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Theorem cxpsqrtlem 26682

Description: Lemma for cxpsqrt 26683. (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 11091	. . 3 ⊢ i ∈ ℂ
2		sqrtcl 15318	. . . 4 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
3	2	ad2antrr 727	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) ∈ ℂ)
4		mulcl 11116	. . 3 ⊢ ((i ∈ ℂ ∧ (√‘𝐴) ∈ ℂ) → (i · (√‘𝐴)) ∈ ℂ)
5	1, 3, 4	sylancr 588	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℂ)
6		imval 15063	. . . 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 11579	. . . . . 6 ⊢ i ≠ 0
9		divcan3 11829	. . . . . 6 ⊢ (((√‘𝐴) ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → ((i · (√‘𝐴)) / i) = (√‘𝐴))
10	1, 8, 9	mp3an23 1456	. . . . 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 6839	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘((i · (√‘𝐴)) / i)) = (ℜ‘(√‘𝐴)))
13		halfre 12384	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ
14	13	recni 11153	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
15		logcl 26548	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
16		mulcl 11116	. . . . . . . . . . . 12 ⊢ (((1 / 2) ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → ((1 / 2) · (log‘𝐴)) ∈ ℂ)
17	14, 15, 16	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((1 / 2) · (log‘𝐴)) ∈ ℂ)
18	17	recld 15150	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘((1 / 2) · (log‘𝐴))) ∈ ℝ)
19	18	reefcld 16047	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
20	17	imcld 15151	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) ∈ ℝ)
21	20	recoscld 16105	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
22	18	rpefcld 16066	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ⁺)
23	22	rpge0d 12984	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))))
24		immul2 15093	. . . . . . . . . . . . 13 ⊢ (((1 / 2) ∈ ℝ ∧ (log‘𝐴) ∈ ℂ) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((1 / 2) · (ℑ‘(log‘𝐴))))
25	13, 15, 24	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((1 / 2) · (ℑ‘(log‘𝐴))))
26	15	imcld 15151	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
27	26	recnd 11167	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ)
28		mulcom 11118	. . . . . . . . . . . . 13 ⊢ (((1 / 2) ∈ ℂ ∧ (ℑ‘(log‘𝐴)) ∈ ℂ) → ((1 / 2) · (ℑ‘(log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
29	14, 27, 28	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((1 / 2) · (ℑ‘(log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
30	25, 29	eqtrd 2772	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
31		logimcl 26549	. . . . . . . . . . . . . . 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 26437	. . . . . . . . . . . . . . . 16 ⊢ π ∈ ℝ
34	33	renegcli 11449	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℝ
35		ltle 11228	. . . . . . . . . . . . . . 15 ⊢ ((-π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
36	34, 26, 35	sylancr 588	. . . . . . . . . . . . . 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 13359	. . . . . . . . . . . . 13 ⊢ ((ℑ‘(log‘𝐴)) ∈ (-π[,]π) ↔ ((ℑ‘(log‘𝐴)) ∈ ℝ ∧ -π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
40	26, 37, 38, 39	syl3anbrc 1345	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ (-π[,]π))
41		halfgt0 12386	. . . . . . . . . . . . . 14 ⊢ 0 < (1 / 2)
42	13, 41	elrpii 12939	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ⁺
43	33	recni 11153	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℂ
44		2cn 12250	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
45		2ne0 12279	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
46		divneg 11840	. . . . . . . . . . . . . . 15 ⊢ ((π ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(π / 2) = (-π / 2))
47	43, 44, 45, 46	mp3an 1464	. . . . . . . . . . . . . 14 ⊢ -(π / 2) = (-π / 2)
48	34	recni 11153	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℂ
49	48, 44, 45	divreci 11894	. . . . . . . . . . . . . 14 ⊢ (-π / 2) = (-π · (1 / 2))
50	47, 49	eqtr2i 2761	. . . . . . . . . . . . 13 ⊢ (-π · (1 / 2)) = -(π / 2)
51	43, 44, 45	divreci 11894	. . . . . . . . . . . . . 14 ⊢ (π / 2) = (π · (1 / 2))
52	51	eqcomi 2746	. . . . . . . . . . . . 13 ⊢ (π · (1 / 2)) = (π / 2)
53	34, 33, 42, 50, 52	iccdili 13438	. . . . . . . . . . . 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 2837	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) ∈ (-(π / 2)[,](π / 2)))
