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Theorem cxpsqrtlem 25762

Description: Lemma for cxpsqrt 25763. (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 10861	. . 3 ⊢ i ∈ ℂ
2		sqrtcl 15001	. . . 4 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
3	2	ad2antrr 722	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) ∈ ℂ)
4		mulcl 10886	. . 3 ⊢ ((i ∈ ℂ ∧ (√‘𝐴) ∈ ℂ) → (i · (√‘𝐴)) ∈ ℂ)
5	1, 3, 4	sylancr 586	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℂ)
6		imval 14746	. . . 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 11340	. . . . . 6 ⊢ i ≠ 0
9		divcan3 11589	. . . . . 6 ⊢ (((√‘𝐴) ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → ((i · (√‘𝐴)) / i) = (√‘𝐴))
10	1, 8, 9	mp3an23 1451	. . . . 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 6760	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘((i · (√‘𝐴)) / i)) = (ℜ‘(√‘𝐴)))
13		halfre 12117	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ
14	13	recni 10920	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
15		logcl 25629	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
16		mulcl 10886	. . . . . . . . . . . 12 ⊢ (((1 / 2) ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → ((1 / 2) · (log‘𝐴)) ∈ ℂ)
17	14, 15, 16	sylancr 586	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((1 / 2) · (log‘𝐴)) ∈ ℂ)
18	17	recld 14833	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘((1 / 2) · (log‘𝐴))) ∈ ℝ)
19	18	reefcld 15725	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
20	17	imcld 14834	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) ∈ ℝ)
21	20	recoscld 15781	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
22	18	rpefcld 15742	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ⁺)
23	22	rpge0d 12705	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))))
24		immul2 14776	. . . . . . . . . . . . 13 ⊢ (((1 / 2) ∈ ℝ ∧ (log‘𝐴) ∈ ℂ) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((1 / 2) · (ℑ‘(log‘𝐴))))
25	13, 15, 24	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((1 / 2) · (ℑ‘(log‘𝐴))))
26	15	imcld 14834	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
27	26	recnd 10934	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ)
28		mulcom 10888	. . . . . . . . . . . . 13 ⊢ (((1 / 2) ∈ ℂ ∧ (ℑ‘(log‘𝐴)) ∈ ℂ) → ((1 / 2) · (ℑ‘(log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
29	14, 27, 28	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((1 / 2) · (ℑ‘(log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
30	25, 29	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
31		logimcl 25630	. . . . . . . . . . . . . . 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 25520	. . . . . . . . . . . . . . . 16 ⊢ π ∈ ℝ
34	33	renegcli 11212	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℝ
35		ltle 10994	. . . . . . . . . . . . . . 15 ⊢ ((-π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
36	34, 26, 35	sylancr 586	. . . . . . . . . . . . . 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 13074	. . . . . . . . . . . . 13 ⊢ ((ℑ‘(log‘𝐴)) ∈ (-π[,]π) ↔ ((ℑ‘(log‘𝐴)) ∈ ℝ ∧ -π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
40	26, 37, 38, 39	syl3anbrc 1341	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ (-π[,]π))
41		halfgt0 12119	. . . . . . . . . . . . . 14 ⊢ 0 < (1 / 2)
42	13, 41	elrpii 12662	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ⁺
43	33	recni 10920	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℂ
44		2cn 11978	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
45		2ne0 12007	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
46		divneg 11597	. . . . . . . . . . . . . . 15 ⊢ ((π ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(π / 2) = (-π / 2))
47	43, 44, 45, 46	mp3an 1459	. . . . . . . . . . . . . 14 ⊢ -(π / 2) = (-π / 2)
48	34	recni 10920	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℂ
49	48, 44, 45	divreci 11650	. . . . . . . . . . . . . 14 ⊢ (-π / 2) = (-π · (1 / 2))
50	47, 49	eqtr2i 2767	. . . . . . . . . . . . 13 ⊢ (-π · (1 / 2)) = -(π / 2)
51	43, 44, 45	divreci 11650	. . . . . . . . . . . . . 14 ⊢ (π / 2) = (π · (1 / 2))
52	51	eqcomi 2747	. . . . . . . . . . . . 13 ⊢ (π · (1 / 2)) = (π / 2)
53	34, 33, 42, 50, 52	iccdili 13152	. . . . . . . . . . . 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 2839	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) ∈ (-(π / 2)[,](π / 2)))
56		cosq14ge0 25573	. . . . . . . . . 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 11482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ ((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · (cos‘(ℑ‘((1 / 2) · (log‘𝐴))))))
59		cxpef 25725	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
60	14, 59	mp3an3 1448	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
61		efeul 15799	. . . . . . . . . . . 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 2778	. . . . . . . . . 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 6760	. . . . . . . . 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 10934	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ)
66	20	resincld 15780	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
67	66	recnd 10934	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ)
68		mulcl 10886	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ) → (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴))))) ∈ ℂ)
69	1, 67, 68	sylancr 586	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴))))) ∈ ℂ)
70	65, 69	addcld 10925	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))))) ∈ ℂ)
71	19, 70	remul2d 14866	. . . . . . . . 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 14870	. . . . . . . . . 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 7271	. . . . . . . . 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 2782	. . . . . . . 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 5098	. . . . . . 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 6760	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(𝐴↑_𝑐(1 / 2))) = (ℜ‘-(√‘𝐴)))
79	3	renegd 14848	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘-(√‘𝐴)) = -(ℜ‘(√‘𝐴)))
80	78, 79	eqtrd 2778	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(𝐴↑_𝑐(1 / 2))) = -(ℜ‘(√‘𝐴)))
81	76, 80	breqtrd 5096	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ -(ℜ‘(√‘𝐴)))
82	3	recld 14833	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(√‘𝐴)) ∈ ℝ)
83	82	le0neg1d 11476	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((ℜ‘(√‘𝐴)) ≤ 0 ↔ 0 ≤ -(ℜ‘(√‘𝐴))))
84	81, 83	mpbird 256	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(√‘𝐴)) ≤ 0)
85		sqrtrege0 15005	. . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ≤ (ℜ‘(√‘𝐴)))
86	85	ad2antrr 722	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ (ℜ‘(√‘𝐴)))
87		0re 10908	. . . . 5 ⊢ 0 ∈ ℝ
88		letri3 10991	. . . . 5 ⊢ (((ℜ‘(√‘𝐴)) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘(√‘𝐴)) = 0 ↔ ((ℜ‘(√‘𝐴)) ≤ 0 ∧ 0 ≤ (ℜ‘(√‘𝐴)))))
89	82, 87, 88	sylancl 585	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((ℜ‘(√‘𝐴)) = 0 ↔ ((ℜ‘(√‘𝐴)) ≤ 0 ∧ 0 ≤ (ℜ‘(√‘𝐴)))))
90	84, 86, 89	mpbir2and 709	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(√‘𝐴)) = 0)
91	7, 12, 90	3eqtrd 2782	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℑ‘(i · (√‘𝐴))) = 0)
92	5, 91	reim0bd 14839	1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 ici 10804 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 -cneg 11136 / cdiv 11562 2c2 11958 [,]cicc 13011 ℜcre 14736 ℑcim 14737 √csqrt 14872 expce 15699 sincsin 15701 cosccos 15702 πcpi 15704 logclog 25615 ↑_𝑐ccxp 25616

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-cxp 25618

This theorem is referenced by: cxpsqrt 25763

W3C validator