Theorem resinf1o 26443

Description: The sine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)

Assertion

Ref	Expression
resinf1o	⊢ (sin ↾ (-(π / 2)[,](π / 2))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1)

Proof of Theorem resinf1o

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		recosf1o 26442	. . 3 ⊢ (cos ↾ (0[,]π)):(0[,]π)–1-1-onto→(-1[,]1)
2		eqid 2729	. . . . 5 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥)) = (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))
3		halfpire 26371	. . . . . . . 8 ⊢ (π / 2) ∈ ℝ
4		neghalfpire 26372	. . . . . . . . . 10 ⊢ -(π / 2) ∈ ℝ
5		iccssre 13332	. . . . . . . . . 10 ⊢ ((-(π / 2) ∈ ℝ ∧ (π / 2) ∈ ℝ) → (-(π / 2)[,](π / 2)) ⊆ ℝ)
6	4, 3, 5	mp2an 692	. . . . . . . . 9 ⊢ (-(π / 2)[,](π / 2)) ⊆ ℝ
7	6	sseli 3931	. . . . . . . 8 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) → 𝑥 ∈ ℝ)
8		resubcl 11428	. . . . . . . 8 ⊢ (((π / 2) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((π / 2) − 𝑥) ∈ ℝ)
9	3, 7, 8	sylancr 587	. . . . . . 7 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) → ((π / 2) − 𝑥) ∈ ℝ)
10	4, 3	elicc2i 13315	. . . . . . . . 9 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↔ (𝑥 ∈ ℝ ∧ -(π / 2) ≤ 𝑥 ∧ 𝑥 ≤ (π / 2)))
11	10	simp3bi 1147	. . . . . . . 8 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) → 𝑥 ≤ (π / 2))
12		subge0 11633	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (0 ≤ ((π / 2) − 𝑥) ↔ 𝑥 ≤ (π / 2)))
13	3, 7, 12	sylancr 587	. . . . . . . 8 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) → (0 ≤ ((π / 2) − 𝑥) ↔ 𝑥 ≤ (π / 2)))
14	11, 13	mpbird 257	. . . . . . 7 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) → 0 ≤ ((π / 2) − 𝑥))
15	3	recni 11129	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ
16		picn 26365	. . . . . . . . . 10 ⊢ π ∈ ℂ
17	15	negcli 11432	. . . . . . . . . 10 ⊢ -(π / 2) ∈ ℂ
18	16, 15	negsubi 11442	. . . . . . . . . . 11 ⊢ (π + -(π / 2)) = (π − (π / 2))
19		pidiv2halves 26374	. . . . . . . . . . . 12 ⊢ ((π / 2) + (π / 2)) = π
20	16, 15, 15, 19	subaddrii 11453	. . . . . . . . . . 11 ⊢ (π − (π / 2)) = (π / 2)
21	18, 20	eqtri 2752	. . . . . . . . . 10 ⊢ (π + -(π / 2)) = (π / 2)
22	15, 16, 17, 21	subaddrii 11453	. . . . . . . . 9 ⊢ ((π / 2) − π) = -(π / 2)
23	10	simp2bi 1146	. . . . . . . . 9 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) → -(π / 2) ≤ 𝑥)
24	22, 23	eqbrtrid 5127	. . . . . . . 8 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) → ((π / 2) − π) ≤ 𝑥)
25		pire 26364	. . . . . . . . 9 ⊢ π ∈ ℝ
26		suble 11598	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℝ ∧ π ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((π / 2) − π) ≤ 𝑥 ↔ ((π / 2) − 𝑥) ≤ π))
