cnreeu - Mathbox for Steven Nguyen

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Theorem cnreeu 40087

Description: The reals in the expression given by cnre 10795 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 7198	. . . 4 ⊢ ((𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)) → ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠))) = ((𝑡 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))))
2	1	oveq2d 7207	. . 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 10826	. . . . . . . . . 10 ⊢ (𝜑 → 𝑟 ∈ ℂ)
5		ax-icn 10753	. . . . . . . . . . . 12 ⊢ i ∈ ℂ
6	5	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → i ∈ ℂ)
7		cnreeu.s	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑠 ∈ ℝ)
8	7	recnd 10826	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑠 ∈ ℂ)
9	6, 8	mulcld 10818	. . . . . . . . . 10 ⊢ (𝜑 → (i · 𝑠) ∈ ℂ)
10		rernegcl 40003	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → (0 −_ℝ 𝑠) ∈ ℝ)
11	7, 10	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 −_ℝ 𝑠) ∈ ℝ)
12	11	recnd 10826	. . . . . . . . . . 11 ⊢ (𝜑 → (0 −_ℝ 𝑠) ∈ ℂ)
13	6, 12	mulcld 10818	. . . . . . . . . 10 ⊢ (𝜑 → (i · (0 −_ℝ 𝑠)) ∈ ℂ)
14	4, 9, 13	addassd 10820	. . . . . . . . 9 ⊢ (𝜑 → ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠))) = (𝑟 + ((i · 𝑠) + (i · (0 −_ℝ 𝑠)))))
15		renegid 40005	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → (𝑠 + (0 −_ℝ 𝑠)) = 0)
16	7, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑠 + (0 −_ℝ 𝑠)) = 0)
17	16	oveq2d 7207	. . . . . . . . . . 11 ⊢ (𝜑 → (i · (𝑠 + (0 −_ℝ 𝑠))) = (i · 0))
18	6, 8, 12	adddid 10822	. . . . . . . . . . 11 ⊢ (𝜑 → (i · (𝑠 + (0 −_ℝ 𝑠))) = ((i · 𝑠) + (i · (0 −_ℝ 𝑠))))
19		sn-it0e0 40046	. . . . . . . . . . . 12 ⊢ (i · 0) = 0
20	19	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (i · 0) = 0)
21	17, 18, 20	3eqtr3d 2779	. . . . . . . . . 10 ⊢ (𝜑 → ((i · 𝑠) + (i · (0 −_ℝ 𝑠))) = 0)
22	21	oveq2d 7207	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 + ((i · 𝑠) + (i · (0 −_ℝ 𝑠)))) = (𝑟 + 0))
23		readdid1 40041	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ → (𝑟 + 0) = 𝑟)
24	3, 23	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 + 0) = 𝑟)
25	14, 22, 24	3eqtrd 2775	. . . . . . . 8 ⊢ (𝜑 → ((𝑟 + (i · 𝑠)) + (i · (0 −_ℝ 𝑠))) = 𝑟)
26	25	oveq2d 7207	. . . . . . 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 40003	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → (0 −_ℝ 𝑡) ∈ ℝ)
29	27, 28	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (0 −_ℝ 𝑡) ∈ ℝ)
30	29	recnd 10826	. . . . . . . . . 10 ⊢ (𝜑 → (0 −_ℝ 𝑡) ∈ ℂ)
31	27	recnd 10826	. . . . . . . . . 10 ⊢ (𝜑 → 𝑡 ∈ ℂ)
32		cnreeu.u	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑢 ∈ ℝ)
33	32	recnd 10826	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑢 ∈ ℂ)
34	6, 33	mulcld 10818	. . . . . . . . . 10 ⊢ (𝜑 → (i · 𝑢) ∈ ℂ)
35	30, 31, 34	addassd 10820	. . . . . . . . 9 ⊢ (𝜑 → (((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) = ((0 −_ℝ 𝑡) + (𝑡 + (i · 𝑢))))
36	35	oveq1d 7206	. . . . . . . 8 ⊢ (𝜑 → ((((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))) = (((0 −_ℝ 𝑡) + (𝑡 + (i · 𝑢))) + (i · (0 −_ℝ 𝑠))))
