wilthlem1 - Metamath Proof Explorer

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Theorem wilthlem1 27008

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 16701, (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 13428	. . . . . . . . . 10 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ∈ ℤ)
2	1	adantl 481	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℤ)
3		peano2zm 12523	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
4	2, 3	syl 17	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 − 1) ∈ ℤ)
5	4	zcnd 12586	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 − 1) ∈ ℂ)
6	2	peano2zd 12588	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 + 1) ∈ ℤ)
7	6	zcnd 12586	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 + 1) ∈ ℂ)
8	5, 7	mulcomd 11142	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 − 1) · (𝑁 + 1)) = ((𝑁 + 1) · (𝑁 − 1)))
9	2	zcnd 12586	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℂ)
10		ax-1cn 11073	. . . . . . 7 ⊢ 1 ∈ ℂ
11		subsq 14121	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁↑2) − (1↑2)) = ((𝑁 + 1) · (𝑁 − 1)))
12	9, 10, 11	sylancl 586	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁↑2) − (1↑2)) = ((𝑁 + 1) · (𝑁 − 1)))
13	9	sqvald 14054	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁↑2) = (𝑁 · 𝑁))
14		sq1 14106	. . . . . . . 8 ⊢ (1↑2) = 1
15	14	a1i 11	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (1↑2) = 1)
16	13, 15	oveq12d 7372	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁↑2) − (1↑2)) = ((𝑁 · 𝑁) − 1))
17	8, 12, 16	3eqtr2d 2774	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 − 1) · (𝑁 + 1)) = ((𝑁 · 𝑁) − 1))
18	17	breq2d 5107	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ ((𝑁 − 1) · (𝑁 + 1)) ↔ 𝑃 ∥ ((𝑁 · 𝑁) − 1)))
19		fz1ssfz0 13527	. . . . . 6 ⊢ (1...(𝑃 − 1)) ⊆ (0...(𝑃 − 1))
20		simpr 484	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ (1...(𝑃 − 1)))
21	19, 20	sselid 3928	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ (0...(𝑃 − 1)))
22	21	biantrurd 532	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ ((𝑁 · 𝑁) − 1) ↔ (𝑁 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝑁 · 𝑁) − 1))))
23	18, 22	bitrd 279	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ ((𝑁 − 1) · (𝑁 + 1)) ↔ (𝑁 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝑁 · 𝑁) − 1))))
24		simpl 482	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℙ)
25		euclemma 16628	. . . 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 1373	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ ((𝑁 − 1) · (𝑁 + 1)) ↔ (𝑃 ∥ (𝑁 − 1) ∨ 𝑃 ∥ (𝑁 + 1))))
27		prmnn 16589	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
28		fzm1ndvds 16237	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝑁)
29	27, 28	sylan 580	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝑁)
30		eqid 2733	. . . . 5 ⊢ ((𝑁↑(𝑃 − 2)) mod 𝑃) = ((𝑁↑(𝑃 − 2)) mod 𝑃)
31	30	prmdiveq 16701	. . . 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 1373	. . 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 310	. 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 12511	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 1 ∈ ℤ)
36		moddvds 16178	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑁 mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ (𝑁 − 1)))
37	34, 2, 35, 36	syl3anc 1373	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ (𝑁 − 1)))
38		elfznn 13457	. . . . . . . 8 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ∈ ℕ)
39	38	adantl 481	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℕ)
40	39	nnred 12149	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℝ)
41	34	nnrpd 12936	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℝ⁺)
42	39	nnnn0d 12451	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℕ₀)
43	42	nn0ge0d 12454	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 0 ≤ 𝑁)
44		elfzle2 13432	. . . . . . . 8 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ≤ (𝑃 − 1))
45	44	adantl 481	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ≤ (𝑃 − 1))
46		prmz 16590	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
47		zltlem1 12533	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 < 𝑃 ↔ 𝑁 ≤ (𝑃 − 1)))
48	1, 46, 47	syl2anr 597	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 < 𝑃 ↔ 𝑁 ≤ (𝑃 − 1)))
49	45, 48	mpbird 257	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 < 𝑃)
50		modid 13804	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ 𝑁 ∧ 𝑁 < 𝑃)) → (𝑁 mod 𝑃) = 𝑁)
51	40, 41, 43, 49, 50	syl22anc 838	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 mod 𝑃) = 𝑁)
52	34	nnred 12149	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℝ)
53		prmuz2 16611	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
54	24, 53	syl 17	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ (ℤ_≥‘2))
