cnreeu - Mathbox for Steven Nguyen

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Theorem cnreeu 43191

Description: The reals in the expression given by cnre 11207 uniquely define a complex number. (Contributed by SN, 27-Jun-2024.)

**Hypotheses**
Ref	Expression
cnreeu.r	⊢ (𝜑 → 𝑟 ∈ ℝ)
cnreeu.s	⊢ (𝜑 → 𝑠 ∈ ℝ)
cnreeu.t	⊢ (𝜑 → 𝑡 ∈ ℝ)
cnreeu.u	⊢ (𝜑 → 𝑢 ∈ ℝ)

**Assertion**
Ref	Expression
cnreeu	⊢ (𝜑 → ((𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)) ↔ (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))

**Proof of Theorem cnreeu**
Step	Hyp	Ref	Expression
1		oveq1 7420	. . . 4 ⊢ ((𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)) → ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠))) = ((𝑡 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))))
2	1	oveq2d 7429	. . 3 ⊢ ((𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)) → ((0 −_ℝ 𝑡) + ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠)))) = ((0 −_ℝ 𝑡) + ((𝑡 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠)))))
3		cnreeu.r	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑟 ∈ ℝ)
4	3	recnd 11239	. . . . . . . . . 10 ⊢ (𝜑 → 𝑟 ∈ ℂ)
5		ax-icn 11161	. . . . . . . . . . . 12 ⊢ i ∈ ℂ
6	5	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → i ∈ ℂ)
7		cnreeu.s	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑠 ∈ ℝ)
8	7	recnd 11239	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑠 ∈ ℂ)
9	6, 8	mulcld 11231	. . . . . . . . . 10 ⊢ (𝜑 → (i · 𝑠) ∈ ℂ)
10		rernegcl 43059	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → (0 −_ℝ 𝑠) ∈ ℝ)
11	7, 10	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 −_ℝ 𝑠) ∈ ℝ)
12	11	recnd 11239	. . . . . . . . . . 11 ⊢ (𝜑 → (0 −_ℝ 𝑠) ∈ ℂ)
13	6, 12	mulcld 11231	. . . . . . . . . 10 ⊢ (𝜑 → (i · (0 −_ℝ 𝑠)) ∈ ℂ)
14	4, 9, 13	addassd 11233	. . . . . . . . 9 ⊢ (𝜑 → ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠))) = (𝑟 + ((i · 𝑠) + (i · (0 −_ℝ 𝑠)))))
15		renegid 43061	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → (𝑠 + (0 −_ℝ 𝑠)) = 0)
16	7, 15	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑠 + (0 −_ℝ 𝑠)) = 0)
17	16	oveq2d 7429	. . . . . . . . . . 11 ⊢ (𝜑 → (i · (𝑠 + (0 −_ℝ 𝑠))) = (i · 0))
18	6, 8, 12	adddid 11235	. . . . . . . . . . 11 ⊢ (𝜑 → (i · (𝑠 + (0 −_ℝ 𝑠))) = ((i · 𝑠) + (i · (0 −_ℝ 𝑠))))
19		sn-it0e0 43104	. . . . . . . . . . . 12 ⊢ (i · 0) = 0
20	19	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (i · 0) = 0)
21	17, 18, 20	3eqtr3d 2812	. . . . . . . . . 10 ⊢ (𝜑 → ((i · 𝑠) + (i · (0 −_ℝ 𝑠))) = 0)
22	21	oveq2d 7429	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 + ((i · 𝑠) + (i · (0 −_ℝ 𝑠)))) = (𝑟 + 0))
23		readdrid 43098	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ → (𝑟 + 0) = 𝑟)
24	3, 23	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 + 0) = 𝑟)
25	14, 22, 24	3eqtrd 2808	. . . . . . . 8 ⊢ (𝜑 → ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠))) = 𝑟)
26	25	oveq2d 7429	. . . . . . 7 ⊢ (𝜑 → ((0 −_ℝ 𝑡) + ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠)))) = ((0 −_ℝ 𝑡) + 𝑟))
27		cnreeu.t	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑡 ∈ ℝ)
28		rernegcl 43059	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → (0 −_ℝ 𝑡) ∈ ℝ)
29	27, 28	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (0 −_ℝ 𝑡) ∈ ℝ)
30	29	recnd 11239	. . . . . . . . . 10 ⊢ (𝜑 → (0 −_ℝ 𝑡) ∈ ℂ)
31	27	recnd 11239	. . . . . . . . . 10 ⊢ (𝜑 → 𝑡 ∈ ℂ)
32		cnreeu.u	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑢 ∈ ℝ)
33	32	recnd 11239	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑢 ∈ ℂ)
34	6, 33	mulcld 11231	. . . . . . . . . 10 ⊢ (𝜑 → (i · 𝑢) ∈ ℂ)
35	30, 31, 34	addassd 11233	. . . . . . . . 9 ⊢ (𝜑 → (((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) = ((0 −_ℝ 𝑡) + (𝑡 + (i · 𝑢))))
36	35	oveq1d 7428	. . . . . . . 8 ⊢ (𝜑 → ((((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))) = (((0 −_ℝ 𝑡) + (𝑡 + (i · 𝑢))) + (i · (0 −_ℝ 𝑠))))
