:: SIN_COS6 semantic presentation
theorem Th1: :: SIN_COS6:1
theorem Th2: :: SIN_COS6:2
theorem Th3: :: SIN_COS6:3
theorem Th4: :: SIN_COS6:4
theorem Th5: :: SIN_COS6:5
theorem Th6: :: SIN_COS6:6
deffunc H1( Integer) -> set = (2 * PI ) * $1;
Lm1:
dom sin = REAL
by SIN_COS:def 20;
Lm2:
dom cos = REAL
by SIN_COS:def 22;
Lm3:
[.((- (PI / 2)) + H1(0)),((PI / 2) + H1(0)).] = [.(- (PI / 2)),(PI / 2).]
;
Lm4:
[.((PI / 2) + H1(0)),(((3 / 2) * PI ) + H1(0)).] = [.(PI / 2),((3 / 2) * PI ).]
;
Lm5:
[.H1(0),(PI + H1(0)).] = [.0,PI .]
;
Lm6:
[.(PI + H1(0)),((2 * PI ) + H1(0)).] = [.PI ,(2 * PI ).]
;
Lm7:
for r, s being real number st (r ^2 ) + (s ^2 ) = 1 holds
( - 1 <= r & r <= 1 )
Lm8:
- (PI / 2) < - 0
by XREAL_1:26;
Lm9:
PI / 2 < PI / 1
by REAL_2:200;
Lm10:
1 * PI < (3 / 2) * PI
by XREAL_1:70;
Lm11:
0 + H1(1) < (PI / 2) + H1(1)
by XREAL_1:8;
Lm12:
(3 / 2) * PI < 2 * PI
by XREAL_1:70;
Lm13:
1 * PI < 2 * PI
by XREAL_1:70;
Lm14:
].(- 1),1.[ c= [.(- 1),1.]
by RCOMP_1:15;
Lm15:
].(- (PI / 2)),0.[ c= [.(- (PI / 2)),0.]
by RCOMP_1:15;
Lm16:
].(- (PI / 2)),(PI / 2).[ c= [.(- (PI / 2)),(PI / 2).]
by RCOMP_1:15;
Lm17:
].0,(PI / 2).[ c= [.0,(PI / 2).]
by RCOMP_1:15;
Lm18:
].0,PI .[ c= [.0,PI .]
by RCOMP_1:15;
Lm19:
].(PI / 2),PI .[ c= [.(PI / 2),PI .]
by RCOMP_1:15;
Lm20:
].(PI / 2),((3 / 2) * PI ).[ c= [.(PI / 2),((3 / 2) * PI ).]
by RCOMP_1:15;
Lm21:
].PI ,((3 / 2) * PI ).[ c= [.PI ,((3 / 2) * PI ).]
by RCOMP_1:15;
Lm22:
].PI ,(2 * PI ).[ c= [.PI ,(2 * PI ).]
by RCOMP_1:15;
Lm23:
].((3 / 2) * PI ),(2 * PI ).[ c= [.((3 / 2) * PI ),(2 * PI ).]
by RCOMP_1:15;
Lm24:
[.(- (PI / 2)),0.] c= [.(- (PI / 2)),(PI / 2).]
by RCOMP_1:49;
Lm25:
[.0,(PI / 2).] c= [.(- (PI / 2)),(PI / 2).]
by Lm8, RCOMP_1:49;
Lm26:
[.0,(PI / 2).] c= [.0,PI .]
by Lm9, RCOMP_1:49;
Lm27:
[.(PI / 2),PI .] c= [.(PI / 2),((3 / 2) * PI ).]
by Lm10, RCOMP_1:49;
Lm28:
[.(PI / 2),PI .] c= [.0,PI .]
by RCOMP_1:49;
Lm29:
[.PI ,((3 / 2) * PI ).] c= [.(PI / 2),((3 / 2) * PI ).]
by Lm9, RCOMP_1:49;
Lm30:
[.PI ,((3 / 2) * PI ).] c= [.PI ,(2 * PI ).]
by Lm12, RCOMP_1:49;
Lm31:
[.((3 / 2) * PI ),(2 * PI ).] c= [.((- (PI / 2)) + H1(1)),((PI / 2) + H1(1)).]
