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Theorem cxpsqrtlem 26940

Description: Lemma for cxpsqrt 26941. (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 11184	. . 3 ⊢ i ∈ ℂ
2		sqrtcl 15450	. . . 4 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
3	2	ad2antrr 739	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) ∈ ℂ)
4		mulcl 11209	. . 3 ⊢ ((i ∈ ℂ ∧ (√‘𝐴) ∈ ℂ) → (i · (√‘𝐴)) ∈ ℂ)
5	1, 3, 4	sylancr 599	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℂ)
6		imval 15195	. . . 4 ⊢ ((i · (√‘𝐴)) ∈ ℂ → (ℑ‘(i · (√‘𝐴))) = (ℜ‘((i · (√‘𝐴)) / i)))
7	5, 6	syl 18	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℑ‘(i · (√‘𝐴))) = (ℜ‘((i · (√‘𝐴)) / i)))
8		ine0 11674	. . . . . 6 ⊢ i ≠ 0
9		divcan3 11923	. . . . . 6 ⊢ (((√‘𝐴) ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → ((i · (√‘𝐴)) / i) = (√‘𝐴))
10	1, 8, 9	mp3an23 1482	. . . . 5 ⊢ ((√‘𝐴) ∈ ℂ → ((i · (√‘𝐴)) / i) = (√‘𝐴))
11	3, 10	syl 18	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((i · (√‘𝐴)) / i) = (√‘𝐴))
12	11	fveq2d 6883	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘((i · (√‘𝐴)) / i)) = (ℜ‘(√‘𝐴)))
13		halfre 12482	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ
14	13	recni 11248	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
15		logcl 26806	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
16		mulcl 11209	. . . . . . . . . . . 12 ⊢ (((1 / 2) ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → ((1 / 2) · (log‘𝐴)) ∈ ℂ)
17	14, 15, 16	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((1 / 2) · (log‘𝐴)) ∈ ℂ)
18	17	recld 15282	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘((1 / 2) · (log‘𝐴))) ∈ ℝ)
19	18	reefcld 16175	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
20	17	imcld 15283	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) ∈ ℝ)
21	20	recoscld 16233	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
22	18	rpefcld 16194	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ⁺)
23	22	rpge0d 13091	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))))
24		immul2 15225	. . . . . . . . . . . . 13 ⊢ (((1 / 2) ∈ ℝ ∧ (log‘𝐴) ∈ ℂ) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((1 / 2) · (ℑ‘(log‘𝐴))))
25	13, 15, 24	sylancr 599	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((1 / 2) · (ℑ‘(log‘𝐴))))
26	15	imcld 15283	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
27	26	recnd 11262	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ)
28		mulcom 11211	. . . . . . . . . . . . 13 ⊢ (((1 / 2) ∈ ℂ ∧ (ℑ‘(log‘𝐴)) ∈ ℂ) → ((1 / 2) · (ℑ‘(log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
29	14, 27, 28	sylancr 599	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((1 / 2) · (ℑ‘(log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
30	25, 29	eqtrd 2795	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
31		logimcl 26807	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
32	31	simpld 500	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -π < (ℑ‘(log‘𝐴)))
33		pire 26693	. . . . . . . . . . . . . . . 16 ⊢ π ∈ ℝ
34	33	renegcli 11544	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℝ
35		ltle 11323	. . . . . . . . . . . . . . 15 ⊢ ((-π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
36	34, 26, 35	sylancr 599	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
37	32, 36	mpd 16	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -π ≤ (ℑ‘(log‘𝐴)))
38	31	simprd 501	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ≤ π)
39	34, 33	elicc2i 13466	. . . . . . . . . . . . 13 ⊢ ((ℑ‘(log‘𝐴)) ∈ (-π[,]π) ↔ ((ℑ‘(log‘𝐴)) ∈ ℝ ∧ -π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
40	26, 37, 38, 39	syl3anbrc 1362	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ (-π[,]π))
41		halfgt0 12484	. . . . . . . . . . . . . 14 ⊢ 0 < (1 / 2)
42	13, 41	elrpii 13046	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ⁺
43	33	recni 11248	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℂ
44		2cn 12341	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
45		2ne0 12372	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
46		divneg 11931	. . . . . . . . . . . . . . 15 ⊢ ((π ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(π / 2) = (-π / 2))
47	43, 44, 45, 46	mp3an 1490	. . . . . . . . . . . . . 14 ⊢ -(π / 2) = (-π / 2)
48	34	recni 11248	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℂ
49	48, 44, 45	divreci 11985	. . . . . . . . . . . . . 14 ⊢ (-π / 2) = (-π · (1 / 2))
50	47, 49	eqtr2i 2784	. . . . . . . . . . . . 13 ⊢ (-π · (1 / 2)) = -(π / 2)
51	43, 44, 45	divreci 11985	. . . . . . . . . . . . . 14 ⊢ (π / 2) = (π · (1 / 2))
52	51	eqcomi 2769	. . . . . . . . . . . . 13 ⊢ (π · (1 / 2)) = (π / 2)
53	34, 33, 42, 50, 52	iccdili 13545	. . . . . . . . . . . 12 ⊢ ((ℑ‘(log‘𝐴)) ∈ (-π[,]π) → ((ℑ‘(log‘𝐴)) · (1 / 2)) ∈ (-(π / 2)[,](π / 2)))
54	40, 53	syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((ℑ‘(log‘𝐴)) · (1 / 2)) ∈ (-(π / 2)[,](π / 2)))
55	30, 54	eqeltrd 2860	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) ∈ (-(π / 2)[,](π / 2)))
56		cosq14ge0 26750	. . . . . . . . . 10 ⊢ ((ℑ‘((1 / 2) · (log‘𝐴))) ∈ (-(π / 2)[,](π / 2)) → 0 ≤ (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))))
57	55, 56	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))))
58	19, 21, 23, 57	mulge0d 11816	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ ((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · (cos‘(ℑ‘((1 / 2) · (log‘𝐴))))))
59		cxpef 26903	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
60	14, 59	mp3an3 1479	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
61		efeul 16251	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 2795	. . . . . . . . . 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 6883	. . . . . . . . 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 11262	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ)
66	20	resincld 16232	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
67	66	recnd 11262	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ)
68		mulcl 11209	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ) → (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴))))) ∈ ℂ)
69	1, 67, 68	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴))))) ∈ ℂ)
70	65, 69	addcld 11253	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))))) ∈ ℂ)
71	19, 70	remul2d 15315	. . . . . . . . 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 15319	. . . . . . . . . 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 7430	. . . . . . . . 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 2799	. . . . . . . 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 5133	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ (ℜ‘(𝐴↑_𝑐(1 / 2))))
76	75	adantr 486	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ (ℜ‘(𝐴↑_𝑐(1 / 2))))
77		simpr 490	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴))
78	77	fveq2d 6883	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(𝐴↑_𝑐(1 / 2))) = (ℜ‘-(√‘𝐴)))
79	3	renegd 15297	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘-(√‘𝐴)) = -(ℜ‘(√‘𝐴)))
80	78, 79	eqtrd 2795	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(𝐴↑_𝑐(1 / 2))) = -(ℜ‘(√‘𝐴)))
81	76, 80	breqtrd 5131	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ -(ℜ‘(√‘𝐴)))
82	3	recld 15282	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(√‘𝐴)) ∈ ℝ)
83	82	le0neg1d 11810	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((ℜ‘(√‘𝐴)) ≤ 0 ↔ 0 ≤ -(ℜ‘(√‘𝐴))))
84	81, 83	mpbird 260	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(√‘𝐴)) ≤ 0)
85		sqrtrege0 15454	. . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ≤ (ℜ‘(√‘𝐴)))
86	85	ad2antrr 739	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ (ℜ‘(√‘𝐴)))
87		0re 11235	. . . . 5 ⊢ 0 ∈ ℝ
88		letri3 11320	. . . . 5 ⊢ (((ℜ‘(√‘𝐴)) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘(√‘𝐴)) = 0 ↔ ((ℜ‘(√‘𝐴)) ≤ 0 ∧ 0 ≤ (ℜ‘(√‘𝐴)))))
89	82, 87, 88	sylancl 598	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → ((ℜ‘(√‘𝐴)) = 0 ↔ ((ℜ‘(√‘𝐴)) ≤ 0 ∧ 0 ≤ (ℜ‘(√‘𝐴)))))
90	84, 86, 89	mpbir2and 726	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(√‘𝐴)) = 0)
91	7, 12, 90	3eqtrd 2799	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℑ‘(i · (√‘𝐴))) = 0)
92	5, 91	reim0bd 15288	1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℝ)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 class class class wbr 5103 ‘cfv 6533 (class class class)co 7414 ℂcc 11123 ℝcr 11124 0cc0 11125 1c1 11126 ici 11127 + caddc 11128 · cmul 11130 < clt 11268 ≤ cle 11269 -cneg 11467 / cdiv 11896 2c2 12320 [,]cicc 13402 ℜcre 15185 ℑcim 15186 √csqrt 15321 expce 16148 sincsin 16150 cosccos 16151 πcpi 16153 logclog 26792 ↑_𝑐ccxp 26793

