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Theorem ostthlem1 27137

Description: Lemma for ostth 27149. If two absolute values agree on the positive integers greater than one, then they agree for all rational numbers and thus are equal as functions. (Contributed by Mario Carneiro, 9-Sep-2014.)

**Hypotheses**
Ref	Expression
qrng.q	⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
qabsabv.a	⊢ 𝐴 = (AbsVal‘𝑄)
ostthlem1.1	⊢ (𝜑 → 𝐹 ∈ 𝐴)
ostthlem1.2	⊢ (𝜑 → 𝐺 ∈ 𝐴)
ostthlem1.3	⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘2)) → (𝐹‘𝑛) = (𝐺‘𝑛))

**Assertion**
Ref	Expression
ostthlem1	⊢ (𝜑 → 𝐹 = 𝐺)

Distinct variable groups: 𝑛,𝐺 𝜑,𝑛 𝐴,𝑛 𝑄,𝑛 𝑛,𝐹

Proof of Theorem ostthlem1 Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ostthlem1.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
2		qabsabv.a	. . . 4 ⊢ 𝐴 = (AbsVal‘𝑄)
3		qrng.q	. . . . 5 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
4	3	qrngbas 27129	. . . 4 ⊢ ℚ = (Base‘𝑄)
5	2, 4	abvf 20435	. . 3 ⊢ (𝐹 ∈ 𝐴 → 𝐹:ℚ⟶ℝ)
6		ffn 6717	. . 3 ⊢ (𝐹:ℚ⟶ℝ → 𝐹 Fn ℚ)
7	1, 5, 6	3syl 18	. 2 ⊢ (𝜑 → 𝐹 Fn ℚ)
8		ostthlem1.2	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐴)
9	2, 4	abvf 20435	. . 3 ⊢ (𝐺 ∈ 𝐴 → 𝐺:ℚ⟶ℝ)
10		ffn 6717	. . 3 ⊢ (𝐺:ℚ⟶ℝ → 𝐺 Fn ℚ)
11	8, 9, 10	3syl 18	. 2 ⊢ (𝜑 → 𝐺 Fn ℚ)
12		elq 12936	. . . 4 ⊢ (𝑦 ∈ ℚ ↔ ∃𝑘 ∈ ℤ ∃𝑛 ∈ ℕ 𝑦 = (𝑘 / 𝑛))
13	3	qdrng 27130	. . . . . . . . . 10 ⊢ 𝑄 ∈ DivRing
14	13	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → 𝑄 ∈ DivRing)
15	1	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → 𝐹 ∈ 𝐴)
16		zq 12940	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℚ)
17	16	ad2antrl 726	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → 𝑘 ∈ ℚ)
18		nnq 12948	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℚ)
19	18	ad2antll 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → 𝑛 ∈ ℚ)
20		nnne0 12248	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
21	20	ad2antll 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → 𝑛 ≠ 0)
22	3	qrng0 27131	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝑄)
23		eqid 2732	. . . . . . . . . 10 ⊢ (/_r‘𝑄) = (/_r‘𝑄)
24	2, 4, 22, 23	abvdiv 20449	. . . . . . . . 9 ⊢ (((𝑄 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑘 ∈ ℚ ∧ 𝑛 ∈ ℚ ∧ 𝑛 ≠ 0)) → (𝐹‘(𝑘(/_r‘𝑄)𝑛)) = ((𝐹‘𝑘) / (𝐹‘𝑛)))
25	14, 15, 17, 19, 21, 24	syl23anc 1377	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → (𝐹‘(𝑘(/_r‘𝑄)𝑛)) = ((𝐹‘𝑘) / (𝐹‘𝑛)))
26	8	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → 𝐺 ∈ 𝐴)
27	2, 4, 22, 23	abvdiv 20449	. . . . . . . . . 10 ⊢ (((𝑄 ∈ DivRing ∧ 𝐺 ∈ 𝐴) ∧ (𝑘 ∈ ℚ ∧ 𝑛 ∈ ℚ ∧ 𝑛 ≠ 0)) → (𝐺‘(𝑘(/_r‘𝑄)𝑛)) = ((𝐺‘𝑘) / (𝐺‘𝑛)))
28	14, 26, 17, 19, 21, 27	syl23anc 1377	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → (𝐺‘(𝑘(/_r‘𝑄)𝑛)) = ((𝐺‘𝑘) / (𝐺‘𝑛)))
29	2, 22	abv0 20443	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘0) = 0)
30	1, 29	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹‘0) = 0)
31	2, 22	abv0 20443	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ 𝐴 → (𝐺‘0) = 0)
32	8, 31	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺‘0) = 0)
33	30, 32	eqtr4d 2775	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0))
34		fveq2 6891	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → (𝐹‘𝑘) = (𝐹‘0))
35		fveq2 6891	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → (𝐺‘𝑘) = (𝐺‘0))
36	34, 35	eqeq12d 2748	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘0) = (𝐺‘0)))
37	33, 36	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 = 0 → (𝐹‘𝑘) = (𝐺‘𝑘)))
