cnreeu - Mathbox for Steven Nguyen

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Theorem cnreeu 42473

Description: The reals in the expression given by cnre 11112 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 7356	. . . 4 ⊢ ((𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)) → ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠))) = ((𝑡 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))))
2	1	oveq2d 7365	. . 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 11143	. . . . . . . . . 10 ⊢ (𝜑 → 𝑟 ∈ ℂ)
5		ax-icn 11068	. . . . . . . . . . . 12 ⊢ i ∈ ℂ
6	5	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → i ∈ ℂ)
7		cnreeu.s	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑠 ∈ ℝ)
8	7	recnd 11143	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑠 ∈ ℂ)
9	6, 8	mulcld 11135	. . . . . . . . . 10 ⊢ (𝜑 → (i · 𝑠) ∈ ℂ)
10		rernegcl 42354	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → (0 −_ℝ 𝑠) ∈ ℝ)
11	7, 10	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 −_ℝ 𝑠) ∈ ℝ)
12	11	recnd 11143	. . . . . . . . . . 11 ⊢ (𝜑 → (0 −_ℝ 𝑠) ∈ ℂ)
13	6, 12	mulcld 11135	. . . . . . . . . 10 ⊢ (𝜑 → (i · (0 −_ℝ 𝑠)) ∈ ℂ)
14	4, 9, 13	addassd 11137	. . . . . . . . 9 ⊢ (𝜑 → ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠))) = (𝑟 + ((i · 𝑠) + (i · (0 −_ℝ 𝑠)))))
15		renegid 42356	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → (𝑠 + (0 −_ℝ 𝑠)) = 0)
16	7, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑠 + (0 −_ℝ 𝑠)) = 0)
17	16	oveq2d 7365	. . . . . . . . . . 11 ⊢ (𝜑 → (i · (𝑠 + (0 −_ℝ 𝑠))) = (i · 0))
18	6, 8, 12	adddid 11139	. . . . . . . . . . 11 ⊢ (𝜑 → (i · (𝑠 + (0 −_ℝ 𝑠))) = ((i · 𝑠) + (i · (0 −_ℝ 𝑠))))
19		sn-it0e0 42399	. . . . . . . . . . . 12 ⊢ (i · 0) = 0
20	19	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (i · 0) = 0)
21	17, 18, 20	3eqtr3d 2772	. . . . . . . . . 10 ⊢ (𝜑 → ((i · 𝑠) + (i · (0 −_ℝ 𝑠))) = 0)
22	21	oveq2d 7365	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 + ((i · 𝑠) + (i · (0 −_ℝ 𝑠)))) = (𝑟 + 0))
23		readdrid 42393	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ → (𝑟 + 0) = 𝑟)
24	3, 23	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 + 0) = 𝑟)
25	14, 22, 24	3eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠))) = 𝑟)
26	25	oveq2d 7365	. . . . . . 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 42354	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → (0 −_ℝ 𝑡) ∈ ℝ)
29	27, 28	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (0 −_ℝ 𝑡) ∈ ℝ)
30	29	recnd 11143	. . . . . . . . . 10 ⊢ (𝜑 → (0 −_ℝ 𝑡) ∈ ℂ)
31	27	recnd 11143	. . . . . . . . . 10 ⊢ (𝜑 → 𝑡 ∈ ℂ)
32		cnreeu.u	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑢 ∈ ℝ)
33	32	recnd 11143	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑢 ∈ ℂ)
34	6, 33	mulcld 11135	. . . . . . . . . 10 ⊢ (𝜑 → (i · 𝑢) ∈ ℂ)
35	30, 31, 34	addassd 11137	. . . . . . . . 9 ⊢ (𝜑 → (((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) = ((0 −_ℝ 𝑡) + (𝑡 + (i · 𝑢))))
36	35	oveq1d 7364	. . . . . . . 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 42387	. . . . . . . . . . 11 ⊢ ((i · 𝑢) ∈ ℂ → (0 + (i · 𝑢)) = (i · 𝑢))
