wilthlem1 - Metamath Proof Explorer

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Theorem wilthlem1 25645

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 16123, (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 12909	. . . . . . . . . 10 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ∈ ℤ)
2	1	adantl 484	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℤ)
3		peano2zm 12026	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
4	2, 3	syl 17	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 − 1) ∈ ℤ)
5	4	zcnd 12089	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 − 1) ∈ ℂ)
6	2	peano2zd 12091	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 + 1) ∈ ℤ)
7	6	zcnd 12089	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 + 1) ∈ ℂ)
8	5, 7	mulcomd 10662	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 − 1) · (𝑁 + 1)) = ((𝑁 + 1) · (𝑁 − 1)))
9	2	zcnd 12089	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℂ)
10		ax-1cn 10595	. . . . . . 7 ⊢ 1 ∈ ℂ
11		subsq 13573	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁↑2) − (1↑2)) = ((𝑁 + 1) · (𝑁 − 1)))
12	9, 10, 11	sylancl 588	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁↑2) − (1↑2)) = ((𝑁 + 1) · (𝑁 − 1)))
13	9	sqvald 13508	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁↑2) = (𝑁 · 𝑁))
14		sq1 13559	. . . . . . . 8 ⊢ (1↑2) = 1
15	14	a1i 11	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (1↑2) = 1)
16	13, 15	oveq12d 7174	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁↑2) − (1↑2)) = ((𝑁 · 𝑁) − 1))
17	8, 12, 16	3eqtr2d 2862	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 − 1) · (𝑁 + 1)) = ((𝑁 · 𝑁) − 1))
18	17	breq2d 5078	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ ((𝑁 − 1) · (𝑁 + 1)) ↔ 𝑃 ∥ ((𝑁 · 𝑁) − 1)))
19		fz1ssfz0 13004	. . . . . 6 ⊢ (1...(𝑃 − 1)) ⊆ (0...(𝑃 − 1))
20		simpr 487	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ (1...(𝑃 − 1)))
21	19, 20	sseldi 3965	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ (0...(𝑃 − 1)))
22	21	biantrurd 535	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ ((𝑁 · 𝑁) − 1) ↔ (𝑁 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝑁 · 𝑁) − 1))))
23	18, 22	bitrd 281	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ ((𝑁 − 1) · (𝑁 + 1)) ↔ (𝑁 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝑁 · 𝑁) − 1))))
24		simpl 485	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℙ)
25		euclemma 16057	. . . 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 1367	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ ((𝑁 − 1) · (𝑁 + 1)) ↔ (𝑃 ∥ (𝑁 − 1) ∨ 𝑃 ∥ (𝑁 + 1))))
27		prmnn 16018	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
28		fzm1ndvds 15672	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝑁)
29	27, 28	sylan 582	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝑁)
30		eqid 2821	. . . . 5 ⊢ ((𝑁↑(𝑃 − 2)) mod 𝑃) = ((𝑁↑(𝑃 − 2)) mod 𝑃)
31	30	prmdiveq 16123	. . . 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 1367	. . 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 312	. 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 12014	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 1 ∈ ℤ)
36		moddvds 15618	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑁 mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ (𝑁 − 1)))
37	34, 2, 35, 36	syl3anc 1367	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ (𝑁 − 1)))
38		elfznn 12937	. . . . . . . 8 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ∈ ℕ)
39	38	adantl 484	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℕ)
40	39	nnred 11653	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℝ)
41	34	nnrpd 12430	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℝ⁺)
42	39	nnnn0d 11956	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℕ₀)
43	42	nn0ge0d 11959	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 0 ≤ 𝑁)
44		elfzle2 12912	. . . . . . . 8 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ≤ (𝑃 − 1))
45	44	adantl 484	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ≤ (𝑃 − 1))
46		prmz 16019	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
47		zltlem1 12036	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 < 𝑃 ↔ 𝑁 ≤ (𝑃 − 1)))
48	1, 46, 47	syl2anr 598	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 < 𝑃 ↔ 𝑁 ≤ (𝑃 − 1)))
49	45, 48	mpbird 259	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 < 𝑃)