55		eluz2gt1 12822	. . . . . . 7 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
56	54, 55	syl 17	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 1 < 𝑃)
57		1mod 13811	. . . . . 6 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
58	52, 56, 57	syl2anc 584	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (1 mod 𝑃) = 1)
59	51, 58	eqeq12d 2749	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 mod 𝑃) = (1 mod 𝑃) ↔ 𝑁 = 1))
60	37, 59	bitr3d 281	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ (𝑁 − 1) ↔ 𝑁 = 1))
61	35	znegcld 12587	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → -1 ∈ ℤ)
62		moddvds 16178	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ -1 ∈ ℤ) → ((𝑁 mod 𝑃) = (-1 mod 𝑃) ↔ 𝑃 ∥ (𝑁 − -1)))
63	34, 2, 61, 62	syl3anc 1373	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 mod 𝑃) = (-1 mod 𝑃) ↔ 𝑃 ∥ (𝑁 − -1)))
64	34	nncnd 12150	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℂ)
65	64	mullidd 11139	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (1 · 𝑃) = 𝑃)
66	65	oveq2d 7370	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (-1 + (1 · 𝑃)) = (-1 + 𝑃))
67		neg1cn 12119	. . . . . . . . 9 ⊢ -1 ∈ ℂ
68		addcom 11308	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (-1 + 𝑃) = (𝑃 + -1))
69	67, 64, 68	sylancr 587	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (-1 + 𝑃) = (𝑃 + -1))
70		negsub 11418	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑃 + -1) = (𝑃 − 1))
71	64, 10, 70	sylancl 586	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 + -1) = (𝑃 − 1))
72	66, 69, 71	3eqtrd 2772	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (-1 + (1 · 𝑃)) = (𝑃 − 1))
73	72	oveq1d 7369	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((-1 + (1 · 𝑃)) mod 𝑃) = ((𝑃 − 1) mod 𝑃))
74		neg1rr 12120	. . . . . . . 8 ⊢ -1 ∈ ℝ
75	74	a1i 11	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → -1 ∈ ℝ)
76		modcyc 13814	. . . . . . 7 ⊢ ((-1 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺ ∧ 1 ∈ ℤ) → ((-1 + (1 · 𝑃)) mod 𝑃) = (-1 mod 𝑃))
77	75, 41, 35, 76	syl3anc 1373	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((-1 + (1 · 𝑃)) mod 𝑃) = (-1 mod 𝑃))
78		peano2rem 11437	. . . . . . . 8 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ)
79	52, 78	syl 17	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 1) ∈ ℝ)
80		nnm1nn0 12431	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
81	34, 80	syl 17	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 1) ∈ ℕ₀)
82	81	nn0ge0d 12454	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 0 ≤ (𝑃 − 1))
83	52	ltm1d 12063	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 1) < 𝑃)
84		modid 13804	. . . . . . 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 838	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑃 − 1) mod 𝑃) = (𝑃 − 1))
86	73, 77, 85	3eqtr3d 2776	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (-1 mod 𝑃) = (𝑃 − 1))
87	51, 86	eqeq12d 2749	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 mod 𝑃) = (-1 mod 𝑃) ↔ 𝑁 = (𝑃 − 1)))
88		subneg 11419	. . . . . 6 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 − -1) = (𝑁 + 1))
89	9, 10, 88	sylancl 586	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 − -1) = (𝑁 + 1))
90	89	breq2d 5107	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ (𝑁 − -1) ↔ 𝑃 ∥ (𝑁 + 1)))
91	63, 87, 90	3bitr3rd 310	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ (𝑁 + 1) ↔ 𝑁 = (𝑃 − 1)))
92	60, 91	orbi12d 918	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑃 ∥ (𝑁 − 1) ∨ 𝑃 ∥ (𝑁 + 1)) ↔ (𝑁 = 1 ∨ 𝑁 = (𝑃 − 1))))
93	33, 92	bitrd 279	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 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 class class class wbr 5095 ‘cfv 6488 (class class class)co 7354 ℂcc 11013 ℝcr 11014 0cc0 11015 1c1 11016 + caddc 11018 · cmul 11020 < clt 11155 ≤ cle 11156 − cmin 11353 -cneg 11354 ℕcn 12134 2c2 12189 ℕ₀cn0 12390 ℤcz 12477 ℤ_≥cuz 12740 ℝ⁺crp 12894 ...cfz 13411 mod cmo 13777 ↑cexp 13972 ∥ cdvds 16167 ℙcprime 16586

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092 ax-pre-sup 11093

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7805 df-1st 7929 df-2nd 7930 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-2o 8394 df-oadd 8397 df-er 8630 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-sup 9335 df-inf 9336 df-dju 9803 df-card 9841 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-div 11784 df-nn 12135 df-2 12197 df-3 12198 df-n0 12391 df-xnn0 12464 df-z 12478 df-uz 12741 df-rp 12895 df-fz 13412 df-fzo 13559 df-fl 13700 df-mod 13778 df-seq 13913 df-exp 13973 df-hash 14242 df-cj 15010 df-re 15011 df-im 15012 df-sqrt 15146 df-abs 15147 df-dvds 16168 df-gcd 16410 df-prm 16587 df-phi 16681

This theorem is referenced by: wilthlem2 27009

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