37		sn-addid2 40036	. . . . . . . . . . 11 ⊢ ((i · 𝑢) ∈ ℂ → (0 + (i · 𝑢)) = (i · 𝑢))
38	34, 37	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0 + (i · 𝑢)) = (i · 𝑢))
39	38	oveq1d 7206	. . . . . . . . 9 ⊢ (𝜑 → ((0 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))) = ((i · 𝑢) + (i · (0 −_ℝ 𝑠))))
40		renegid2 40045	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → ((0 −_ℝ 𝑡) + 𝑡) = 0)
41	27, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((0 −_ℝ 𝑡) + 𝑡) = 0)
42	41	oveq1d 7206	. . . . . . . . . 10 ⊢ (𝜑 → (((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) = (0 + (i · 𝑢)))
43	42	oveq1d 7206	. . . . . . . . 9 ⊢ (𝜑 → ((((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))) = ((0 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))))
44	6, 33, 12	adddid 10822	. . . . . . . . 9 ⊢ (𝜑 → (i · (𝑢 + (0 −_ℝ 𝑠))) = ((i · 𝑢) + (i · (0 −_ℝ 𝑠))))
45	39, 43, 44	3eqtr4d 2781	. . . . . . . 8 ⊢ (𝜑 → ((((0 −_ℝ 𝑡) + 𝑡) + (i · 𝑢)) + (i · (0 −_ℝ 𝑠))) = (i · (𝑢 + (0 −_ℝ 𝑠))))
46	31, 34	addcld 10817	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 + (i · 𝑢)) ∈ ℂ)
47	30, 46, 13	addassd 10820	. . . . . . . 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 2780	. . . . . . 7 ⊢ (𝜑 → ((0 −_ℝ 𝑡) + ((𝑡 + (i · 𝑢)) + (i · (0 −_ℝ 𝑠)))) = (i · (𝑢 + (0 −_ℝ 𝑠))))
49	26, 48	eqeq12d 2752	. . . . . 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 480	. . . . 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 488	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠))))
52	32	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → 𝑢 ∈ ℝ)
53	11	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (0 −_ℝ 𝑠) ∈ ℝ)
54	52, 53	readdcld 10827	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (𝑢 + (0 −_ℝ 𝑠)) ∈ ℝ)
55	29	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (0 −_ℝ 𝑡) ∈ ℝ)
56	3	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → 𝑟 ∈ ℝ)
57	55, 56	readdcld 10827	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → ((0 −_ℝ 𝑡) + 𝑟) ∈ ℝ)
58	51, 57	eqeltrrd 2832	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (i · (𝑢 + (0 −_ℝ 𝑠))) ∈ ℝ)
59		itrere 40085	. . . . . . . . . . 11 ⊢ ((𝑢 + (0 −_ℝ 𝑠)) ∈ ℝ → ((i · (𝑢 + (0 −_ℝ 𝑠))) ∈ ℝ ↔ (𝑢 + (0 −_ℝ 𝑠)) = 0))
60	59	biimpa 480	. . . . . . . . . 10 ⊢ (((𝑢 + (0 −_ℝ 𝑠)) ∈ ℝ ∧ (i · (𝑢 + (0 −_ℝ 𝑠))) ∈ ℝ) → (𝑢 + (0 −_ℝ 𝑠)) = 0)
61	54, 58, 60	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (𝑢 + (0 −_ℝ 𝑠)) = 0)
62	61	oveq2d 7207	. . . . . . . 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 2775	. . . . . . 7 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → ((0 −_ℝ 𝑡) + 𝑟) = 0)
65		oveq2 7199	. . . . . . . . 9 ⊢ (((0 −_ℝ 𝑡) + 𝑟) = 0 → (𝑡 + ((0 −_ℝ 𝑡) + 𝑟)) = (𝑡 + 0))
66	65	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (𝑡 + ((0 −_ℝ 𝑡) + 𝑟)) = (𝑡 + 0))
67		renegid 40005	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → (𝑡 + (0 −_ℝ 𝑡)) = 0)
68	27, 67	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 + (0 −_ℝ 𝑡)) = 0)
69	68	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (𝑡 + (0 −_ℝ 𝑡)) = 0)