38	34, 37	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0 + (i · 𝑢)) = (i · 𝑢))
39	38	oveq1d 7364	. . . . . . . . 9 ⊢ (𝜑 → ((0 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))) = ((i · 𝑢) + (i · (0 −_ℝ 𝑠))))
40		renegid2 42397	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → ((0 −_ℝ 𝑡) + 𝑡) = 0)
41	27, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((0 −_ℝ 𝑡) + 𝑡) = 0)
42	41	oveq1d 7364	. . . . . . . . . 10 ⊢ (𝜑 → (((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) = (0 + (i · 𝑢)))
43	42	oveq1d 7364	. . . . . . . . 9 ⊢ (𝜑 → ((((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))) = ((0 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))))
44	6, 33, 12	adddid 11139	. . . . . . . . 9 ⊢ (𝜑 → (i · (𝑢 + (0 −_ℝ 𝑠))) = ((i · 𝑢) + (i · (0 −_ℝ 𝑠))))
45	39, 43, 44	3eqtr4d 2774	. . . . . . . 8 ⊢ (𝜑 → ((((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))) = (i · (𝑢 + (0 −_ℝ 𝑠))))
46	31, 34	addcld 11134	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 + (i · 𝑢)) ∈ ℂ)
47	30, 46, 13	addassd 11137	. . . . . . . 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 2773	. . . . . . 7 ⊢ (𝜑 → ((0 −_ℝ 𝑡) + ((𝑡 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠)))) = (i · (𝑢 + (0 −_ℝ 𝑠))))
49	26, 48	eqeq12d 2745	. . . . . 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 476	. . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠))))
52	32	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → 𝑢 ∈ ℝ)
53	11	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (0 −_ℝ 𝑠) ∈ ℝ)
54	52, 53	readdcld 11144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (𝑢 + (0 −_ℝ 𝑠)) ∈ ℝ)
55	29	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (0 −_ℝ 𝑡) ∈ ℝ)
56	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → 𝑟 ∈ ℝ)
57	55, 56	readdcld 11144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → ((0 −_ℝ 𝑡) + 𝑟) ∈ ℝ)
58	51, 57	eqeltrrd 2829	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (i · (𝑢 + (0 −_ℝ 𝑠))) ∈ ℝ)
59		sn-itrere 42471	. . . . . . . . . . 11 ⊢ ((𝑢 + (0 −_ℝ 𝑠)) ∈ ℝ → ((i · (𝑢 + (0 −_ℝ 𝑠))) ∈ ℝ ↔ (𝑢 + (0 −_ℝ 𝑠)) = 0))
60	59	biimpa 476	. . . . . . . . . 10 ⊢ (((𝑢 + (0 −_ℝ 𝑠)) ∈ ℝ ∧ (i · (𝑢 + (0 −_ℝ 𝑠))) ∈ ℝ) → (𝑢 + (0 −_ℝ 𝑠)) = 0)
61	54, 58, 60	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (𝑢 + (0 −_ℝ 𝑠)) = 0)
62	61	oveq2d 7365	. . . . . . . 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 2768	. . . . . . 7 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → ((0 −_ℝ 𝑡) + 𝑟) = 0)
65		oveq2 7357	. . . . . . . . 9 ⊢ (((0 −_ℝ 𝑡) + 𝑟) = 0 → (𝑡 + ((0 −_ℝ 𝑡) + 𝑟)) = (𝑡 + 0))
66	65	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (𝑡 + ((0 −_ℝ 𝑡) + 𝑟)) = (𝑡 + 0))
67		renegid 42356	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → (𝑡 + (0 −_ℝ 𝑡)) = 0)
68	27, 67	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 + (0 −_ℝ 𝑡)) = 0)
69	68	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (𝑡 + (0 −_ℝ 𝑡)) = 0)
70	69	oveq1d 7364	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → ((𝑡 + (0 −_ℝ 𝑡)) + 𝑟) = (0 + 𝑟))
71	31	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → 𝑡 ∈ ℂ)
72	30	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (0 −_ℝ 𝑡) ∈ ℂ)
