wilthlem1 - Metamath Proof Explorer

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Theorem wilthlem1 25653

Description: The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in ℤ / 𝑃ℤ are 1 and -1≡𝑃 − 1. (Note that from prmdiveq 16113, (𝑁↑(𝑃 − 2)) mod 𝑃 is the modular inverse of 𝑁 in ℤ / 𝑃ℤ. (Contributed by Mario Carneiro, 24-Jan-2015.)

**Assertion**
Ref	Expression
wilthlem1	⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 = ((𝑁↑(𝑃 − 2)) mod 𝑃) ↔ (𝑁 = 1 ∨ 𝑁 = (𝑃 − 1))))

**Proof of Theorem wilthlem1**
Step	Hyp	Ref	Expression
1		elfzelz 12902	. . . . . . . . . 10 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ∈ ℤ)
2	1	adantl 485	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℤ)
3		peano2zm 12013	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
4	2, 3	syl 17	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 − 1) ∈ ℤ)
5	4	zcnd 12076	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 − 1) ∈ ℂ)
6	2	peano2zd 12078	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 + 1) ∈ ℤ)
7	6	zcnd 12076	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 + 1) ∈ ℂ)
8	5, 7	mulcomd 10651	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 − 1) · (𝑁 + 1)) = ((𝑁 + 1) · (𝑁 − 1)))
9	2	zcnd 12076	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℂ)
10		ax-1cn 10584	. . . . . . 7 ⊢ 1 ∈ ℂ
11		subsq 13568	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁↑2) − (1↑2)) = ((𝑁 + 1) · (𝑁 − 1)))
12	9, 10, 11	sylancl 589	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁↑2) − (1↑2)) = ((𝑁 + 1) · (𝑁 − 1)))
13	9	sqvald 13503	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁↑2) = (𝑁 · 𝑁))
14		sq1 13554	. . . . . . . 8 ⊢ (1↑2) = 1
15	14	a1i 11	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (1↑2) = 1)
16	13, 15	oveq12d 7153	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁↑2) − (1↑2)) = ((𝑁 · 𝑁) − 1))
17	8, 12, 16	3eqtr2d 2839	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 − 1) · (𝑁 + 1)) = ((𝑁 · 𝑁) − 1))
18	17	breq2d 5042	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ ((𝑁 − 1) · (𝑁 + 1)) ↔ 𝑃 ∥ ((𝑁 · 𝑁) − 1)))
19		fz1ssfz0 12998	. . . . . 6 ⊢ (1...(𝑃 − 1)) ⊆ (0...(𝑃 − 1))
20		simpr 488	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ (1...(𝑃 − 1)))
21	19, 20	sseldi 3913	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ (0...(𝑃 − 1)))
22	21	biantrurd 536	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ ((𝑁 · 𝑁) − 1) ↔ (𝑁 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝑁 · 𝑁) − 1))))
23	18, 22	bitrd 282	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ ((𝑁 − 1) · (𝑁 + 1)) ↔ (𝑁 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝑁 · 𝑁) − 1))))
24		simpl 486	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℙ)
25		euclemma 16047	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 − 1) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → (𝑃 ∥ ((𝑁 − 1) · (𝑁 + 1)) ↔ (𝑃 ∥ (𝑁 − 1) ∨ 𝑃 ∥ (𝑁 + 1))))
26	24, 4, 6, 25	syl3anc 1368	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ ((𝑁 − 1) · (𝑁 + 1)) ↔ (𝑃 ∥ (𝑁 − 1) ∨ 𝑃 ∥ (𝑁 + 1))))
27		prmnn 16008	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
28		fzm1ndvds 15664	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝑁)
29	27, 28	sylan 583	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝑁)
30		eqid 2798	. . . . 5 ⊢ ((𝑁↑(𝑃 − 2)) mod 𝑃) = ((𝑁↑(𝑃 − 2)) mod 𝑃)
31	30	prmdiveq 16113	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ ¬ 𝑃 ∥ 𝑁) → ((𝑁 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝑁 · 𝑁) − 1)) ↔ 𝑁 = ((𝑁↑(𝑃 − 2)) mod 𝑃)))
32	24, 2, 29, 31	syl3anc 1368	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝑁 · 𝑁) − 1)) ↔ 𝑁 = ((𝑁↑(𝑃 − 2)) mod 𝑃)))
33	23, 26, 32	3bitr3rd 313	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 = ((𝑁↑(𝑃 − 2)) mod 𝑃) ↔ (𝑃 ∥ (𝑁 − 1) ∨ 𝑃 ∥ (𝑁 + 1))))
34	24, 27	syl 17	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℕ)
35		1zzd 12001	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 1 ∈ ℤ)
36		moddvds 15610	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑁 mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ (𝑁 − 1)))
37	34, 2, 35, 36	syl3anc 1368	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ (𝑁 − 1)))
38		elfznn 12931	. . . . . . . 8 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ∈ ℕ)
39	38	adantl 485	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℕ)
40	39	nnred 11640	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℝ)
41	34	nnrpd 12417	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℝ⁺)
42	39	nnnn0d 11943	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℕ₀)
43	42	nn0ge0d 11946	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 0 ≤ 𝑁)
44		elfzle2 12906	. . . . . . . 8 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ≤ (𝑃 − 1))
45	44	adantl 485	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ≤ (𝑃 − 1))
46		prmz 16009	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