70	69	oveq1d 7206	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → ((𝑡 + (0 −_ℝ 𝑡)) + 𝑟) = (0 + 𝑟))
71	31	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → 𝑡 ∈ ℂ)
72	30	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (0 −_ℝ 𝑡) ∈ ℂ)
73	4	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → 𝑟 ∈ ℂ)
74	71, 72, 73	addassd 10820	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → ((𝑡 + (0 −_ℝ 𝑡)) + 𝑟) = (𝑡 + ((0 −_ℝ 𝑡) + 𝑟)))
75		readdid2 40035	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ → (0 + 𝑟) = 𝑟)
76	3, 75	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0 + 𝑟) = 𝑟)
77	76	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (0 + 𝑟) = 𝑟)
78	70, 74, 77	3eqtr3d 2779	. . . . . . . 8 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → (𝑡 + ((0 −_ℝ 𝑡) + 𝑟)) = 𝑟)
79	27	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → 𝑡 ∈ ℝ)
80		readdid1 40041	. . . . . . . . 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 2779	. . . . . . 7 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = 0) → 𝑟 = 𝑡)
83	64, 82	syldan 594	. . . . . 6 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → 𝑟 = 𝑡)
84	33, 12, 8	addassd 10820	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑢 + (0 −_ℝ 𝑠)) + 𝑠) = (𝑢 + ((0 −_ℝ 𝑠) + 𝑠)))
85		renegid2 40045	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℝ → ((0 −_ℝ 𝑠) + 𝑠) = 0)
86	7, 85	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((0 −_ℝ 𝑠) + 𝑠) = 0)
87	86	oveq2d 7207	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 + ((0 −_ℝ 𝑠) + 𝑠)) = (𝑢 + 0))
88		readdid1 40041	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ℝ → (𝑢 + 0) = 𝑢)
89	32, 88	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑢 + 0) = 𝑢)
90	84, 87, 89	3eqtrd 2775	. . . . . . . . 9 ⊢ (𝜑 → ((𝑢 + (0 −_ℝ 𝑠)) + 𝑠) = 𝑢)
91		oveq1 7198	. . . . . . . . 9 ⊢ ((𝑢 + (0 −_ℝ 𝑠)) = 0 → ((𝑢 + (0 −_ℝ 𝑠)) + 𝑠) = (0 + 𝑠))
92	90, 91	sylan9req 2792	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 + (0 −_ℝ 𝑠)) = 0) → 𝑢 = (0 + 𝑠))
93		readdid2 40035	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → (0 + 𝑠) = 𝑠)
94	7, 93	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0 + 𝑠) = 𝑠)
95	94	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 + (0 −_ℝ 𝑠)) = 0) → (0 + 𝑠) = 𝑠)
96	92, 95	eqtr2d 2772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 + (0 −_ℝ 𝑠)) = 0) → 𝑠 = 𝑢)
97	61, 96	syldan 594	. . . . . 6 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → 𝑠 = 𝑢)
98	83, 97	jca 515	. . . . 5 ⊢ ((𝜑 ∧ ((0 −_ℝ 𝑡) + 𝑟) = (i · (𝑢 + (0 −_ℝ 𝑠)))) → (𝑟 = 𝑡 ∧ 𝑠 = 𝑢))
99	50, 98	syldan 594	. . . 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 416	. . 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 7199	. . 3 ⊢ (𝑠 = 𝑢 → (i · 𝑠) = (i · 𝑢))
104	102, 103	oveqan12d 7210	. 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 399 = wceq 1543 ∈ wcel 2112 (class class class)co 7191 ℂcc 10692 ℝcr 10693 0cc0 10694 ici 10696 + caddc 10697 · cmul 10699 −_ℝ cresub 39997

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-2 11858 df-3 11859 df-resub 39998

This theorem is referenced by: (None)

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