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Theorem cxpaddle 26733

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 11163	. . . . . . 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 11715	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵))
7		cxpaddle.5	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
8	7	rpred 12975	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
9	3, 6, 8	recxpcld 26703	. . . . . 6 ⊢ (𝜑 → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℝ)
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℝ)
11	10	recnd 11162	. . . 4 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℂ)
12	11	mullidd 11152	. . 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 616	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵)))
15		elrp 12933	. . . . . . . 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 13006	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ∈ ℝ)
18	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ∈ ℝ)
19	18, 16	rerpdivcld 13006	. . . . . 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 12014	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) → 0 ≤ (𝐴 / (𝐴 + 𝐵)))
24	13, 20, 21, 22, 23	syl22anc 839	. . . . . . 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 26703	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) ∈ ℝ)
27	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ 𝐵)
28		divge0 12014	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) → 0 ≤ (𝐵 / (𝐴 + 𝐵)))
29	18, 27, 21, 22, 28	syl22anc 839	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ (𝐵 / (𝐴 + 𝐵)))
30	19, 29, 25	recxpcld 26703	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶) ∈ ℝ)
31	1, 2	addge01d 11727	. . . . . . . . . . 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 11162	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈ ℂ)
35	34	mulridd 11151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) · 1) = (𝐴 + 𝐵))
36	33, 35	breqtrrd 5114	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ≤ ((𝐴 + 𝐵) · 1))
37		1red 11134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 1 ∈ ℝ)
38		ledivmul 12021	. . . . . . . . 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 1377	. . . . . . . 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 26732	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ≤ ((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶))
45	2, 1	addge02d 11728	. . . . . . . . . . 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 5114	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ≤ ((𝐴 + 𝐵) · 1))
49		ledivmul 12021	. . . . . . . . 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 1377	. . . . . . . 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 26732	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ≤ ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶))
53	17, 19, 26, 30, 44, 52	le2addd 11758	. . . . 5 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))) ≤ (((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) + ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶)))
54	13	recnd 11162	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ∈ ℂ)
55	18	recnd 11162	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ∈ ℂ)
56	16	rpne0d 12980	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ≠ 0)
57	54, 55, 34, 56	divdird 11958	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) / (𝐴 + 𝐵)) = ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))))
58	34, 56	dividd 11918	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) / (𝐴 + 𝐵)) = 1)
59	57, 58	eqtr3d 2774	. . . . 5 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))) = 1)
60	8	recnd 11162	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
61	60	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈ ℂ)
62	13, 20, 16, 61	divcxpd 26702	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) / ((𝐴 + 𝐵)↑_𝑐𝐶)))
63	18, 27, 16, 61	divcxpd 26702	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶) = ((𝐵↑_𝑐𝐶) / ((𝐴 + 𝐵)↑_𝑐𝐶)))
64	62, 63	oveq12d 7376	. . . . . 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 26703	. . . . . . . . 9 ⊢ (𝜑 → (𝐴↑_𝑐𝐶) ∈ ℝ)
66	65	recnd 11162	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑_𝑐𝐶) ∈ ℂ)
67	66	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴↑_𝑐𝐶) ∈ ℂ)
68	2, 5, 8	recxpcld 26703	. . . . . . . . 9 ⊢ (𝜑 → (𝐵↑_𝑐𝐶) ∈ ℝ)
69	68	recnd 11162	. . . . . . . 8 ⊢ (𝜑 → (𝐵↑_𝑐𝐶) ∈ ℂ)
70	69	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵↑_𝑐𝐶) ∈ ℂ)
71	16, 25	rpcxpcld 26713	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℝ⁺)
72	71	rpne0d 12980	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ≠ 0)
73	67, 70, 11, 72	divdird 11958	. . . . . 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 2775	. . . . 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 5117	. . . 4 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 1 ≤ (((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)) / ((𝐴 + 𝐵)↑_𝑐𝐶)))
76	65, 68	readdcld 11163	. . . . . 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 13024	. . . 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 5110	. 2 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
81	7	rpne0d 12980	. . . . . 6 ⊢ (𝜑 → 𝐶 ≠ 0)
82	60, 81	0cxpd 26690	. . . . 5 ⊢ (𝜑 → (0↑_𝑐𝐶) = 0)
83	1, 4, 8	cxpge0d 26704	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴↑_𝑐𝐶))
84	2, 5, 8	cxpge0d 26704	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐵↑_𝑐𝐶))
85	65, 68, 83, 84	addge0d 11715	. . . . 5 ⊢ (𝜑 → 0 ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
86	82, 85	eqbrtrd 5108	. . . 4 ⊢ (𝜑 → (0↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
87		oveq1 7365	. . . . 5 ⊢ (0 = (𝐴 + 𝐵) → (0↑_𝑐𝐶) = ((𝐴 + 𝐵)↑_𝑐𝐶))
88	87	breq1d 5096	. . . 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 11135	. . . 4 ⊢ 0 ∈ ℝ
92		leloe 11221	. . . 4 ⊢ ((0 ∈ ℝ ∧ (𝐴 + 𝐵) ∈ ℝ) → (0 ≤ (𝐴 + 𝐵) ↔ (0 < (𝐴 + 𝐵) ∨ 0 = (𝐴 + 𝐵))))
93	91, 3, 92	sylancr 588	. . 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 961	1 ⊢ (𝜑 → ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 class class class wbr 5086 (class class class)co 7358 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 < clt 11168 ≤ cle 11169 / cdiv 11796 ℝ⁺crp 12931 ↑_𝑐ccxp 26535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15638 df-ef 16021 df-sin 16023 df-cos 16024 df-pi 16026 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18741 df-mulg 19033 df-cntz 19281 df-cmn 19746 df-psmet 21334 df-xmet 21335 df-met 21336 df-bl 21337 df-mopn 21338 df-fbas 21339 df-fg 21340 df-cnfld 21343 df-top 22868 df-topon 22885 df-topsp 22907 df-bases 22920 df-cld 22993 df-ntr 22994 df-cls 22995 df-nei 23072 df-lp 23110 df-perf 23111 df-cn 23201 df-cnp 23202 df-haus 23289 df-tx 23536 df-hmeo 23729 df-fil 23820 df-fm 23912 df-flim 23913 df-flf 23914 df-xms 24294 df-ms 24295 df-tms 24296 df-cncf 24854 df-limc 25842 df-dv 25843 df-log 26536 df-cxp 26537

This theorem is referenced by: abvcxp 27597

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