Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ftc1anclem3 Structured version   Unicode version

Theorem ftc1anclem3 29685
Description: Lemma for ftc1anc 29691- the absolute value of the sum of a simple function and  _i times another simple function is itself a simple function. (Contributed by Brendan Leahy, 27-May-2018.)
Assertion
Ref Expression
ftc1anclem3  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( abs  o.  ( F  oF  +  ( ( RR 
X.  { _i }
)  oF  x.  G ) ) )  e.  dom  S.1 )

Proof of Theorem ftc1anclem3
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1ff 21834 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
21ffvelrnda 6020 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( F `  x )  e.  RR )
3 i1ff 21834 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
43ffvelrnda 6020 . . . . . . 7  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( G `  x )  e.  RR )
5 absreim 13088 . . . . . . 7  |-  ( ( ( F `  x
)  e.  RR  /\  ( G `  x )  e.  RR )  -> 
( abs `  (
( F `  x
)  +  ( _i  x.  ( G `  x ) ) ) )  =  ( sqr `  ( ( ( F `
 x ) ^
2 )  +  ( ( G `  x
) ^ 2 ) ) ) )
62, 4, 5syl2an 477 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  x  e.  RR )  /\  ( G  e. 
dom  S.1  /\  x  e.  RR ) )  -> 
( abs `  (
( F `  x
)  +  ( _i  x.  ( G `  x ) ) ) )  =  ( sqr `  ( ( ( F `
 x ) ^
2 )  +  ( ( G `  x
) ^ 2 ) ) ) )
76anandirs 829 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  ( abs `  ( ( F `
 x )  +  ( _i  x.  ( G `  x )
) ) )  =  ( sqr `  (
( ( F `  x ) ^ 2 )  +  ( ( G `  x ) ^ 2 ) ) ) )
82recnd 9621 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( F `  x )  e.  CC )
98sqvald 12274 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( F `
 x ) ^
2 )  =  ( ( F `  x
)  x.  ( F `
 x ) ) )
104recnd 9621 . . . . . . . . 9  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( G `  x )  e.  CC )
1110sqvald 12274 . . . . . . . 8  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( G `
 x ) ^
2 )  =  ( ( G `  x
)  x.  ( G `
 x ) ) )
129, 11oveqan12d 6302 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  x  e.  RR )  /\  ( G  e. 
dom  S.1  /\  x  e.  RR ) )  -> 
( ( ( F `
 x ) ^
2 )  +  ( ( G `  x
) ^ 2 ) )  =  ( ( ( F `  x
)  x.  ( F `
 x ) )  +  ( ( G `
 x )  x.  ( G `  x
) ) ) )
1312anandirs 829 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  (
( ( F `  x ) ^ 2 )  +  ( ( G `  x ) ^ 2 ) )  =  ( ( ( F `  x )  x.  ( F `  x ) )  +  ( ( G `  x )  x.  ( G `  x )
) ) )
1413fveq2d 5869 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  ( sqr `  ( ( ( F `  x ) ^ 2 )  +  ( ( G `  x ) ^ 2 ) ) )  =  ( sqr `  (
( ( F `  x )  x.  ( F `  x )
)  +  ( ( G `  x )  x.  ( G `  x ) ) ) ) )
157, 14eqtrd 2508 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  ( abs `  ( ( F `
 x )  +  ( _i  x.  ( G `  x )
) ) )  =  ( sqr `  (
( ( F `  x )  x.  ( F `  x )
)  +  ( ( G `  x )  x.  ( G `  x ) ) ) ) )
1615mpteq2dva 4533 . . 3  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( x  e.  RR  |->  ( abs `  (
( F `  x
)  +  ( _i  x.  ( G `  x ) ) ) ) )  =  ( x  e.  RR  |->  ( sqr `  ( ( ( F `  x
)  x.  ( F `
 x ) )  +  ( ( G `
 x )  x.  ( G `  x
) ) ) ) ) )
17 ax-icn 9550 . . . . . . 7  |-  _i  e.  CC
18 mulcl 9575 . . . . . . 7  |-  ( ( _i  e.  CC  /\  ( G `  x )  e.  CC )  -> 
( _i  x.  ( G `  x )
)  e.  CC )
1917, 10, 18sylancr 663 . . . . . 6  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( _i  x.  ( G `  x ) )  e.  CC )
20 addcl 9573 . . . . . 6  |-  ( ( ( F `  x
)  e.  CC  /\  ( _i  x.  ( G `  x )
)  e.  CC )  ->  ( ( F `
 x )  +  ( _i  x.  ( G `  x )
) )  e.  CC )
218, 19, 20syl2an 477 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  x  e.  RR )  /\  ( G  e. 
dom  S.1  /\  x  e.  RR ) )  -> 
( ( F `  x )  +  ( _i  x.  ( G `
 x ) ) )  e.  CC )
2221anandirs 829 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  (
( F `  x
)  +  ( _i  x.  ( G `  x ) ) )  e.  CC )
23 reex 9582 . . . . . 6  |-  RR  e.  _V
2423a1i 11 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  RR  e.  _V )
252adantlr 714 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  ( F `  x )  e.  RR )
26 ovex 6308 . . . . . 6  |-  ( _i  x.  ( G `  x ) )  e. 
_V
2726a1i 11 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  (
_i  x.  ( G `  x ) )  e. 
