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Theorem cxpaddle 26678

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 11714	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵))
7		cxpaddle.5	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
8	7	rpred 12955	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
9	3, 6, 8	recxpcld 26648	. . . . . 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 615	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵)))
15		elrp 12913	. . . . . . . 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 12986	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ∈ ℝ)
18	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ∈ ℝ)
19	18, 16	rerpdivcld 12986	. . . . . 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 12012	. . . . . . . 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 26648	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) ∈ ℝ)
27	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ 𝐵)
28		divge0 12012	. . . . . . . 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 26648	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶) ∈ ℝ)
31	1, 2	addge01d 11726	. . . . . . . . . . 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 5123	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ≤ ((𝐴 + 𝐵) · 1))
37		1red 11135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 1 ∈ ℝ)
38		ledivmul 12019	. . . . . . . . 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 26677	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ≤ ((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶))
45	2, 1	addge02d 11727	. . . . . . . . . . 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 5123	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ≤ ((𝐴 + 𝐵) · 1))
49		ledivmul 12019	. . . . . . . . 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 26677	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ≤ ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶))
53	17, 19, 26, 30, 44, 52	le2addd 11757	. . . . 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 12960	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ≠ 0)
57	54, 55, 34, 56	divdird 11956	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) / (𝐴 + 𝐵)) = ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))))
58	34, 56	dividd 11916	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) / (𝐴 + 𝐵)) = 1)
59	57, 58	eqtr3d 2766	. . . . 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 26647	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵))↑_𝑐𝐶) = ((𝐴↑_𝑐𝐶) / ((𝐴 + 𝐵)↑_𝑐𝐶)))
63	18, 27, 16, 61	divcxpd 26647	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵))↑_𝑐𝐶) = ((𝐵↑_𝑐𝐶) / ((𝐴 + 𝐵)↑_𝑐𝐶)))
64	62, 63	oveq12d 7371	. . . . . 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 26648	. . . . . . . . 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 26648	. . . . . . . . 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 26658	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ∈ ℝ⁺)
72	71	rpne0d 12960	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ≠ 0)
73	67, 70, 11, 72	divdird 11956	. . . . . 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 2767	. . . . 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 5126	. . . 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 13004	. . . 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 5119	. 2 ⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
81	7	rpne0d 12960	. . . . . 6 ⊢ (𝜑 → 𝐶 ≠ 0)
82	60, 81	0cxpd 26635	. . . . 5 ⊢ (𝜑 → (0↑_𝑐𝐶) = 0)
83	1, 4, 8	cxpge0d 26649	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴↑_𝑐𝐶))
84	2, 5, 8	cxpge0d 26649	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐵↑_𝑐𝐶))
85	65, 68, 83, 84	addge0d 11714	. . . . 5 ⊢ (𝜑 → 0 ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
86	82, 85	eqbrtrd 5117	. . . 4 ⊢ (𝜑 → (0↑_𝑐𝐶) ≤ ((𝐴↑_𝑐𝐶) + (𝐵↑_𝑐𝐶)))
87		oveq1 7360	. . . . 5 ⊢ (0 = (𝐴 + 𝐵) → (0↑_𝑐𝐶) = ((𝐴 + 𝐵)↑_𝑐𝐶))
88	87	breq1d 5105	. . . 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 11136	. . . 4 ⊢ 0 ∈ ℝ
92		leloe 11220	. . . 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 2109 class class class wbr 5095 (class class class)co 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 < clt 11168 ≤ cle 11169 / cdiv 11795 ℝ⁺crp 12911 ↑_𝑐ccxp 26480

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-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

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 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-ef 15992 df-sin 15994 df-cos 15995 df-pi 15997 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-mulg 18965 df-cntz 19214 df-cmn 19679 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24224 df-ms 24225 df-tms 24226 df-cncf 24787 df-limc 25783 df-dv 25784 df-log 26481 df-cxp 26482

This theorem is referenced by: abvcxp 27542

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