38	37	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝑘 = 0 → (𝐹‘𝑘) = (𝐺‘𝑘)))
39	38	imp 407	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑘 = 0) → (𝐹‘𝑘) = (𝐺‘𝑘))
40		elnn1uz2 12911	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 = 1 ∨ 𝑛 ∈ (ℤ_≥‘2)))
41	3	qrng1 27132	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 = (1_r‘𝑄)
42	2, 41	abv1 20445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄 ∈ DivRing ∧ 𝐹 ∈ 𝐴) → (𝐹‘1) = 1)
43	13, 1, 42	sylancr 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹‘1) = 1)
44	2, 41	abv1 20445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄 ∈ DivRing ∧ 𝐺 ∈ 𝐴) → (𝐺‘1) = 1)
45	13, 8, 44	sylancr 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺‘1) = 1)
46	43, 45	eqtr4d 2775	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐹‘1) = (𝐺‘1))
47		fveq2 6891	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1))
48		fveq2 6891	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → (𝐺‘𝑛) = (𝐺‘1))
49	47, 48	eqeq12d 2748	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → ((𝐹‘𝑛) = (𝐺‘𝑛) ↔ (𝐹‘1) = (𝐺‘1)))
50	46, 49	syl5ibrcom 246	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑛 = 1 → (𝐹‘𝑛) = (𝐺‘𝑛)))
51	50	imp 407	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 = 1) → (𝐹‘𝑛) = (𝐺‘𝑛))
52		ostthlem1.3	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘2)) → (𝐹‘𝑛) = (𝐺‘𝑛))
53	51, 52	jaodan 956	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 = 1 ∨ 𝑛 ∈ (ℤ_≥‘2))) → (𝐹‘𝑛) = (𝐺‘𝑛))
54	40, 53	sylan2b 594	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = (𝐺‘𝑛))
55	54	ralrimiva 3146	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) = (𝐺‘𝑛))
56	55	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ∀𝑛 ∈ ℕ (𝐹‘𝑛) = (𝐺‘𝑛))
57		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
58		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
59	57, 58	eqeq12d 2748	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛) = (𝐺‘𝑛) ↔ (𝐹‘𝑘) = (𝐺‘𝑘)))
60	59	rspccva 3611	. . . . . . . . . . . . 13 ⊢ ((∀𝑛 ∈ ℕ (𝐹‘𝑛) = (𝐺‘𝑛) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (𝐺‘𝑘))
61	56, 60	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (𝐺‘𝑘))
62		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((inv_g‘𝑄)‘𝑘) → (𝐹‘𝑛) = (𝐹‘((inv_g‘𝑄)‘𝑘)))
63		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((inv_g‘𝑄)‘𝑘) → (𝐺‘𝑛) = (𝐺‘((inv_g‘𝑄)‘𝑘)))
64	62, 63	eqeq12d 2748	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ((inv_g‘𝑄)‘𝑘) → ((𝐹‘𝑛) = (𝐺‘𝑛) ↔ (𝐹‘((inv_g‘𝑄)‘𝑘)) = (𝐺‘((inv_g‘𝑄)‘𝑘))))
65	55	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ -𝑘 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝐹‘𝑛) = (𝐺‘𝑛))
66	16	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℚ)
67	3	qrngneg 27133	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℚ → ((inv_g‘𝑄)‘𝑘) = -𝑘)
68	66, 67	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((inv_g‘𝑄)‘𝑘) = -𝑘)
69	68	eleq1d 2818	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (((inv_g‘𝑄)‘𝑘) ∈ ℕ ↔ -𝑘 ∈ ℕ))
70	69	biimpar 478	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ -𝑘 ∈ ℕ) → ((inv_g‘𝑄)‘𝑘) ∈ ℕ)
71	64, 65, 70	rspcdva 3613	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ -𝑘 ∈ ℕ) → (𝐹‘((inv_g‘𝑄)‘𝑘)) = (𝐺‘((inv_g‘𝑄)‘𝑘)))
72	1	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ -𝑘 ∈ ℕ) → 𝐹 ∈ 𝐴)
73	16	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ -𝑘 ∈ ℕ) → 𝑘 ∈ ℚ)
74		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (inv_g‘𝑄) = (inv_g‘𝑄)