50		modid 13265	. . . . . 6 ⊢ (((𝑁 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ 𝑁 ∧ 𝑁 < 𝑃)) → (𝑁 mod 𝑃) = 𝑁)
51	40, 41, 43, 49, 50	syl22anc 836	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 mod 𝑃) = 𝑁)
52	34	nnred 11653	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℝ)
53		prmuz2 16040	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
54	24, 53	syl 17	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ (ℤ_≥‘2))
55		eluz2gt1 12321	. . . . . . 7 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 1 < 𝑃)
56	54, 55	syl 17	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 1 < 𝑃)
57		1mod 13272	. . . . . 6 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
58	52, 56, 57	syl2anc 586	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (1 mod 𝑃) = 1)
59	51, 58	eqeq12d 2837	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 mod 𝑃) = (1 mod 𝑃) ↔ 𝑁 = 1))
60	37, 59	bitr3d 283	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ (𝑁 − 1) ↔ 𝑁 = 1))
61	35	znegcld 12090	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → -1 ∈ ℤ)
62		moddvds 15618	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ -1 ∈ ℤ) → ((𝑁 mod 𝑃) = (-1 mod 𝑃) ↔ 𝑃 ∥ (𝑁 − -1)))
63	34, 2, 61, 62	syl3anc 1367	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 mod 𝑃) = (-1 mod 𝑃) ↔ 𝑃 ∥ (𝑁 − -1)))
64	34	nncnd 11654	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℂ)
65	64	mulid2d 10659	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (1 · 𝑃) = 𝑃)
66	65	oveq2d 7172	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (-1 + (1 · 𝑃)) = (-1 + 𝑃))
67		neg1cn 11752	. . . . . . . . 9 ⊢ -1 ∈ ℂ
68		addcom 10826	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (-1 + 𝑃) = (𝑃 + -1))
69	67, 64, 68	sylancr 589	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (-1 + 𝑃) = (𝑃 + -1))
70		negsub 10934	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑃 + -1) = (𝑃 − 1))
71	64, 10, 70	sylancl 588	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 + -1) = (𝑃 − 1))
72	66, 69, 71	3eqtrd 2860	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (-1 + (1 · 𝑃)) = (𝑃 − 1))
73	72	oveq1d 7171	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((-1 + (1 · 𝑃)) mod 𝑃) = ((𝑃 − 1) mod 𝑃))
74		neg1rr 11753	. . . . . . . 8 ⊢ -1 ∈ ℝ
75	74	a1i 11	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → -1 ∈ ℝ)
76		modcyc 13275	. . . . . . 7 ⊢ ((-1 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺ ∧ 1 ∈ ℤ) → ((-1 + (1 · 𝑃)) mod 𝑃) = (-1 mod 𝑃))
77	75, 41, 35, 76	syl3anc 1367	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((-1 + (1 · 𝑃)) mod 𝑃) = (-1 mod 𝑃))
78		peano2rem 10953	. . . . . . . 8 ⊢ (𝑃 ∈ ℝ → (𝑃 − 1) ∈ ℝ)
79	52, 78	syl 17	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 1) ∈ ℝ)
80		nnm1nn0 11939	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
81	34, 80	syl 17	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 1) ∈ ℕ₀)
82	81	nn0ge0d 11959	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 0 ≤ (𝑃 − 1))
83	52	ltm1d 11572	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 1) < 𝑃)
84		modid 13265	. . . . . . 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 836	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑃 − 1) mod 𝑃) = (𝑃 − 1))
86	73, 77, 85	3eqtr3d 2864	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (-1 mod 𝑃) = (𝑃 − 1))
87	51, 86	eqeq12d 2837	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑁 mod 𝑃) = (-1 mod 𝑃) ↔ 𝑁 = (𝑃 − 1)))
88		subneg 10935	. . . . . 6 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 − -1) = (𝑁 + 1))
89	9, 10, 88	sylancl 588	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 − -1) = (𝑁 + 1))
90	89	breq2d 5078	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ (𝑁 − -1) ↔ 𝑃 ∥ (𝑁 + 1)))
91	63, 87, 90	3bitr3rd 312	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 ∥ (𝑁 + 1) ↔ 𝑁 = (𝑃 − 1)))
92	60, 91	orbi12d 915	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((𝑃 ∥ (𝑁 − 1) ∨ 𝑃 ∥ (𝑁 + 1)) ↔ (𝑁 = 1 ∨ 𝑁 = (𝑃 − 1))))
93	33, 92	bitrd 281	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 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 < clt 10675 ≤ cle 10676 − cmin 10870 -cneg 10871 ℕcn 11638 2c2 11693 ℕ₀cn0 11898 ℤcz 11982 ℤ_≥cuz 12244 ℝ⁺crp 12390 ...cfz 12893 mod cmo 13238 ↑cexp 13430 ∥ cdvds 15607 ℙcprime 16015

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-gcd 15844 df-prm 16016 df-phi 16103

This theorem is referenced by: wilthlem2 25646

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