37		sn-addlid 43092	. . . . . . . . . . 11 ⊢ ((i · 𝑢) ∈ ℂ → (0 + (i · 𝑢)) = (i · 𝑢))
38	34, 37	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0 + (i · 𝑢)) = (i · 𝑢))
39	38	oveq1d 7428	. . . . . . . . 9 ⊢ (𝜑 → ((0 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))) = ((i · 𝑢) + (i · (0 −_ℝ 𝑠))))
40		renegid2 43102	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → ((0 −_ℝ 𝑡) + 𝑡) = 0)
41	27, 40	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((0 −_ℝ 𝑡) + 𝑡) = 0)
42	41	oveq1d 7428	. . . . . . . . . 10 ⊢ (𝜑 → (((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) = (0 + (i · 𝑢)))
43	42	oveq1d 7428	. . . . . . . . 9 ⊢ (𝜑 → ((((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))) = ((0 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))))
44	6, 33, 12	adddid 11235	. . . . . . . . 9 ⊢ (𝜑 → (i · (𝑢 + (0 −_ℝ 𝑠))) = ((i · 𝑢) + (i · (0 −_ℝ 𝑠))))
45	39, 43, 44	3eqtr4d 2814	. . . . . . . 8 ⊢ (𝜑 → ((((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))) = (i · (𝑢 + (0 −_ℝ 𝑠))))
46	31, 34	addcld 11230	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 + (i · 𝑢)) ∈ ℂ)
47	30, 46, 13	addassd 11233	. . . . . . . 8 ⊢ (𝜑 → (((0 −_ℝ 𝑡) + (𝑡 + (i · 𝑢))) + (i · (0 −_ℝ 𝑠))) = ((0 −_ℝ 𝑡) + ((𝑡 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠)))))
48	36, 45, 47	3eqtr3rd 2813	. . . . . . 7 ⊢ (𝜑 → ((0 −_ℝ 𝑡) + ((𝑡 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠)))) = (i · (𝑢 + (0 −_ℝ 𝑠))))
49	26, 48	eqeq12d 2785	. . . . . 6 ⊢ (𝜑 → (((0 −_ℝ 𝑡) + ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠)))) = ((0 −_ℝ 𝑡) + ((𝑡 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠)))) ↔ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))))
50	49	biimpa 481	. . . . 5 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠)))) = ((0 −_ℝ 𝑡) + ((𝑡 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))))) → ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠))))
51		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠))))
52	32	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → 𝑢 ∈ ℝ)
53	11	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (0 −_ℝ 𝑠) ∈ ℝ)
54	52, 53	readdcld 11240	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (𝑢 + (0 −_ℝ 𝑠)) ∈ ℝ)
55	29	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (0 −_ℝ 𝑡) ∈ ℝ)
56	3	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → 𝑟 ∈ ℝ)
57	55, 56	readdcld 11240	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → ((0 −_ℝ 𝑡) + 𝑟) ∈ ℝ)
58	51, 57	eqeltrrd 2870	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (i · (𝑢 + (0 −_ℝ 𝑠))) ∈ ℝ)
59		sn-itrere 43189	. . . . . . . . . . 11 ⊢ ((𝑢 + (0 −_ℝ 𝑠)) ∈ ℝ → ((i · (𝑢 + (0 −_ℝ 𝑠))) ∈ ℝ ↔ (𝑢 + (0 −_ℝ 𝑠)) = 0))
60	59	biimpa 481	. . . . . . . . . 10 ⊢ (((𝑢 + (0 −_ℝ 𝑠)) ∈ ℝ ∧ (i · (𝑢 + (0 −_ℝ 𝑠))) ∈ ℝ) → (𝑢 + (0 −_ℝ 𝑠)) = 0)
61	54, 58, 60	syl2anc 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (𝑢 + (0 −_ℝ 𝑠)) = 0)
62	61	oveq2d 7429	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (i · (𝑢 + (0 −_ℝ 𝑠))) = (i · 0))
63	19	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (i · 0) = 0)
64	51, 62, 63	3eqtrd 2808	. . . . . . 7 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → ((0 −_ℝ 𝑡) + 𝑟) = 0)
65		oveq2 7421	. . . . . . . . 9 ⊢ (((0 −_ℝ 𝑡) + 𝑟) = 0 → (𝑡 + ((0 −_ℝ 𝑡) + 𝑟)) = (𝑡 + 0))
66	65	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (𝑡 + ((0 −_ℝ 𝑡) + 𝑟)) = (𝑡 + 0))
67		renegid 43061	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → (𝑡 + (0 −_ℝ 𝑡)) = 0)
68	27, 67	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 + (0 −_ℝ 𝑡)) = 0)
69	68	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (𝑡 + (0 −_ℝ 𝑡)) = 0)
70	69	oveq1d 7428	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → ((𝑡 + (0 −_ℝ 𝑡)) + 𝑟) = (0 + 𝑟))
71	31	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → 𝑡 ∈ ℂ)
72	30	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (0 −_ℝ 𝑡) ∈ ℂ)
73	4	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → 𝑟 ∈ ℂ)