by Lm11, RCOMP_1:49;
Lm32:
[.((3 / 2) * PI ),(2 * PI ).] c= [.PI ,(2 * PI ).]
by Lm10, RCOMP_1:49;
theorem Th7: :: SIN_COS6:7
theorem Th8: :: SIN_COS6:8
theorem Th9: :: SIN_COS6:9
theorem Th10: :: SIN_COS6:10
theorem Th11: :: SIN_COS6:11
theorem Th12: :: SIN_COS6:12
theorem Th13: :: SIN_COS6:13
theorem Th14: :: SIN_COS6:14
theorem Th15: :: SIN_COS6:15
theorem :: SIN_COS6:16
theorem :: SIN_COS6:17
theorem :: SIN_COS6:18
theorem :: SIN_COS6:19
theorem :: SIN_COS6:20
theorem Th21: :: SIN_COS6:21
theorem Th22: :: SIN_COS6:22
theorem Th23: :: SIN_COS6:23
theorem Th24: :: SIN_COS6:24
theorem Th25: :: SIN_COS6:25
theorem Th26: :: SIN_COS6:26
theorem Th27: :: SIN_COS6:27
theorem Th28: :: SIN_COS6:28
theorem Th29: :: SIN_COS6:29
theorem Th30: :: SIN_COS6:30
theorem Th31: :: SIN_COS6:31
theorem Th32: :: SIN_COS6:32
theorem Th33: :: SIN_COS6:33
theorem Th34: :: SIN_COS6:34
theorem Th35: :: SIN_COS6:35
theorem Th36: :: SIN_COS6:36
theorem :: SIN_COS6:37
theorem :: SIN_COS6:38
theorem :: SIN_COS6:39
theorem :: SIN_COS6:40
theorem :: SIN_COS6:41
theorem :: SIN_COS6:42
theorem :: SIN_COS6:43
theorem :: SIN_COS6:44
theorem Th45: :: SIN_COS6:45
theorem Th46: :: SIN_COS6:46
theorem :: SIN_COS6:47
theorem :: SIN_COS6:48
theorem Th49: :: SIN_COS6:49
theorem Th50: :: SIN_COS6:50
theorem :: SIN_COS6:51
theorem :: SIN_COS6:52
theorem Th53: :: SIN_COS6:53
theorem Th54: :: SIN_COS6:54
theorem Th55: :: SIN_COS6:55
theorem Th56: :: SIN_COS6:56
theorem Th57: :: SIN_COS6:57
theorem Th58: :: SIN_COS6:58
registration
cluster sin | [.(- (PI / 2)),0.] -> one-to-one ;
coherence
sin | [.(- (PI / 2)),0.] is one-to-one
cluster sin | [.0,(PI / 2).] -> one-to-one ;
coherence
sin | [.0,(PI / 2).] is one-to-one
cluster sin | [.(PI / 2),PI .] -> one-to-one ;
coherence
sin | [.(PI / 2),PI .] is one-to-one
cluster sin | [.PI ,((3 / 2) * PI ).] -> one-to-one ;
coherence
sin | [.PI ,((3 / 2) * PI ).] is one-to-one
cluster sin | [.((3 / 2) * PI ),(2 * PI ).] -> one-to-one ;
coherence
sin | [.((3 / 2) * PI ),(2 * PI ).] is one-to-one
end;
registration
cluster sin | ].(- (PI / 2)),(PI / 2).[ -> one-to-one ;
coherence
sin | ].(- (PI / 2)),(PI / 2).[ is one-to-one
cluster sin | ].(PI / 2),((3 / 2) * PI ).[ -> one-to-one ;
coherence
sin | ].(PI / 2),((3 / 2) * PI ).[ is one-to-one
cluster sin | ].(- (PI / 2)),0.[ -> one-to-one ;
coherence
sin | ].(- (PI / 2)),0.[ is one-to-one
cluster sin | ].0,(PI / 2).[ -> one-to-one ;
coherence
sin | ].0,(PI / 2).[ is one-to-one
cluster sin | ].(PI / 2),PI .[ -> one-to-one ;
coherence
sin | ].(PI / 2),PI .[ is one-to-one