27	3, 25, 7, 26	mp3an12i 1467	. . . . . . . 8 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) → (((π / 2) − π) ≤ 𝑥 ↔ ((π / 2) − 𝑥) ≤ π))
28	24, 27	mpbid 232	. . . . . . 7 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) → ((π / 2) − 𝑥) ≤ π)
29		0re 11117	. . . . . . . 8 ⊢ 0 ∈ ℝ
30	29, 25	elicc2i 13315	. . . . . . 7 ⊢ (((π / 2) − 𝑥) ∈ (0[,]π) ↔ (((π / 2) − 𝑥) ∈ ℝ ∧ 0 ≤ ((π / 2) − 𝑥) ∧ ((π / 2) − 𝑥) ≤ π))
31	9, 14, 28, 30	syl3anbrc 1344	. . . . . 6 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) → ((π / 2) − 𝑥) ∈ (0[,]π))
32	31	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (-(π / 2)[,](π / 2))) → ((π / 2) − 𝑥) ∈ (0[,]π))
33	29, 25	elicc2i 13315	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]π) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ∧ 𝑦 ≤ π))
34	33	simp1bi 1145	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]π) → 𝑦 ∈ ℝ)
35		resubcl 11428	. . . . . . . 8 ⊢ (((π / 2) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((π / 2) − 𝑦) ∈ ℝ)
36	3, 34, 35	sylancr 587	. . . . . . 7 ⊢ (𝑦 ∈ (0[,]π) → ((π / 2) − 𝑦) ∈ ℝ)
37	33	simp3bi 1147	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]π) → 𝑦 ≤ π)
38	15, 15	subnegi 11443	. . . . . . . . . 10 ⊢ ((π / 2) − -(π / 2)) = ((π / 2) + (π / 2))
39	38, 19	eqtri 2752	. . . . . . . . 9 ⊢ ((π / 2) − -(π / 2)) = π
40	37, 39	breqtrrdi 5134	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]π) → 𝑦 ≤ ((π / 2) − -(π / 2)))
41		lesub 11599	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ -(π / 2) ∈ ℝ) → (𝑦 ≤ ((π / 2) − -(π / 2)) ↔ -(π / 2) ≤ ((π / 2) − 𝑦)))
42	3, 4, 41	mp3an23 1455	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (𝑦 ≤ ((π / 2) − -(π / 2)) ↔ -(π / 2) ≤ ((π / 2) − 𝑦)))
43	34, 42	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]π) → (𝑦 ≤ ((π / 2) − -(π / 2)) ↔ -(π / 2) ≤ ((π / 2) − 𝑦)))
44	40, 43	mpbid 232	. . . . . . 7 ⊢ (𝑦 ∈ (0[,]π) → -(π / 2) ≤ ((π / 2) − 𝑦))
45	15	subidi 11435	. . . . . . . . 9 ⊢ ((π / 2) − (π / 2)) = 0
46	33	simp2bi 1146	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]π) → 0 ≤ 𝑦)
47	45, 46	eqbrtrid 5127	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]π) → ((π / 2) − (π / 2)) ≤ 𝑦)
48		suble 11598	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((π / 2) − (π / 2)) ≤ 𝑦 ↔ ((π / 2) − 𝑦) ≤ (π / 2)))
49	3, 3, 34, 48	mp3an12i 1467	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]π) → (((π / 2) − (π / 2)) ≤ 𝑦 ↔ ((π / 2) − 𝑦) ≤ (π / 2)))
50	47, 49	mpbid 232	. . . . . . 7 ⊢ (𝑦 ∈ (0[,]π) → ((π / 2) − 𝑦) ≤ (π / 2))
51	4, 3	elicc2i 13315	. . . . . . 7 ⊢ (((π / 2) − 𝑦) ∈ (-(π / 2)[,](π / 2)) ↔ (((π / 2) − 𝑦) ∈ ℝ ∧ -(π / 2) ≤ ((π / 2) − 𝑦) ∧ ((π / 2) − 𝑦) ≤ (π / 2)))
52	36, 44, 50, 51	syl3anbrc 1344	. . . . . 6 ⊢ (𝑦 ∈ (0[,]π) → ((π / 2) − 𝑦) ∈ (-(π / 2)[,](π / 2)))
53	52	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (0[,]π)) → ((π / 2) − 𝑦) ∈ (-(π / 2)[,](π / 2)))
