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Theorem cxpsqrtlem 27087

Description: Lemma for cxpsqrt 27088. (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 11298	. . 3 ⊢ i ∈ ℂ
2		sqrtcl 15573	. . . 4 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
3	2	ad2antrr 739	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (√‘𝐴) ∈ ℂ)
4		mulcl 11323	. . 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 15317	. . . 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 11790	. . . . . 6 ⊢ i ≠ 0
9		divcan3 12040	. . . . . 6 ⊢ (((√‘𝐴) ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → ((i · (√‘𝐴)) / i) = (√‘𝐴))
10	1, 8, 9	mp3an23 1483	. . . . 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 6920	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘((i · (√‘𝐴)) / i)) = (ℜ‘(√‘𝐴)))
13		halfre 12599	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ
14	13	recni 11362	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℂ
15		logcl 26953	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
16		mulcl 11323	. . . . . . . . . . . 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 15404	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘((1 / 2) · (log‘𝐴))) ∈ ℝ)
19	18	reefcld 16298	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
20	17	imcld 15405	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) ∈ ℝ)
21	20	recoscld 16356	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
22	18	rpefcld 16317	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ⁺)
23	22	rpge0d 13208	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ (exp‘(ℜ‘((1 / 2) · (log‘𝐴)))))
24		immul2 15347	. . . . . . . . . . . . 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 15405	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
27	26	recnd 11376	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ)
28		mulcom 11325	. . . . . . . . . . . . 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 2798	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) = ((ℑ‘(log‘𝐴)) · (1 / 2)))
31		logimcl 26954	. . . . . . . . . . . . . . 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 26840	. . . . . . . . . . . . . . . 16 ⊢ π ∈ ℝ
34	33	renegcli 11658	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℝ
35		ltle 11437	. . . . . . . . . . . . . . 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 13583	. . . . . . . . . . . . 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 12601	. . . . . . . . . . . . . 14 ⊢ 0 < (1 / 2)
42	13, 41	elrpii 13163	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ⁺
43	33	recni 11362	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℂ
44		2cn 12458	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
45		2ne0 12489	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
46		divneg 12048	. . . . . . . . . . . . . . 15 ⊢ ((π ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(π / 2) = ( -π / 2))
47	43, 44, 45, 46	mp3an 1491	. . . . . . . . . . . . . 14 ⊢ -(π / 2) = ( -π / 2)
48	34	recni 11362	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℂ
49	48, 44, 45	divreci 12102	. . . . . . . . . . . . . 14 ⊢ ( -π / 2) = ( -π · (1 / 2))
50	47, 49	eqtr2i 2787	. . . . . . . . . . . . 13 ⊢ ( -π · (1 / 2)) = -(π / 2)
51	43, 44, 45	divreci 12102	. . . . . . . . . . . . . 14 ⊢ (π / 2) = (π · (1 / 2))
52	51	eqcomi 2772	. . . . . . . . . . . . 13 ⊢ (π · (1 / 2)) = (π / 2)
53	34, 33, 42, 50, 52	iccdili 13662	. . . . . . . . . . . 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 2864	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘((1 / 2) · (log‘𝐴))) ∈ ( -(π / 2)[,](π / 2)))
56		cosq14ge0 26897	. . . . . . . . . 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 11933	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 ≤ ((exp‘(ℜ‘((1 / 2) · (log‘𝐴)))) · (cos‘(ℑ‘((1 / 2) · (log‘𝐴))))))
59		cxpef 27050	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (1 / 2) ∈ ℂ) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
60	14, 59	mp3an3 1480	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘𝐴))))
61		efeul 16374	. . . . . . . . . . . 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 2798	. . . . . . . . . 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 6920	. . . . . . . . 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 11376	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ)
66	20	resincld 16355	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℝ)
67	66	recnd 11376	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))) ∈ ℂ)
68		mulcl 11323	. . . . . . . . . . . 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 11367	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((cos‘(ℑ‘((1 / 2) · (log‘𝐴)))) + (i · (sin‘(ℑ‘((1 / 2) · (log‘𝐴)))))) ∈ ℂ)
71	19, 70	remul2d 15437	. . . . . . . . 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 15441	. . . . . . . . . 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 7467	. . . . . . . . 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 2802	. . . . . . . 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 5139	. . . . . . 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 6920	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(𝐴↑_𝑐(1 / 2))) = (ℜ‘ -(√‘𝐴)))
79	3	renegd 15419	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘ -(√‘𝐴)) = -(ℜ‘(√‘𝐴)))
80	78, 79	eqtrd 2798	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(𝐴↑_𝑐(1 / 2))) = -(ℜ‘(√‘𝐴)))
81	76, 80	breqtrd 5137	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → 0 ≤ -(ℜ‘(√‘𝐴)))
82	3	recld 15404	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℜ‘(√‘𝐴)) ∈ ℝ)
83	82	le0neg1d 11927	. . . . 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 15577	. . . . 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 11349	. . . . 5 ⊢ 0 ∈ ℝ
88		letri3 11434	. . . . 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 2802	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑_𝑐(1 / 2)) = -(√‘𝐴)) → (ℑ‘(i · (√‘𝐴))) = 0)
92	5, 91	reim0bd 15410	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 2147 ≠ wne 2959 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 ici 11241 + caddc 11242 · cmul 11244 < clt 11382 ≤ cle 11383 -cneg 11581 / cdiv 12013 2c2 12437 [,]cicc 13519 ℜcre 15307 ℑcim 15308 √csqrt 15443 expce 16271 sincsin 16273 cosccos 16274 πcpi 16276 logclog 26939 ↑_𝑐ccxp 26940

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-sum 15898 df-ef 16277 df-sin 16279 df-cos 16280 df-pi 16282 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242 df-log 26941 df-cxp 26942

This theorem is used by: cxpsqrt 27088

W3C validator