cluster sin | ].PI ,((3 / 2) * PI ).[ -> one-to-one ;
coherence
sin | ].PI ,((3 / 2) * PI ).[ is one-to-one
cluster sin | ].((3 / 2) * PI ),(2 * PI ).[ -> one-to-one ;
coherence
sin | ].((3 / 2) * PI ),(2 * PI ).[ is one-to-one
end;
theorem Th59: :: SIN_COS6:59
theorem Th60: :: SIN_COS6:60
registration
cluster cos | ].0,PI .[ -> one-to-one ;
coherence
cos | ].0,PI .[ is one-to-one
cluster cos | ].PI ,(2 * PI ).[ -> one-to-one ;
coherence
cos | ].PI ,(2 * PI ).[ is one-to-one
cluster cos | ].0,(PI / 2).[ -> one-to-one ;
coherence
cos | ].0,(PI / 2).[ is one-to-one
cluster cos | ].(PI / 2),PI .[ -> one-to-one ;
coherence
cos | ].(PI / 2),PI .[ is one-to-one
cluster cos | ].PI ,((3 / 2) * PI ).[ -> one-to-one ;
coherence
cos | ].PI ,((3 / 2) * PI ).[ is one-to-one
cluster cos | ].((3 / 2) * PI ),(2 * PI ).[ -> one-to-one ;
coherence
cos | ].((3 / 2) * PI ),(2 * PI ).[ is one-to-one
end;
theorem :: SIN_COS6:61
:: deftheorem defines arcsin SIN_COS6:def 1 :
:: deftheorem defines arcsin SIN_COS6:def 2 :
Lm37:
arcsin " = sin | [.(- (PI / 2)),(PI / 2).]
by FUNCT_1:65;
theorem :: SIN_COS6:62
canceled;
theorem Th63: :: SIN_COS6:63
theorem Th64: :: SIN_COS6:64
theorem Th65: :: SIN_COS6:65
theorem :: SIN_COS6:66
theorem Th67: :: SIN_COS6:67
theorem :: SIN_COS6:68
theorem Th69: :: SIN_COS6:69
theorem Th70: :: SIN_COS6:70
theorem :: SIN_COS6:71
theorem :: SIN_COS6:72
theorem :: SIN_COS6:73
theorem :: SIN_COS6:74
theorem :: SIN_COS6:75
theorem :: SIN_COS6:76
theorem Th77: :: SIN_COS6:77
theorem Th78: :: SIN_COS6:78
theorem Th79: :: SIN_COS6:79
theorem Th80: :: SIN_COS6:80
theorem :: SIN_COS6:81
theorem Th82: :: SIN_COS6:82
theorem :: SIN_COS6:83
theorem :: SIN_COS6:84
theorem :: SIN_COS6:85
:: deftheorem defines arccos SIN_COS6:def 3 :
:: deftheorem defines arccos SIN_COS6:def 4 :
Lm38:
arccos " = cos | [.0,PI .]
by FUNCT_1:65;
theorem :: SIN_COS6:86
canceled;
theorem Th87: :: SIN_COS6:87
theorem Th88: :: SIN_COS6:88
theorem Th89: :: SIN_COS6:89
theorem :: SIN_COS6:90
theorem Th91: :: SIN_COS6:91
theorem :: SIN_COS6:92
theorem Th93: :: SIN_COS6:93
theorem Th94: :: SIN_COS6:94
theorem :: SIN_COS6:95
theorem :: SIN_COS6:96
theorem :: SIN_COS6:97
theorem :: SIN_COS6:98
theorem :: SIN_COS6:99
theorem :: SIN_COS6:100
theorem Th101: :: SIN_COS6:101
theorem Th102: :: SIN_COS6:102
theorem Th103: :: SIN_COS6:103
theorem Th104: :: SIN_COS6:104
theorem :: SIN_COS6:105
theorem Th106: :: SIN_COS6:106
theorem :: SIN_COS6:107
theorem :: SIN_COS6:108
theorem :: SIN_COS6:109
theorem Th110: :: SIN_COS6:110
theorem :: SIN_COS6:111
theorem :: SIN_COS6:112