54		iccssre 13332	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
55	29, 25, 54	mp2an 692	. . . . . . . . . 10 ⊢ (0[,]π) ⊆ ℝ
56		ax-resscn 11066	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
57	55, 56	sstri 3945	. . . . . . . . 9 ⊢ (0[,]π) ⊆ ℂ
58	57	sseli 3931	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]π) → 𝑦 ∈ ℂ)
59	6, 56	sstri 3945	. . . . . . . . 9 ⊢ (-(π / 2)[,](π / 2)) ⊆ ℂ
60	59	sseli 3931	. . . . . . . 8 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) → 𝑥 ∈ ℂ)
61		subsub23 11368	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (((π / 2) − 𝑦) = 𝑥 ↔ ((π / 2) − 𝑥) = 𝑦))
62	15, 61	mp3an1 1450	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (((π / 2) − 𝑦) = 𝑥 ↔ ((π / 2) − 𝑥) = 𝑦))
63	58, 60, 62	syl2anr 597	. . . . . . 7 ⊢ ((𝑥 ∈ (-(π / 2)[,](π / 2)) ∧ 𝑦 ∈ (0[,]π)) → (((π / 2) − 𝑦) = 𝑥 ↔ ((π / 2) − 𝑥) = 𝑦))
64	63	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ (-(π / 2)[,](π / 2)) ∧ 𝑦 ∈ (0[,]π))) → (((π / 2) − 𝑦) = 𝑥 ↔ ((π / 2) − 𝑥) = 𝑦))
65		eqcom 2736	. . . . . 6 ⊢ (𝑥 = ((π / 2) − 𝑦) ↔ ((π / 2) − 𝑦) = 𝑥)
66		eqcom 2736	. . . . . 6 ⊢ (𝑦 = ((π / 2) − 𝑥) ↔ ((π / 2) − 𝑥) = 𝑦)
67	64, 65, 66	3bitr4g 314	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ (-(π / 2)[,](π / 2)) ∧ 𝑦 ∈ (0[,]π))) → (𝑥 = ((π / 2) − 𝑦) ↔ 𝑦 = ((π / 2) − 𝑥)))
68	2, 32, 53, 67	f1o2d 7603	. . . 4 ⊢ (⊤ → (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥)):(-(π / 2)[,](π / 2))–1-1-onto→(0[,]π))
69	68	mptru 1547	. . 3 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥)):(-(π / 2)[,](π / 2))–1-1-onto→(0[,]π)
70		f1oco 6787	. . 3 ⊢ (((cos ↾ (0[,]π)):(0[,]π)–1-1-onto→(-1[,]1) ∧ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥)):(-(π / 2)[,](π / 2))–1-1-onto→(0[,]π)) → ((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1))
71	1, 69, 70	mp2an 692	. 2 ⊢ ((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1)
72		cosf 16034	. . . . . . . 8 ⊢ cos:ℂ⟶ℂ
73		ffn 6652	. . . . . . . 8 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ)
74	72, 73	ax-mp 5	. . . . . . 7 ⊢ cos Fn ℂ
75		fnssres 6605	. . . . . . 7 ⊢ ((cos Fn ℂ ∧ (0[,]π) ⊆ ℂ) → (cos ↾ (0[,]π)) Fn (0[,]π))
76	74, 57, 75	mp2an 692	. . . . . 6 ⊢ (cos ↾ (0[,]π)) Fn (0[,]π)
77	2, 31	fmpti 7046	. . . . . 6 ⊢ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥)):(-(π / 2)[,](π / 2))⟶(0[,]π)
78		fnfco 6689	. . . . . 6 ⊢ (((cos ↾ (0[,]π)) Fn (0[,]π) ∧ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥)):(-(π / 2)[,](π / 2))⟶(0[,]π)) → ((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))) Fn (-(π / 2)[,](π / 2)))
79	76, 77, 78	mp2an 692	. . . . 5 ⊢ ((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))) Fn (-(π / 2)[,](π / 2))
80		sinf 16033	. . . . . . 7 ⊢ sin:ℂ⟶ℂ
