cxpsqrtlem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cxpsqrtlem		Structured version Visualization version GIF version

Theorem cxpsqrtlem 26665

Description: Lemma for cxpsqrt 26666. (Contributed by Mario Carneiro, 2-Aug-2014.)

**Assertion**
Ref	Expression
cxpsqrtlem	⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℝ)

**Proof of Theorem cxpsqrtlem**
Step	Hyp	Ref	Expression
1		ax-icn 11083	. . 3 ⊢ i ∈ ℂ
2		sqrtcl 15283	. . . 4 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
3	2	ad2antrr 726	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) ∈ ℂ)
4		mulcl 11108	. . 3 ⊢ ((i ∈ ℂ ∧ (√‘𝐴) ∈ ℂ) → (i · (√‘𝐴)) ∈ ℂ)
5	1, 3, 4	sylancr 587	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℂ)
6		imval 15028	. . . 4 ⊢ ((i · (√‘𝐴)) ∈ ℂ → (ℑ‘(i · (√‘𝐴))) = (ℜ‘((i · (√‘𝐴)) / i)))
7	5, 6	syl 17	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℑ‘(i · (√‘𝐴))) = (ℜ‘((i · (√‘𝐴)) / i)))
8		ine0 11570	. . . . . 6 ⊢ i ≠ 0
9		divcan3 11820	. . . . . 6 ⊢ (((√‘𝐴) ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → ((i · (√‘𝐴)) / i) = (√‘𝐴))
10	1, 8, 9	mp3an23 1455	. . . . 5 ⊢ ((√‘𝐴) ∈ ℂ → ((i · (√‘𝐴)) / i) = (√‘𝐴))
11	3, 10	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴)) / i) = (√‘𝐴))
12	11	fveq2d 6836	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘((i · (√‘𝐴)) / i)) = (ℜ‘(√‘𝐴)))
13		halfre 12352	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ
14	13	recni 11144	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
15		logcl 26531	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
16		mulcl 11108	. . . . . . . . . . . 12 ⊢ (((1 / 2) ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → ((1 / 2) · (log‘𝐴)) ∈ ℂ)
17	14, 15, 16	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((1 / 2) · (log‘𝐴)) ∈ ℂ)
18	17	recld 15115	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘((1 / 2) · (log‘𝐴))) ∈ ℝ)
19	18	reefcld 16009	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
20	17	imcld 15116	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) ∈ ℝ)
21	20	recoscld 16067	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
22	18	rpefcld 16028	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ⁺)
23	22	rpge0d 12951	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))))
24		immul2 15058	. . . . . . . . . . . . 13 ⊢ (((1 / 2) ∈ ℝ ∧ (log‘𝐴) ∈ ℂ) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((1 / 2) · (ℑ‘(log‘𝐴))))
25	13, 15, 24	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((1 / 2) · (ℑ‘(log‘𝐴))))
26	15	imcld 15116	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
27	26	recnd 11158	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ)
28		mulcom 11110	. . . . . . . . . . . . 13 ⊢ (((1 / 2) ∈ ℂ ∧ (ℑ‘(log‘𝐴)) ∈ ℂ) → ((1 / 2) · (ℑ‘(log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
29	14, 27, 28	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((1 / 2) · (ℑ‘(log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
30	25, 29	eqtrd 2769	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
31		logimcl 26532	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
32	31	simpld 494	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -π < (ℑ‘(log‘𝐴)))
33		pire 26420	. . . . . . . . . . . . . . . 16 ⊢ π ∈ ℝ
34	33	renegcli 11440	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℝ
35		ltle 11219	. . . . . . . . . . . . . . 15 ⊢ ((-π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
36	34, 26, 35	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
37	32, 36	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -π ≤ (ℑ‘(log‘𝐴)))
38	31	simprd 495	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ≤ π)
39	34, 33	elicc2i 13326	. . . . . . . . . . . . 13 ⊢ ((ℑ‘(log‘𝐴)) ∈ (-π[,]π) ↔ ((ℑ‘(log‘𝐴)) ∈ ℝ ∧ -π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
40	26, 37, 38, 39	syl3anbrc 1344	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ (-π[,]π))
41		halfgt0 12354	. . . . . . . . . . . . . 14 ⊢ 0 < (1 / 2)
42	13, 41	elrpii 12906	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ⁺
43	33	recni 11144	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℂ
44		2cn 12218	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
45		2ne0 12247	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
46		divneg 11831	. . . . . . . . . . . . . . 15 ⊢ ((π ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(π / 2) = (-π / 2))
47	43, 44, 45, 46	mp3an 1463	. . . . . . . . . . . . . 14 ⊢ -(π / 2) = (-π / 2)
48	34	recni 11144	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℂ
49	48, 44, 45	divreci 11884	. . . . . . . . . . . . . 14 ⊢ (-π / 2) = (-π · (1 / 2))
50	47, 49	eqtr2i 2758	. . . . . . . . . . . . 13 ⊢ (-π · (1 / 2)) = -(π / 2)
51	43, 44, 45	divreci 11884	. . . . . . . . . . . . . 14 ⊢ (π / 2) = (π · (1 / 2))
52	51	eqcomi 2743	. . . . . . . . . . . . 13 ⊢ (π · (1 / 2)) = (π / 2)
53	34, 33, 42, 50, 52	iccdili 13405	. . . . . . . . . . . 12 ⊢ ((ℑ‘(log‘𝐴)) ∈ (-π[,]π) → ((ℑ‘(log‘𝐴)) · (1 / 2)) ∈ (-(π / 2)[,](π / 2)))
54	40, 53	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((ℑ‘(log‘𝐴)) · (1 / 2)) ∈ (-(π / 2)[,](π / 2)))
55	30, 54	eqeltrd 2834	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) ∈ (-(π / 2)[,](π / 2)))
56		cosq14ge0 26474	. . . . . . . . . 10 ⊢ ((ℑ‘((1 / 2) · (log‘𝐴))) ∈ (-(π / 2)[,](π / 2)) → 0 ≤ (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))))