75	2, 4, 74	abvneg 20446	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑘 ∈ ℚ) → (𝐹‘((inv_g‘𝑄)‘𝑘)) = (𝐹‘𝑘))
76	72, 73, 75	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ -𝑘 ∈ ℕ) → (𝐹‘((inv_g‘𝑄)‘𝑘)) = (𝐹‘𝑘))
77	8	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ -𝑘 ∈ ℕ) → 𝐺 ∈ 𝐴)
78	2, 4, 74	abvneg 20446	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑘 ∈ ℚ) → (𝐺‘((inv_g‘𝑄)‘𝑘)) = (𝐺‘𝑘))
79	77, 73, 78	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ -𝑘 ∈ ℕ) → (𝐺‘((inv_g‘𝑄)‘𝑘)) = (𝐺‘𝑘))
80	71, 76, 79	3eqtr3d 2780	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ -𝑘 ∈ ℕ) → (𝐹‘𝑘) = (𝐺‘𝑘))
81		elz 12562	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↔ (𝑘 ∈ ℝ ∧ (𝑘 = 0 ∨ 𝑘 ∈ ℕ ∨ -𝑘 ∈ ℕ)))
82	81	simprbi 497	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤ → (𝑘 = 0 ∨ 𝑘 ∈ ℕ ∨ -𝑘 ∈ ℕ))
83	82	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝑘 = 0 ∨ 𝑘 ∈ ℕ ∨ -𝑘 ∈ ℕ))
84	39, 61, 80, 83	mpjao3dan 1431	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐺‘𝑘))
85	84	adantrr 715	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → (𝐹‘𝑘) = (𝐺‘𝑘))
86	54	adantrl 714	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → (𝐹‘𝑛) = (𝐺‘𝑛))
87	85, 86	oveq12d 7429	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → ((𝐹‘𝑘) / (𝐹‘𝑛)) = ((𝐺‘𝑘) / (𝐺‘𝑛)))
88	28, 87	eqtr4d 2775	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → (𝐺‘(𝑘(/_r‘𝑄)𝑛)) = ((𝐹‘𝑘) / (𝐹‘𝑛)))
89	25, 88	eqtr4d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → (𝐹‘(𝑘(/_r‘𝑄)𝑛)) = (𝐺‘(𝑘(/_r‘𝑄)𝑛)))
90	3	qrngdiv 27134	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℚ ∧ 𝑛 ∈ ℚ ∧ 𝑛 ≠ 0) → (𝑘(/_r‘𝑄)𝑛) = (𝑘 / 𝑛))
91	17, 19, 21, 90	syl3anc 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → (𝑘(/_r‘𝑄)𝑛) = (𝑘 / 𝑛))
92	91	fveq2d 6895	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → (𝐹‘(𝑘(/_r‘𝑄)𝑛)) = (𝐹‘(𝑘 / 𝑛)))
93	91	fveq2d 6895	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → (𝐺‘(𝑘(/_r‘𝑄)𝑛)) = (𝐺‘(𝑘 / 𝑛)))
94	89, 92, 93	3eqtr3d 2780	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → (𝐹‘(𝑘 / 𝑛)) = (𝐺‘(𝑘 / 𝑛)))
95		fveq2 6891	. . . . . . 7 ⊢ (𝑦 = (𝑘 / 𝑛) → (𝐹‘𝑦) = (𝐹‘(𝑘 / 𝑛)))
96		fveq2 6891	. . . . . . 7 ⊢ (𝑦 = (𝑘 / 𝑛) → (𝐺‘𝑦) = (𝐺‘(𝑘 / 𝑛)))
97	95, 96	eqeq12d 2748	. . . . . 6 ⊢ (𝑦 = (𝑘 / 𝑛) → ((𝐹‘𝑦) = (𝐺‘𝑦) ↔ (𝐹‘(𝑘 / 𝑛)) = (𝐺‘(𝑘 / 𝑛))))
98	94, 97	syl5ibrcom 246	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ)) → (𝑦 = (𝑘 / 𝑛) → (𝐹‘𝑦) = (𝐺‘𝑦)))
99	98	rexlimdvva 3211	. . . 4 ⊢ (𝜑 → (∃𝑘 ∈ ℤ ∃𝑛 ∈ ℕ 𝑦 = (𝑘 / 𝑛) → (𝐹‘𝑦) = (𝐺‘𝑦)))
100	12, 99	biimtrid 241	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℚ → (𝐹‘𝑦) = (𝐺‘𝑦)))
101	100	imp 407	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℚ) → (𝐹‘𝑦) = (𝐺‘𝑦))
102	7, 11, 101	eqfnfvd 7035	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ℝcr 11111 0cc0 11112 1c1 11113 -cneg 11447 / cdiv 11873 ℕcn 12214 2c2 12269 ℤcz 12560 ℤ_≥cuz 12824 ℚcq 12934 ↾_s cress 17175 inv_gcminusg 18822 /_rcdvr 20218 DivRingcdr 20361 AbsValcabv 20428 ℂ_fldccnfld 20950

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-ico 13332 df-fz 13487 df-seq 13969 df-exp 14030 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-0g 17389 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-minusg 18825 df-subg 19005 df-cmn 19652 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-dvr 20219 df-subrg 20321 df-drng 20363 df-abv 20429 df-cnfld 20951

This theorem is referenced by: ostthlem2 27138 ostth2 27147

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