81		ffn 6652	. . . . . . 7 ⊢ (sin:ℂ⟶ℂ → sin Fn ℂ)
82	80, 81	ax-mp 5	. . . . . 6 ⊢ sin Fn ℂ
83		fnssres 6605	. . . . . 6 ⊢ ((sin Fn ℂ ∧ (-(π / 2)[,](π / 2)) ⊆ ℂ) → (sin ↾ (-(π / 2)[,](π / 2))) Fn (-(π / 2)[,](π / 2)))
84	82, 59, 83	mp2an 692	. . . . 5 ⊢ (sin ↾ (-(π / 2)[,](π / 2))) Fn (-(π / 2)[,](π / 2))
85		eqfnfv 6965	. . . . 5 ⊢ ((((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))) Fn (-(π / 2)[,](π / 2)) ∧ (sin ↾ (-(π / 2)[,](π / 2))) Fn (-(π / 2)[,](π / 2))) → (((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))) = (sin ↾ (-(π / 2)[,](π / 2))) ↔ ∀𝑦 ∈ (-(π / 2)[,](π / 2))(((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥)))‘𝑦) = ((sin ↾ (-(π / 2)[,](π / 2)))‘𝑦)))
86	79, 84, 85	mp2an 692	. . . 4 ⊢ (((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))) = (sin ↾ (-(π / 2)[,](π / 2))) ↔ ∀𝑦 ∈ (-(π / 2)[,](π / 2))(((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥)))‘𝑦) = ((sin ↾ (-(π / 2)[,](π / 2)))‘𝑦))
87	77	ffvelcdmi 7017	. . . . . . 7 ⊢ (𝑦 ∈ (-(π / 2)[,](π / 2)) → ((𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))‘𝑦) ∈ (0[,]π))
88	87	fvresd 6842	. . . . . 6 ⊢ (𝑦 ∈ (-(π / 2)[,](π / 2)) → ((cos ↾ (0[,]π))‘((𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))‘𝑦)) = (cos‘((𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))‘𝑦)))
89		oveq2 7357	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((π / 2) − 𝑥) = ((π / 2) − 𝑦))
90		ovex 7382	. . . . . . . 8 ⊢ ((π / 2) − 𝑦) ∈ V
91	89, 2, 90	fvmpt 6930	. . . . . . 7 ⊢ (𝑦 ∈ (-(π / 2)[,](π / 2)) → ((𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))‘𝑦) = ((π / 2) − 𝑦))
92	91	fveq2d 6826	. . . . . 6 ⊢ (𝑦 ∈ (-(π / 2)[,](π / 2)) → (cos‘((𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))‘𝑦)) = (cos‘((π / 2) − 𝑦)))
93	59	sseli 3931	. . . . . . 7 ⊢ (𝑦 ∈ (-(π / 2)[,](π / 2)) → 𝑦 ∈ ℂ)
94		coshalfpim 26402	. . . . . . 7 ⊢ (𝑦 ∈ ℂ → (cos‘((π / 2) − 𝑦)) = (sin‘𝑦))
95	93, 94	syl 17	. . . . . 6 ⊢ (𝑦 ∈ (-(π / 2)[,](π / 2)) → (cos‘((π / 2) − 𝑦)) = (sin‘𝑦))
96	88, 92, 95	3eqtrd 2768	. . . . 5 ⊢ (𝑦 ∈ (-(π / 2)[,](π / 2)) → ((cos ↾ (0[,]π))‘((𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))‘𝑦)) = (sin‘𝑦))
97		fvco3 6922	. . . . . 6 ⊢ (((𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥)):(-(π / 2)[,](π / 2))⟶(0[,]π) ∧ 𝑦 ∈ (-(π / 2)[,](π / 2))) → (((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥)))‘𝑦) = ((cos ↾ (0[,]π))‘((𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))‘𝑦)))
98	77, 97	mpan 690	. . . . 5 ⊢ (𝑦 ∈ (-(π / 2)[,](π / 2)) → (((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥)))‘𝑦) = ((cos ↾ (0[,]π))‘((𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))‘𝑦)))
99		fvres 6841	. . . . 5 ⊢ (𝑦 ∈ (-(π / 2)[,](π / 2)) → ((sin ↾ (-(π / 2)[,](π / 2)))‘𝑦) = (sin‘𝑦))