57	55, 56	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))))
58	19, 21, 23, 57	mulge0d 11712	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ ((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · (cos‘(ℑ‘((1 / 2) · (log‘𝐴))))))
59		cxpef 26628	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
60	14, 59	mp3an3 1452	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
61		efeul 16085	. . . . . . . . . . . 12 ⊢ (((1 / 2) · (log‘𝐴)) ∈ ℂ → (exp‘((1 / 2) · (log‘𝐴))) = ((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · ((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴))))))))
62	17, 61	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘((1 / 2) · (log‘𝐴))) = ((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · ((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴))))))))
63	60, 62	eqtrd 2769	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐(1 / 2)) = ((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · ((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴))))))))
64	63	fveq2d 6836	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(𝐴↑_𝑐(1 / 2))) = (ℜ‘((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · ((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))))))))
65	21	recnd 11158	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ)
66	20	resincld 16066	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
67	66	recnd 11158	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ)
68		mulcl 11108	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ) → (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴))))) ∈ ℂ)
69	1, 67, 68	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴))))) ∈ ℂ)
70	65, 69	addcld 11149	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))))) ∈ ℂ)
71	19, 70	remul2d 15148	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · ((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))))))) = ((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · (ℜ‘((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))))))))
72	21, 66	crred 15152	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴))))))) = (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))))
73	72	oveq2d 7372	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · (ℜ‘((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))))))) = ((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · (cos‘(ℑ‘((1 / 2) · (log‘𝐴))))))
74	64, 71, 73	3eqtrd 2773	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(𝐴↑_𝑐(1 / 2))) = ((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · (cos‘(ℑ‘((1 / 2) · (log‘𝐴))))))
75	58, 74	breqtrrd 5124	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ (ℜ‘(𝐴↑_𝑐(1 / 2))))
76	75	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ (ℜ‘(𝐴↑_𝑐(1 / 2))))
77		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴))
78	77	fveq2d 6836	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(𝐴↑_𝑐(1 / 2))) = (ℜ‘-(√‘𝐴)))
79	3	renegd 15130	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘-(√‘𝐴)) = -(ℜ‘(√‘𝐴)))
80	78, 79	eqtrd 2769	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(𝐴↑_𝑐(1 / 2))) = -(ℜ‘(√‘𝐴)))
81	76, 80	breqtrd 5122	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ -(ℜ‘(√‘𝐴)))
82	3	recld 15115	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(√‘𝐴)) ∈ ℝ)
83	82	le0neg1d 11706	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((ℜ‘(√‘𝐴)) ≤ 0 ↔ 0 ≤ -(ℜ‘(√‘𝐴))))
84	81, 83	mpbird 257	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(√‘𝐴)) ≤ 0)
85		sqrtrege0 15287	. . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ≤ (ℜ‘(√‘𝐴)))
86	85	ad2antrr 726	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ (ℜ‘(√‘𝐴)))
87		0re 11132	. . . . 5 ⊢ 0 ∈ ℝ
88		letri3 11216	. . . . 5 ⊢ (((ℜ‘(√‘𝐴)) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘(√‘𝐴)) = 0 ↔ ((ℜ‘(√‘𝐴)) ≤ 0 ∧ 0 ≤ (ℜ‘(√‘𝐴)))))
89	82, 87, 88	sylancl 586	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((ℜ‘(√‘𝐴)) = 0 ↔ ((ℜ‘(√‘𝐴)) ≤ 0 ∧ 0 ≤ (ℜ‘(√‘𝐴)))))
90	84, 86, 89	mpbir2and 713	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(√‘𝐴)) = 0)
91	7, 12, 90	3eqtrd 2773	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℑ‘(i · (√‘𝐴))) = 0)
92	5, 91	reim0bd 15121	1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 class class class wbr 5096 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 ℝcr 11023 0cc0 11024 1c1 11025 ici 11026 + caddc 11027 · cmul 11029 < clt 11164 ≤ cle 11165 -cneg 11363 / cdiv 11792 2c2 12198 [,]cicc 13262 ℜcre 15018 ℑcim 15019 √csqrt 15154 expce 15982 sincsin 15984 cosccos 15985 πcpi 15987 logclog 26517 ↑_𝑐ccxp 26518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 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 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 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 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-fac 14195 df-bc 14224 df-hash 14252 df-shft 14988 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410 df-sum 15608 df-ef 15988 df-sin 15990 df-cos 15991 df-pi 15993 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-mulg 18996 df-cntz 19244 df-cmn 19709 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-xms 24262 df-ms 24263 df-tms 24264 df-cncf 24825 df-limc 25821 df-dv 25822 df-log 26519 df-cxp 26520

This theorem is referenced by: cxpsqrt 26666

W3C validator