74	71, 72, 73	addassd 11233	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → ((𝑡 + (0 −_ℝ 𝑡)) + 𝑟) = (𝑡 + ((0 −_ℝ 𝑡) + 𝑟)))
75		readdlid 43091	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ → (0 + 𝑟) = 𝑟)
76	3, 75	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0 + 𝑟) = 𝑟)
77	76	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (0 + 𝑟) = 𝑟)
78	70, 74, 77	3eqtr3d 2812	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (𝑡 + ((0 −_ℝ 𝑡) + 𝑟)) = 𝑟)
79	27	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → 𝑡 ∈ ℝ)
80		readdrid 43098	. . . . . . . . 9 ⊢ (𝑡 ∈ ℝ → (𝑡 + 0) = 𝑡)
81	79, 80	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (𝑡 + 0) = 𝑡)
82	66, 78, 81	3eqtr3d 2812	. . . . . . 7 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → 𝑟 = 𝑡)
83	64, 82	syldan 602	. . . . . 6 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → 𝑟 = 𝑡)
84	33, 12, 8	addassd 11233	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑢 + (0 −_ℝ 𝑠)) + 𝑠) = (𝑢 + ((0 −_ℝ 𝑠) + 𝑠)))
85		renegid2 43102	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℝ → ((0 −_ℝ 𝑠) + 𝑠) = 0)
86	7, 85	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((0 −_ℝ 𝑠) + 𝑠) = 0)
87	86	oveq2d 7429	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 + ((0 −_ℝ 𝑠) + 𝑠)) = (𝑢 + 0))
88		readdrid 43098	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ℝ → (𝑢 + 0) = 𝑢)
89	32, 88	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 + 0) = 𝑢)
90	84, 87, 89	3eqtrd 2808	. . . . . . . . 9 ⊢ (𝜑 → ((𝑢 + (0 −_ℝ 𝑠)) + 𝑠) = 𝑢)
91		oveq1 7420	. . . . . . . . 9 ⊢ ((𝑢 + (0 −_ℝ 𝑠)) = 0 → ((𝑢 + (0 −_ℝ 𝑠)) + 𝑠) = (0 + 𝑠))
92	90, 91	sylan9req 2825	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 + (0 −_ℝ 𝑠)) = 0) → 𝑢 = (0 + 𝑠))
93		readdlid 43091	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → (0 + 𝑠) = 𝑠)
94	7, 93	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (0 + 𝑠) = 𝑠)
95	94	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 + (0 −_ℝ 𝑠)) = 0) → (0 + 𝑠) = 𝑠)
96	92, 95	eqtr2d 2805	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 + (0 −_ℝ 𝑠)) = 0) → 𝑠 = 𝑢)
97	61, 96	syldan 602	. . . . . 6 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → 𝑠 = 𝑢)
98	83, 97	jca 520	. . . . 5 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢))
99	50, 98	syldan 602	. . . 4 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠)))) = ((0 −_ℝ 𝑡) + ((𝑡 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))))) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢))
100	99	ex 417	. . 3 ⊢ (𝜑 → (((0 −_ℝ 𝑡) + ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠)))) = ((0 −_ℝ 𝑡) + ((𝑡 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠)))) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))
101	2, 100	syl5 35	. 2 ⊢ (𝜑 → ((𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))
102		id 23	. . 3 ⊢ (𝑟 = 𝑡 → 𝑟 = 𝑡)
103		oveq2 7421	. . 3 ⊢ (𝑠 = 𝑢 → (i · 𝑠) = (i · 𝑢))
104	102, 103	oveqan12d 7432	. 2 ⊢ ((𝑟 = 𝑡 ∧ 𝑠 = 𝑢) → (𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)))
105	101, 104	impbid1 228	1 ⊢ (𝜑 → ((𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)) ↔ (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 (class class class)co 7413 ℂcc 11100 ℝcr 11101 0cc0 11102 ici 11104 + caddc 11105 · cmul 11107 −_ℝ cresub 43053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5273 ax-pow 5339 ax-pr 5407 ax-un 7735 ax-resscn 11159 ax-1cn 11160 ax-icn 11161 ax-addcl 11162 ax-addrcl 11163 ax-mulcl 11164 ax-mulrcl 11165 ax-addass 11167 ax-mulass 11168 ax-distr 11169 ax-i2m1 11170 ax-1ne0 11171 ax-1rid 11172 ax-rnegex 11173 ax-rrecex 11174 ax-cnre 11175 ax-pre-lttri 11176 ax-pre-lttrn 11177 ax-pre-ltadd 11178 ax-pre-mulgt0 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5559 df-po 5572 df-so 5573 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-2 12305 df-3 12306 df-resub 43054 df-rediv 43129

This theorem is referenced by: (None)

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