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Theorem cxpaddle 26712

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)

**Hypotheses**
Ref	Expression
cxpaddle.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
cxpaddle.2	⊢ (𝜑 → 0 ≤ 𝐴)
cxpaddle.3	⊢ (𝜑 → 𝐵 ∈ ℝ)
cxpaddle.4	⊢ (𝜑 → 0 ≤ 𝐵)
cxpaddle.5	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
cxpaddle.6	⊢ (𝜑 → 𝐶 ≤ 1)

**Assertion**
Ref	Expression
cxpaddle	⊢ (𝜑 → ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))

**Proof of Theorem cxpaddle**
Step	Hyp	Ref	Expression
1		cxpaddle.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		cxpaddle.3	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3	1, 2	readdcld 11262	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
4		cxpaddle.2	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝐴)
5		cxpaddle.4	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝐵)
6	1, 2, 4, 5	addge0d 11811	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵))
7		cxpaddle.5	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
8	7	rpred 13049	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
9	3, 6, 8	recxpcld 26682	. . . . . 6 ⊢ (𝜑 → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℝ)
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℝ)
11	10	recnd 11261	. . . 4 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℂ)
12	11	mullidd 11251	. . 3 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (1 · ((𝐴 + 𝐵)↑_𝑐𝐶)) = ((𝐴 + 𝐵)↑_𝑐𝐶))
13	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ∈ ℝ)
14	3	anim1i 615	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵)))
15		elrp 13008	. . . . . . . 8 ⊢ ((𝐴 + 𝐵) ∈ ℝ⁺ ↔ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵)))
16	14, 15	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈ ℝ⁺)
17	13, 16	rerpdivcld 13080	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ∈ ℝ)
18	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ∈ ℝ)
19	18, 16	rerpdivcld 13080	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ∈ ℝ)
20	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ 𝐴)
21	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈ ℝ)
22		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 < (𝐴 + 𝐵))
23		divge0 12109	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) → 0 ≤ (𝐴 / (𝐴 + 𝐵)))
24	13, 20, 21, 22, 23	syl22anc 838	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ (𝐴 / (𝐴 + 𝐵)))
25	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈ ℝ)
26	17, 24, 25	recxpcld 26682	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) ∈ ℝ)
27	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ 𝐵)
28		divge0 12109	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) → 0 ≤ (𝐵 / (𝐴 + 𝐵)))
29	18, 27, 21, 22, 28	syl22anc 838	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ (𝐵 / (𝐴 + 𝐵)))
30	19, 29, 25	recxpcld 26682	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶) ∈ ℝ)
31	1, 2	addge01d 11823	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵)))
32	5, 31	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ (𝐴 + 𝐵))
33	32	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ≤ (𝐴 + 𝐵))
34	21	recnd 11261	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈ ℂ)
35	34	mulridd 11250	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) · 1) = (𝐴 + 𝐵))
36	33, 35	breqtrrd 5147	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ≤ ((𝐴 + 𝐵) · 1))
37		1red 11234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 1 ∈ ℝ)
38		ledivmul 12116	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) → ((𝐴 / (𝐴 + 𝐵)) ≤ 1 ↔ 𝐴 ≤ ((𝐴 + 𝐵) · 1)))
39	13, 37, 21, 22, 38	syl112anc 1376	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) ≤ 1 ↔ 𝐴 ≤ ((𝐴 + 𝐵) · 1)))
40	36, 39	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ≤ 1)
41	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈ ℝ⁺)
42		cxpaddle.6	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≤ 1)
43	42	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ≤ 1)
44	17, 24, 40, 41, 43	cxpaddlelem 26711	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ≤ ((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶))
45	2, 1	addge02d 11824	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ 𝐵 ≤ (𝐴 + 𝐵)))
46	4, 45	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≤ (𝐴 + 𝐵))
47	46	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ≤ (𝐴 + 𝐵))
48	47, 35	breqtrrd 5147	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ≤ ((𝐴 + 𝐵) · 1))
49		ledivmul 12116	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 1 ∈ ℝ ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) → ((𝐵 / (𝐴 + 𝐵)) ≤ 1 ↔ 𝐵 ≤ ((𝐴 + 𝐵) · 1)))
50	18, 37, 21, 22, 49	syl112anc 1376	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵)) ≤ 1 ↔ 𝐵 ≤ ((𝐴 + 𝐵) · 1)))
51	48, 50	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ≤ 1)
52	19, 29, 51, 41, 43	cxpaddlelem 26711	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ≤ ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶))
53	17, 19, 26, 30, 44, 52	le2addd 11854	. . . . 5 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))) ≤ (((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) + ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶)))
54	13	recnd 11261	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ∈ ℂ)
