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℕ)
24	20	simprbi 273	. . . . . . . . . 10 ⊢ ((1st ‘(◡𝐹‘𝐴)) ∈ 𝐽 → ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
25	17, 24	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
26		2prm 11797	. . . . . . . . . 10 ⊢ 2 ∈ ℙ
27	22	nnzd 9165	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℤ)
28		euclemma 11813	. . . . . . . . . . 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 746	. . . . . . . . . . 11 ⊢ ((2 ∥ (1st ‘(◡𝐹‘𝐴)) ∨ 2 ∥ (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴)))
30	28, 29	syl6bb 195	. . . . . . . . . 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 1320	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))) ↔ 2 ∥ (1st ‘(◡𝐹‘𝐴))))
32	25, 31	mtbird 662	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
33	22	nncnd 8727	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1st ‘(◡𝐹‘𝐴)) ∈ ℂ)
34	33	sqvald 10414	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) = ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴))))
35	34	breq2d 3936	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2) ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴)) · (1st ‘(◡𝐹‘𝐴)))))
36	32, 35	mtbird 662	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2))
37		breq2 3928	. . . . . . . . 9 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (2 ∥ 𝑧 ↔ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
38	37	notbid 656	. . . . . . . 8 ⊢ (𝑧 = ((1st ‘(◡𝐹‘𝐴))↑2) → (¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
39	38, 1	elrab2 2838	. . . . . . 7 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽 ↔ (((1st ‘(◡𝐹‘𝐴))↑2) ∈ ℕ ∧ ¬ 2 ∥ ((1st ‘(◡𝐹‘𝐴))↑2)))
40	23, 36, 39	sylanbrc 413	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))↑2) ∈ 𝐽)
41	12	nnnn0d 9023	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℕ0)
42	9, 41	nn0mulcld 9028	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((2nd ‘(◡𝐹‘𝐴)) · 2) ∈ ℕ0)
43		opelxp 4564	. . . . . 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 413	. . . . 5 ⊢ (𝐴 ∈ ℕ → ⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩ ∈ (𝐽 × ℕ0))
45	12	nncnd 8727	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 2 ∈ ℂ)
46	45, 41, 9	expmuld 10420	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑(2nd ‘(◡𝐹‘𝐴)))↑2))
47	46	oveq1d 5782	. . . . . . 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 10439	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) ∈ ℕ)
49	48, 23	nnmulcld 8762	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)) ∈ ℕ)
50		oveq2 5775	. . . . . . . . 9 ⊢ (𝑥 = ((1st ‘(◡𝐹‘𝐴))↑2) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · ((1st ‘(◡𝐹‘𝐴))↑2)))
51		oveq2 5775	. . . . . . . . . 10 ⊢ (𝑦 = ((2nd ‘(◡𝐹‘𝐴)) · 2) → (2↑𝑦) = (2↑((2nd ‘(◡𝐹‘𝐴)) · 2)))
52	51	oveq1d 5782	. . . . . . . . 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 5898	. . . . . . . 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 1216	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)) = ((2↑((2nd ‘(◡𝐹‘𝐴)) · 2)) · ((1st ‘(◡𝐹‘𝐴))↑2)))
55		f1ocnvfv2 5672	. . . . . . . . . . . . 13 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝐴 ∈ ℕ) → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
56	3, 55	mpan 420	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = 𝐴)
57		1st2nd2 6066	. . . . . . . . . . . . . 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 5418	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (𝐹‘(◡𝐹‘𝐴)) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
60	56, 59	eqtr3d 2172	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩))
61		df-ov 5770	. . . . . . . . . . 11 ⊢ ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = (𝐹‘⟨(1st ‘(◡𝐹‘𝐴)), (2nd ‘(◡𝐹‘𝐴))⟩)
62	60, 61	syl6eqr 2188	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))))
63	12, 9	nnexpcld 10439	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℕ)
64	63, 22	nnmulcld 8762	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))) ∈ ℕ)
65		oveq2 5775	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))))
66		oveq2 5775	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → (2↑𝑦) = (2↑(2nd ‘(◡𝐹‘𝐴))))
67	66	oveq1d 5782	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝐴)) → ((2↑𝑦) · (1st ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
68	65, 67, 2	ovmpog 5898	. . . . . . . . . . 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 1216	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((1st ‘(◡𝐹‘𝐴))𝐹(2nd ‘(◡𝐹‘𝐴))) = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
70	62, 69	eqtrd 2170	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 = ((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴))))
71	70	oveq1d 5782	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2))
72	63	nncnd 8727	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (2↑(2nd ‘(◡𝐹‘𝐴))) ∈ ℂ)
73	72, 33	sqmuld 10429	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (((2↑(2nd ‘(◡𝐹‘𝐴))) · (1st ‘(◡𝐹‘𝐴)))↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
74	71, 73	eqtrd 2170	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((2↑(2nd ‘(◡𝐹‘𝐴)))↑2) · ((1st ‘(◡𝐹‘𝐴))↑2)))
75	47, 54, 74	3eqtr4rd 2181	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) = (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)))
76		df-ov 5770	. . . . . 6 ⊢ (((1st ‘(◡𝐹‘𝐴))↑2)𝐹((2nd ‘(◡𝐹‘𝐴)) · 2)) = (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩)
77	75, 76	syl6req 2187	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐹‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩) = (𝐴↑2))
78		f1ocnvfv 5673	. . . . . 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 420	. . . . 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 5418	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(𝐴↑2))) = (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩))
82		op2ndg 6042	. . . 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 408	. . 3 ⊢ (𝐴 ∈ ℕ → (2nd ‘⟨((1st ‘(◡𝐹‘𝐴))↑2), ((2nd ‘(◡𝐹‘𝐴)) · 2)⟩) = ((2nd ‘(◡𝐹‘𝐴)) · 2))
84	81, 83	eqtrd 2170	. 2 ⊢ (𝐴 ∈ ℕ → (2nd ‘(◡𝐹‘(𝐴↑2))) = ((2nd ‘(◡𝐹‘𝐴)) · 2))
85	15, 84	breqtrrd 3951	1 ⊢ (𝐴 ∈ ℕ → 2 ∥ (2nd ‘(◡𝐹‘(𝐴↑2))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 697   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  {crab 2418  ⟨cop 3525   class class class wbr 3924   × cxp 4532  ^◡ccnv 4533  ⟶wf 5114  –1-1-onto→wf1o 5117  ‘cfv 5118  (class class class)co 5767   ∈ cmpo 5769  1^st c1st 6029  2^nd c2nd 6030   · cmul 7618  ℕcn 8713  2c2 8764  ℕ₀cn0 8970  ℤcz 9047  ↑cexp 10285   ∥ cdvds 11482  ℙcprime 11777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-1o 6306  df-2o 6307  df-er 6422  df-en 6628  df-sup 6864  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-fl 10036  df-mod 10089  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-dvds 11483  df-gcd 11625  df-prm 11778

This theorem is referenced by:  sqne2sq  11844