100	96, 98, 99	3eqtr4d 2774	. . . 4 ⊢ (𝑦 ∈ (-(π / 2)[,](π / 2)) → (((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥)))‘𝑦) = ((sin ↾ (-(π / 2)[,](π / 2)))‘𝑦))
101	86, 100	mprgbir 3051	. . 3 ⊢ ((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))) = (sin ↾ (-(π / 2)[,](π / 2)))
102		f1oeq1 6752	. . 3 ⊢ (((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))) = (sin ↾ (-(π / 2)[,](π / 2))) → (((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1) ↔ (sin ↾ (-(π / 2)[,](π / 2))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1)))
103	101, 102	ax-mp 5	. 2 ⊢ (((cos ↾ (0[,]π)) ∘ (𝑥 ∈ (-(π / 2)[,](π / 2)) ↦ ((π / 2) − 𝑥))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1) ↔ (sin ↾ (-(π / 2)[,](π / 2))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1))
104	71, 103	mpbi 230	1 ⊢ (sin ↾ (-(π / 2)[,](π / 2))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3903   class class class wbr 5092   ↦ cmpt 5173   ↾ cres 5621   ∘ ccom 5623   Fn wfn 6477  ⟶wf 6478  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349  ℂcc 11007  ℝcr 11008  0cc0 11009  1c1 11010   + caddc 11012   ≤ cle 11150   − cmin 11347  -cneg 11348   / cdiv 11777  2c2 12183  [,]cicc 13251  sincsin 15970  cosccos 15971  πcpi 15973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087  ax-addf 11088

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-fi 9301  df-sup 9332  df-inf 9333  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-q 12850  df-rp 12894  df-xneg 13014  df-xadd 13015  df-xmul 13016  df-ioo 13252  df-ioc 13253  df-ico 13254  df-icc 13255  df-fz 13411  df-fzo 13558  df-fl 13696  df-seq 13909  df-exp 13969  df-fac 14181  df-bc 14210  df-hash 14238  df-shft 14974  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-limsup 15378  df-clim 15395  df-rlim 15396  df-sum 15594  df-ef 15974  df-sin 15976  df-cos 15977  df-pi 15979  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-hom 17185  df-cco 17186  df-rest 17326  df-topn 17327  df-0g 17345  df-gsum 17346  df-topgen 17347  df-pt 17348  df-prds 17351  df-xrs 17406  df-qtop 17411  df-imas 17412  df-xps 17414  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-submnd 18658  df-mulg 18947  df-cntz 19196  df-cmn 19661  df-psmet 21253  df-xmet 21254  df-met 21255  df-bl 21256  df-mopn 21257  df-fbas 21258  df-fg 21259  df-cnfld 21262  df-top 22779  df-topon 22796  df-topsp 22818  df-bases 22831  df-cld 22904  df-ntr 22905  df-cls 22906  df-nei 22983  df-lp 23021  df-perf 23022  df-cn 23112  df-cnp 23113  df-haus 23200  df-tx 23447  df-hmeo 23640  df-fil 23731  df-fm 23823  df-flim 23824  df-flf 23825  df-xms 24206  df-ms 24207  df-tms 24208  df-cncf 24769  df-limc 25765  df-dv 25766

This theorem is referenced by:  efif1olem4  26452  asinrebnd  26809