47		zltlem1 12023	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 < 𝑃 ↔ 𝑁 ≤ (𝑃 − 1)))
48	1, 46, 47	syl2anr 599	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 < 𝑃 ↔ 𝑁 ≤ (𝑃 − 1)))
49	45, 48	mpbird 260	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 < 𝑃)
50		modid 13259	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ 𝑁 ∧ 𝑁 < 𝑃)) → (𝑁 mod 𝑃) = 𝑁)
51	40, 41, 43, 49, 50	syl22anc 837	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 mod 𝑃) = 𝑁)
52	34	nnred 11640	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℝ)
53		prmuz2 16030	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
54	24, 53	syl 17	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ (ℤ_≥‘2))
55		eluz2gt1 12308	. . . . . . 7 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
56	54, 55	syl 17	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 1 < 𝑃)
57		1mod 13266	. . . . . 6 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
58	52, 56, 57	syl2anc 587	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (1 mod 𝑃) = 1)
59	51, 58	eqeq12d 2814	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 mod 𝑃) = (1 mod 𝑃) ↔ 𝑁 = 1))
60	37, 59	bitr3d 284	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ (𝑁 − 1) ↔ 𝑁 = 1))
61	35	znegcld 12077	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → -1 ∈ ℤ)
62		moddvds 15610	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ -1 ∈ ℤ) → ((𝑁 mod 𝑃) = (-1 mod 𝑃) ↔ 𝑃 ∥ (𝑁 − -1)))
63	34, 2, 61, 62	syl3anc 1368	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 mod 𝑃) = (-1 mod 𝑃) ↔ 𝑃 ∥ (𝑁 − -1)))
64	34	nncnd 11641	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℂ)
65	64	mulid2d 10648	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (1 · 𝑃) = 𝑃)
66	65	oveq2d 7151	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (-1 + (1 · 𝑃)) = (-1 + 𝑃))
67		neg1cn 11739	. . . . . . . . 9 ⊢ -1 ∈ ℂ
68		addcom 10815	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (-1 + 𝑃) = (𝑃 + -1))
69	67, 64, 68	sylancr 590	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (-1 + 𝑃) = (𝑃 + -1))
70		negsub 10923	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑃 + -1) = (𝑃 − 1))
71	64, 10, 70	sylancl 589	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 + -1) = (𝑃 − 1))
72	66, 69, 71	3eqtrd 2837	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (-1 + (1 · 𝑃)) = (𝑃 − 1))
73	72	oveq1d 7150	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((-1 + (1 · 𝑃)) mod 𝑃) = ((𝑃 − 1) mod 𝑃))
74		neg1rr 11740	. . . . . . . 8 ⊢ -1 ∈ ℝ
75	74	a1i 11	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → -1 ∈ ℝ)
76		modcyc 13269	. . . . . . 7 ⊢ ((-1 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺ ∧ 1 ∈ ℤ) → ((-1 + (1 · 𝑃)) mod 𝑃) = (-1 mod 𝑃))
77	75, 41, 35, 76	syl3anc 1368	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((-1 + (1 · 𝑃)) mod 𝑃) = (-1 mod 𝑃))
78		peano2rem 10942	. . . . . . . 8 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ)
79	52, 78	syl 17	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 1) ∈ ℝ)
80		nnm1nn0 11926	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
81	34, 80	syl 17	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 1) ∈ ℕ₀)
82	81	nn0ge0d 11946	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 0 ≤ (𝑃 − 1))
83	52	ltm1d 11561	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 1) < 𝑃)
84		modid 13259	. . . . . . 7 ⊢ ((((𝑃 − 1) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ (𝑃 − 1) ∧ (𝑃 − 1) < 𝑃)) → ((𝑃 − 1) mod 𝑃) = (𝑃 − 1))
85	79, 41, 82, 83, 84	syl22anc 837	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑃 − 1) mod 𝑃) = (𝑃 − 1))
86	73, 77, 85	3eqtr3d 2841	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (-1 mod 𝑃) = (𝑃 − 1))
87	51, 86	eqeq12d 2814	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 mod 𝑃) = (-1 mod 𝑃) ↔ 𝑁 = (𝑃 − 1)))
88		subneg 10924	. . . . . 6 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 − -1) = (𝑁 + 1))
89	9, 10, 88	sylancl 589	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 − -1) = (𝑁 + 1))
90	89	breq2d 5042	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ (𝑁 − -1) ↔ 𝑃 ∥ (𝑁 + 1)))
91	63, 87, 90	3bitr3rd 313	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ (𝑁 + 1) ↔ 𝑁 = (𝑃 − 1)))
92	60, 91	orbi12d 916	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑃 ∥ (𝑁 − 1) ∨ 𝑃 ∥ (𝑁 + 1)) ↔ (𝑁 = 1 ∨ 𝑁 = (𝑃 − 1))))
93	33, 92	bitrd 282	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 = ((𝑁↑(𝑃 − 2)) mod 𝑃) ↔ (𝑁 = 1 ∨ 𝑁 = (𝑃 − 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 -cneg 10860 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℤcz 11969 ℤ_≥cuz 12231 ℝ⁺crp 12377 ...cfz 12885 mod cmo 13232 ↑cexp 13425 ∥ cdvds 15599 ℙcprime 16005

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-prm 16006 df-phi 16093

This theorem is referenced by: wilthlem2 25654

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