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9621 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9333 df-fi 9382 df-sup 9413 df-inf 9414 df-oi 9483 df-card 9945 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13044 df-xneg 13164 df-xadd 13165 df-xmul 13166 df-ioo 13403 df-ioc 13404 df-ico 13405 df-icc 13406 df-fz 13563 df-fzo 13711 df-fl 13854 df-mod 13932 df-seq 14067 df-exp 14127 df-fac 14339 df-bc 14368 df-hash 14396 df-shft 15141 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-limsup 15559 df-clim 15576 df-rlim 15577 df-sum 15775 df-ef 16154 df-sin 16156 df-cos 16157 df-pi 16159 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-mulr 17357 df-starv 17358 df-sca 17359 df-vsca 17360 df-ip 17361 df-tset 17362 df-ple 17363 df-ds 17365 df-unif 17366 df-hom 17367 df-cco 17368 df-rest 17508 df-topn 17509 df-0g 17527 df-gsum 17528 df-topgen 17529 df-pt 17530 df-prds 17533 df-xrs 17589 df-qtop 17594 df-imas 17595 df-xps 17597 df-mre 17671 df-mrc 17672 df-acs 17674 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-submnd 18893 df-mulg 19192 df-cntz 19445 df-cmn 19910 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-fbas 21583 df-fg 21584 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-nei 23324 df-lp 23362 df-perf 23363 df-cn 23453 df-cnp 23454 df-haus 23541 df-tx 23789 df-hmeo 23982 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-xms 24547 df-ms 24548 df-tms 24549 df-cncf 25107 df-limc 26094 df-dv 26095 df-log 26794 df-cxp 26795

This theorem is used by: cxpsqrt 26941

W3C validator