55	18	recnd 11261	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ∈ ℂ)
56	16	rpne0d 13054	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ≠ 0)
57	54, 55, 34, 56	divdird 12053	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) / (𝐴 + 𝐵)) = ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))))
58	34, 56	dividd 12013	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) / (𝐴 + 𝐵)) = 1)
59	57, 58	eqtr3d 2772	. . . . 5 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))) = 1)
60	8	recnd 11261	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
61	60	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈ ℂ)
62	13, 20, 16, 61	divcxpd 26681	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) / ((𝐴 + 𝐵)↑_𝑐𝐶)))
63	18, 27, 16, 61	divcxpd 26681	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶) = ((𝐵↑_𝑐𝐶) / ((𝐴 + 𝐵)↑_𝑐𝐶)))
64	62, 63	oveq12d 7421	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) + ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶)) = (((𝐴↑_𝑐𝐶) / ((𝐴 + 𝐵)↑_𝑐𝐶)) + ((𝐵↑_𝑐𝐶) / ((𝐴 + 𝐵)↑_𝑐𝐶))))
65	1, 4, 8	recxpcld 26682	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑_𝑐𝐶) ∈ ℝ)
66	65	recnd 11261	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑_𝑐𝐶) ∈ ℂ)
67	66	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴↑_𝑐𝐶) ∈ ℂ)
68	2, 5, 8	recxpcld 26682	. . . . . . . . 9 ⊢ (𝜑 → (𝐵↑_𝑐𝐶) ∈ ℝ)
69	68	recnd 11261	. . . . . . . 8 ⊢ (𝜑 → (𝐵↑_𝑐𝐶) ∈ ℂ)
70	69	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵↑_𝑐𝐶) ∈ ℂ)
71	16, 25	rpcxpcld 26692	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℝ⁺)
72	71	rpne0d 13054	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ≠ 0)
73	67, 70, 11, 72	divdird 12053	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) / ((𝐴 + 𝐵)↑_𝑐𝐶)) = (((𝐴↑_𝑐𝐶) / ((𝐴 + 𝐵)↑_𝑐𝐶)) + ((𝐵↑_𝑐𝐶) / ((𝐴 + 𝐵)↑_𝑐𝐶))))
74	64, 73	eqtr4d 2773	. . . . 5 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) + ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶)) = (((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) / ((𝐴 + 𝐵)↑_𝑐𝐶)))
75	53, 59, 74	3brtr3d 5150	. . . 4 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 1 ≤ (((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) / ((𝐴 + 𝐵)↑_𝑐𝐶)))
76	65, 68	readdcld 11262	. . . . . 6 ⊢ (𝜑 → ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) ∈ ℝ)
77	76	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) ∈ ℝ)
78	37, 77, 71	lemuldivd 13098	. . . 4 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((1 · ((𝐴 + 𝐵)↑_𝑐𝐶)) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) ↔ 1 ≤ (((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) / ((𝐴 + 𝐵)↑_𝑐𝐶))))
79	75, 78	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (1 · ((𝐴 + 𝐵)↑_𝑐𝐶)) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
80	12, 79	eqbrtrrd 5143	. 2 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
81	7	rpne0d 13054	. . . . . 6 ⊢ (𝜑 → 𝐶 ≠ 0)
82	60, 81	0cxpd 26669	. . . . 5 ⊢ (𝜑 → (0↑_𝑐𝐶) = 0)
83	1, 4, 8	cxpge0d 26683	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴↑_𝑐𝐶))
84	2, 5, 8	cxpge0d 26683	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐵↑_𝑐𝐶))
85	65, 68, 83, 84	addge0d 11811	. . . . 5 ⊢ (𝜑 → 0 ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
86	82, 85	eqbrtrd 5141	. . . 4 ⊢ (𝜑 → (0↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
87		oveq1 7410	. . . . 5 ⊢ (0 = (𝐴 + 𝐵) → (0↑_𝑐𝐶) = ((𝐴 + 𝐵)↑_𝑐𝐶))
88	87	breq1d 5129	. . . 4 ⊢ (0 = (𝐴 + 𝐵) → ((0↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) ↔ ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶))))
89	86, 88	syl5ibcom 245	. . 3 ⊢ (𝜑 → (0 = (𝐴 + 𝐵) → ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶))))
90	89	imp 406	. 2 ⊢ ((𝜑 ∧ 0 = (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
91		0re 11235	. . . 4 ⊢ 0 ∈ ℝ
92		leloe 11319	. . . 4 ⊢ ((0 ∈ ℝ ∧ (𝐴 + 𝐵) ∈ ℝ) → (0 ≤ (𝐴 + 𝐵) ↔ (0 < (𝐴 + 𝐵) ∨ 0 = (𝐴 + 𝐵))))
93	91, 3, 92	sylancr 587	. . 3 ⊢ (𝜑 → (0 ≤ (𝐴 + 𝐵) ↔ (0 < (𝐴 + 𝐵) ∨ 0 = (𝐴 + 𝐵))))
94	6, 93	mpbid 232	. 2 ⊢ (𝜑 → (0 < (𝐴 + 𝐵) ∨ 0 = (𝐴 + 𝐵)))
95	80, 90, 94	mpjaodan 960	1 ⊢ (𝜑 → ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2108 class class class wbr 5119 (class class class)co 7403 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 < clt 11267 ≤ cle 11268 / cdiv 11892 ℝ⁺crp 13006 ↑_𝑐ccxp 26514

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-inf2 9653 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-ioc 13365 df-ico 13366 df-icc 13367 df-fz 13523 df-fzo 13670 df-fl 13807 df-mod 13885 df-seq 14018 df-exp 14078 df-fac 14290 df-bc 14319 df-hash 14347 df-shft 15084 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15485 df-clim 15502 df-rlim 15503 df-sum 15701 df-ef 16081 df-sin 16083 df-cos 16084 df-pi 16086 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-mulg 19049 df-cntz 19298 df-cmn 19761 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-ntr 22956 df-cls 22957 df-nei 23034 df-lp 23072 df-perf 23073 df-cn 23163 df-cnp 23164 df-haus 23251 df-tx 23498 df-hmeo 23691 df-fil 23782 df-fm 23874 df-flim 23875 df-flf 23876 df-xms 24257 df-ms 24258 df-tms 24259 df-cncf 24820 df-limc 25817 df-dv 25818 df-log 26515 df-cxp 26516

This theorem is referenced by: abvcxp 27576

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