56		cosq14ge0 26491	. . . . . . . . . 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 11721	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ ((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · (cos‘(ℑ‘((1 / 2) · (log‘𝐴))))))
59		cxpef 26645	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
60	14, 59	mp3an3 1453	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
61		efeul 16123	. . . . . . . . . . . 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 2772	. . . . . . . . . 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 6839	. . . . . . . . 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 11167	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ)
66	20	resincld 16104	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
67	66	recnd 11167	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ)
68		mulcl 11116	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ) → (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴))))) ∈ ℂ)
69	1, 67, 68	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴))))) ∈ ℂ)
70	65, 69	addcld 11158	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))))) ∈ ℂ)
71	19, 70	remul2d 15183	. . . . . . . . 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 15187	. . . . . . . . . 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 7377	. . . . . . . . 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 2776	. . . . . . . 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 5114	. . . . . . 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 6839	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(𝐴↑_𝑐(1 / 2))) = (ℜ‘-(√‘𝐴)))
79	3	renegd 15165	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘-(√‘𝐴)) = -(ℜ‘(√‘𝐴)))
80	78, 79	eqtrd 2772	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(𝐴↑_𝑐(1 / 2))) = -(ℜ‘(√‘𝐴)))
81	76, 80	breqtrd 5112	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ -(ℜ‘(√‘𝐴)))
82	3	recld 15150	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(√‘𝐴)) ∈ ℝ)
83	82	le0neg1d 11715	. . . . 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 15322	. . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ≤ (ℜ‘(√‘𝐴)))
86	85	ad2antrr 727	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ (ℜ‘(√‘𝐴)))
87		0re 11140	. . . . 5 ⊢ 0 ∈ ℝ
88		letri3 11225	. . . . 5 ⊢ (((ℜ‘(√‘𝐴)) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘(√‘𝐴)) = 0 ↔ ((ℜ‘(√‘𝐴)) ≤ 0 ∧ 0 ≤ (ℜ‘(√‘𝐴)))))
89	82, 87, 88	sylancl 587	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((ℜ‘(√‘𝐴)) = 0 ↔ ((ℜ‘(√‘𝐴)) ≤ 0 ∧ 0 ≤ (ℜ‘(√‘𝐴)))))
90	84, 86, 89	mpbir2and 714	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(√‘𝐴)) = 0)
91	7, 12, 90	3eqtrd 2776	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℑ‘(i · (√‘𝐴))) = 0)
92	5, 91	reim0bd 15156	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 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5086 ‘cfv 6493 (class class class)co 7361 ℂcc 11030 ℝcr 11031 0cc0 11032 1c1 11033 ici 11034 + caddc 11035 · cmul 11037 < clt 11173 ≤ cle 11174 -cneg 11372 / cdiv 11801 2c2 12230 [,]cicc 13295 ℜcre 15053 ℑcim 15054 √csqrt 15189 expce 16020 sincsin 16022 cosccos 16023 πcpi 16025 logclog 26534 ↑_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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111

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 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13296 df-ioc 13297 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-mod 13823 df-seq 13958 df-exp 14018 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15023 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-limsup 15427 df-clim 15444 df-rlim 15445 df-sum 15643 df-ef 16026 df-sin 16028 df-cos 16029 df-pi 16031 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-rest 17379 df-topn 17380 df-0g 17398 df-gsum 17399 df-topgen 17400 df-pt 17401 df-prds 17404 df-xrs 17460 df-qtop 17465 df-imas 17466 df-xps 17468 df-mre 17542 df-mrc 17543 df-acs 17545 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-mulg 19038 df-cntz 19286 df-cmn 19751 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-cld 22997 df-ntr 22998 df-cls 22999 df-nei 23076 df-lp 23114 df-perf 23115 df-cn 23205 df-cnp 23206 df-haus 23293 df-tx 23540 df-hmeo 23733 df-fil 23824 df-fm 23916 df-flim 23917 df-flf 23918 df-xms 24298 df-ms 24299 df-tms 24300 df-cncf 24858 df-limc 25846 df-dv 25847 df-log 26536 df-cxp 26537

This theorem is referenced by: cxpsqrt 26683

W3C validator