73	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → 𝑟 ∈ ℂ)
74	71, 72, 73	addassd 11137	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → ((𝑡 + (0 −_ℝ 𝑡)) + 𝑟) = (𝑡 + ((0 −_ℝ 𝑡) + 𝑟)))
75		readdlid 42386	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ → (0 + 𝑟) = 𝑟)
76	3, 75	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0 + 𝑟) = 𝑟)
77	76	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (0 + 𝑟) = 𝑟)
78	70, 74, 77	3eqtr3d 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (𝑡 + ((0 −_ℝ 𝑡) + 𝑟)) = 𝑟)
79	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → 𝑡 ∈ ℝ)
80		readdrid 42393	. . . . . . . . 9 ⊢ (𝑡 ∈ ℝ → (𝑡 + 0) = 𝑡)
81	79, 80	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (𝑡 + 0) = 𝑡)
82	66, 78, 81	3eqtr3d 2772	. . . . . . 7 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → 𝑟 = 𝑡)
83	64, 82	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → 𝑟 = 𝑡)
84	33, 12, 8	addassd 11137	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑢 + (0 −_ℝ 𝑠)) + 𝑠) = (𝑢 + ((0 −_ℝ 𝑠) + 𝑠)))
85		renegid2 42397	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℝ → ((0 −_ℝ 𝑠) + 𝑠) = 0)
86	7, 85	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((0 −_ℝ 𝑠) + 𝑠) = 0)
87	86	oveq2d 7365	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 + ((0 −_ℝ 𝑠) + 𝑠)) = (𝑢 + 0))
88		readdrid 42393	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ℝ → (𝑢 + 0) = 𝑢)
89	32, 88	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 + 0) = 𝑢)
90	84, 87, 89	3eqtrd 2768	. . . . . . . . 9 ⊢ (𝜑 → ((𝑢 + (0 −_ℝ 𝑠)) + 𝑠) = 𝑢)
91		oveq1 7356	. . . . . . . . 9 ⊢ ((𝑢 + (0 −_ℝ 𝑠)) = 0 → ((𝑢 + (0 −_ℝ 𝑠)) + 𝑠) = (0 + 𝑠))
92	90, 91	sylan9req 2785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 + (0 −_ℝ 𝑠)) = 0) → 𝑢 = (0 + 𝑠))
93		readdlid 42386	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → (0 + 𝑠) = 𝑠)
94	7, 93	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0 + 𝑠) = 𝑠)
95	94	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 + (0 −_ℝ 𝑠)) = 0) → (0 + 𝑠) = 𝑠)
96	92, 95	eqtr2d 2765	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 + (0 −_ℝ 𝑠)) = 0) → 𝑠 = 𝑢)
97	61, 96	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → 𝑠 = 𝑢)
98	83, 97	jca 511	. . . . 5 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢))
99	50, 98	syldan 591	. . . 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 412	. . 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 34	. 2 ⊢ (𝜑 → ((𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))
102		id 22	. . 3 ⊢ (𝑟 = 𝑡 → 𝑟 = 𝑡)
103		oveq2 7357	. . 3 ⊢ (𝑠 = 𝑢 → (i · 𝑠) = (i · 𝑢))
104	102, 103	oveqan12d 7368	. 2 ⊢ ((𝑟 = 𝑡 ∧ 𝑠 = 𝑢) → (𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)))
105	101, 104	impbid1 225	1 ⊢ (𝜑 → ((𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)) ↔ (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 ici 11011 + caddc 11012 · cmul 11014 −_ℝ cresub 42348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-2 12191 df-3 12192 df-resub 42349 df-rediv 42424

This theorem is referenced by: (None)

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