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Theorem cxpaddle 25905

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 11004	. . . . . . 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 11551	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵))
7		cxpaddle.5	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
8	7	rpred 12772	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
9	3, 6, 8	recxpcld 25878	. . . . . 6 ⊢ (𝜑 → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℝ)
10	9	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℝ)
11	10	recnd 11003	. . . 4 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℂ)
12	11	mulid2d 10993	. . 3 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (1 · ((𝐴 + 𝐵)↑_𝑐𝐶)) = ((𝐴 + 𝐵)↑_𝑐𝐶))
13	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ∈ ℝ)
14	3	anim1i 615	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵)))
15		elrp 12732	. . . . . . . 8 ⊢ ((𝐴 + 𝐵) ∈ ℝ⁺ ↔ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵)))
16	14, 15	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈ ℝ⁺)
17	13, 16	rerpdivcld 12803	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ∈ ℝ)
18	2	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ∈ ℝ)
19	18, 16	rerpdivcld 12803	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ∈ ℝ)
20	4	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ 𝐴)
21	3	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈ ℝ)
22		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 < (𝐴 + 𝐵))
23		divge0 11844	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) → 0 ≤ (𝐴 / (𝐴 + 𝐵)))
24	13, 20, 21, 22, 23	syl22anc 836	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ (𝐴 / (𝐴 + 𝐵)))
25	8	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈ ℝ)
26	17, 24, 25	recxpcld 25878	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) ∈ ℝ)
27	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ 𝐵)
28		divge0 11844	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) → 0 ≤ (𝐵 / (𝐴 + 𝐵)))
29	18, 27, 21, 22, 28	syl22anc 836	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ (𝐵 / (𝐴 + 𝐵)))
30	19, 29, 25	recxpcld 25878	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶) ∈ ℝ)
31	1, 2	addge01d 11563	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵)))
32	5, 31	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ (𝐴 + 𝐵))
33	32	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ≤ (𝐴 + 𝐵))
34	21	recnd 11003	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈ ℂ)
35	34	mulid1d 10992	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) · 1) = (𝐴 + 𝐵))
36	33, 35	breqtrrd 5102	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ≤ ((𝐴 + 𝐵) · 1))
37		1red 10976	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 1 ∈ ℝ)
38		ledivmul 11851	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) ≤ 1 ↔ 𝐴 ≤ ((𝐴 + 𝐵) · 1)))
40	36, 39	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ≤ 1)
41	7	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈ ℝ⁺)
42		cxpaddle.6	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≤ 1)
43	42	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ≤ 1)
44	17, 24, 40, 41, 43	cxpaddlelem 25904	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ≤ ((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶))
45	2, 1	addge02d 11564	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ 𝐵 ≤ (𝐴 + 𝐵)))
46	4, 45	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≤ (𝐴 + 𝐵))
47	46	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ≤ (𝐴 + 𝐵))
48	47, 35	breqtrrd 5102	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ≤ ((𝐴 + 𝐵) · 1))
49		ledivmul 11851	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵)) ≤ 1 ↔ 𝐵 ≤ ((𝐴 + 𝐵) · 1)))
51	48, 50	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ≤ 1)
52	19, 29, 51, 41, 43	cxpaddlelem 25904	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ≤ ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶))
53	17, 19, 26, 30, 44, 52	le2addd 11594	. . . . 5 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))) ≤ (((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) + ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶)))
54	13	recnd 11003	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ∈ ℂ)
55	18	recnd 11003	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ∈ ℂ)
56	16	rpne0d 12777	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ≠ 0)
57	54, 55, 34, 56	divdird 11789	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) / (𝐴 + 𝐵)) = ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))))
58	34, 56	dividd 11749	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) / (𝐴 + 𝐵)) = 1)
59	57, 58	eqtr3d 2780	. . . . 5 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))) = 1)
60	8	recnd 11003	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
61	60	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈ ℂ)
62	13, 20, 16, 61	divcxpd 25877	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) / ((𝐴 + 𝐵)↑_𝑐𝐶)))
63	18, 27, 16, 61	divcxpd 25877	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶) = ((𝐵↑_𝑐𝐶) / ((𝐴 + 𝐵)↑_𝑐𝐶)))
64	62, 63	oveq12d 7293	. . . . . 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 25878	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑_𝑐𝐶) ∈ ℝ)
66	65	recnd 11003	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑_𝑐𝐶) ∈ ℂ)
67	66	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴↑_𝑐𝐶) ∈ ℂ)
68	2, 5, 8	recxpcld 25878	. . . . . . . . 9 ⊢ (𝜑 → (𝐵↑_𝑐𝐶) ∈ ℝ)
69	68	recnd 11003	. . . . . . . 8 ⊢ (𝜑 → (𝐵↑_𝑐𝐶) ∈ ℂ)
70	69	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵↑_𝑐𝐶) ∈ ℂ)
71	16, 25	rpcxpcld 25887	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℝ⁺)
72	71	rpne0d 12777	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ≠ 0)
73	67, 70, 11, 72	divdird 11789	. . . . . 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 2781	. . . . 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 5105	. . . 4 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 1 ≤ (((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) / ((𝐴 + 𝐵)↑_𝑐𝐶)))
76	65, 68	readdcld 11004	. . . . . 6 ⊢ (𝜑 → ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) ∈ ℝ)
77	76	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) ∈ ℝ)
78	37, 77, 71	lemuldivd 12821	. . . 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 256	. . 3 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (1 · ((𝐴 + 𝐵)↑_𝑐𝐶)) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
80	12, 79	eqbrtrrd 5098	. 2 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
81	7	rpne0d 12777	. . . . . 6 ⊢ (𝜑 → 𝐶 ≠ 0)
82	60, 81	0cxpd 25865	. . . . 5 ⊢ (𝜑 → (0↑_𝑐𝐶) = 0)
83	1, 4, 8	cxpge0d 25879	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴↑_𝑐𝐶))
84	2, 5, 8	cxpge0d 25879	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐵↑_𝑐𝐶))
85	65, 68, 83, 84	addge0d 11551	. . . . 5 ⊢ (𝜑 → 0 ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
86	82, 85	eqbrtrd 5096	. . . 4 ⊢ (𝜑 → (0↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
87		oveq1 7282	. . . . 5 ⊢ (0 = (𝐴 + 𝐵) → (0↑_𝑐𝐶) = ((𝐴 + 𝐵)↑_𝑐𝐶))
88	87	breq1d 5084	. . . 4 ⊢ (0 = (𝐴 + 𝐵) → ((0↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) ↔ ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶))))
89	86, 88	syl5ibcom 244	. . 3 ⊢ (𝜑 → (0 = (𝐴 + 𝐵) → ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶))))
90	89	imp 407	. 2 ⊢ ((𝜑 ∧ 0 = (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
91		0re 10977	. . . 4 ⊢ 0 ∈ ℝ
92		leloe 11061	. . . 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 231	. 2 ⊢ (𝜑 → (0 < (𝐴 + 𝐵) ∨ 0 = (𝐴 + 𝐵)))
95	80, 90, 94	mpjaodan 956	1 ⊢ (𝜑 → ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 / cdiv 11632 ℝ⁺crp 12730 ↑_𝑐ccxp 25711

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 df-cxp 25713

This theorem is referenced by: abvcxp 26763

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