_V )
281feqmptd 5919 . . . . . 6  |-  ( F  e.  dom  S.1  ->  F  =  ( x  e.  RR  |->  ( F `  x ) ) )
2928adantr 465 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  F  =  ( x  e.  RR  |->  ( F `  x ) ) )
3023a1i 11 . . . . . . 7  |-  ( G  e.  dom  S.1  ->  RR  e.  _V )
3117a1i 11 . . . . . . 7  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  _i  e.  CC )
32 fconstmpt 5042 . . . . . . . 8  |-  ( RR 
X.  { _i }
)  =  ( x  e.  RR  |->  _i )
3332a1i 11 . . . . . . 7  |-  ( G  e.  dom  S.1  ->  ( RR  X.  { _i } )  =  ( x  e.  RR  |->  _i ) )
343feqmptd 5919 . . . . . . 7  |-  ( G  e.  dom  S.1  ->  G  =  ( x  e.  RR  |->  ( G `  x ) ) )
3530, 31, 4, 33, 34offval2 6539 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ( ( RR  X.  {
_i } )  oF  x.  G )  =  ( x  e.  RR  |->  ( _i  x.  ( G `  x ) ) ) )
3635adantl 466 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( ( RR 
X.  { _i }
)  oF  x.  G )  =  ( x  e.  RR  |->  ( _i  x.  ( G `
 x ) ) ) )
3724, 25, 27, 29, 36offval2 6539 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( F  oF  +  ( ( RR  X.  { _i }
)  oF  x.  G ) )  =  ( x  e.  RR  |->  ( ( F `  x )  +  ( _i  x.  ( G `
 x ) ) ) ) )
38 absf 13132 . . . . . 6  |-  abs : CC
--> RR
3938a1i 11 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  abs : CC --> RR )
4039feqmptd 5919 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  abs  =  (
y  e.  CC  |->  ( abs `  y ) ) )
41 fveq2 5865 . . . 4  |-  ( y  =  ( ( F `
 x )  +  ( _i  x.  ( G `  x )
) )  ->  ( abs `  y )  =  ( abs `  (
( F `  x
)  +  ( _i  x.  ( G `  x ) ) ) ) )
4222, 37, 40, 41fmptco 6053 . . 3  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( abs  o.  ( F  oF  +  ( ( RR 
X.  { _i }
)  oF  x.  G ) ) )  =  ( x  e.  RR  |->  ( abs `  (
( F `  x
)  +  ( _i  x.  ( G `  x ) ) ) ) ) )
438, 8mulcld 9615 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( F `
 x )  x.  ( F `  x
) )  e.  CC )
4410, 10mulcld 9615 . . . . . 6  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( G `
 x )  x.  ( G `  x
) )  e.  CC )
45 addcl 9573 . . . . . 6  |-  ( ( ( ( F `  x )  x.  ( F `  x )
)  e.  CC  /\  ( ( G `  x )  x.  ( G `  x )
)  e.  CC )  ->  ( ( ( F `  x )  x.  ( F `  x ) )  +  ( ( G `  x )  x.  ( G `  x )
) )  e.  CC )
4643, 44, 45syl2an 477 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  x  e.  RR )  /\  ( G  e. 
dom  S.1  /\  x  e.  RR ) )  -> 
( ( ( F `
 x )  x.  ( F `  x
) )  +  ( ( G `  x
)  x.  ( G `
 x ) ) )  e.  CC )
4746anandirs 829 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  (
( ( F `  x )  x.  ( F `  x )
)  +  ( ( G `  x )  x.  ( G `  x ) ) )  e.  CC )
4843adantlr 714 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  (
( F `  x
)  x.  ( F `
 x ) )  e.  CC )
4944adantll 713 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  (
( G `  x
)  x.  ( G `
 x ) )  e.  CC )
5023a1i 11 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  RR  e.  _V )
5150, 2, 2, 28, 28offval2 6539 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( F  oF  x.  F )  =  ( x  e.  RR  |->  ( ( F `  x
)  x.  ( F `
 x ) ) ) )
5251adantr 465 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( F  oF  x.  F )  =  ( x  e.  RR  |->  ( ( F `
 x )  x.  ( F `  x
) ) ) )
5330, 4, 4, 34, 34offval2 6539 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ( G  oF  x.  G )  =  ( x  e.  RR  |->  ( ( G `  x
)  x.  ( G `
 x ) ) ) )
5453adantl 466 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( G  oF  x.  G )  =  ( x  e.  RR  |->  ( ( G `
 x )  x.  ( G `  x
) ) ) )
5524, 48, 49, 52, 54offval2 6539 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  =  ( x  e.  RR  |->  ( ( ( F `
 x )  x.  ( F `  x
) )  +  ( ( G `  x
)  x.  ( G `
 x ) ) ) ) )
56 sqrtf 13158 . . . . . 6  |-  sqr : CC
--> CC
5756a1i 11 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  sqr : CC --> CC )
5857feqmptd 5919 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  sqr  =  (
y  e.  CC  |->  ( sqr `  y ) ) )
59 fveq2 5865 . . . 4  |-  ( y  =  ( ( ( F `  x )  x.  ( F `  x ) )  +  ( ( G `  x )  x.  ( G `  x )
) )  ->  ( sqr `  y )  =  ( sqr `  (
( ( F `  x )  x.  ( F `  x )
)  +  ( ( G `  x )  x.  ( G `  x ) ) ) ) )
6047, 55, 58, 59fmptco 6053 . . 3  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( sqr  o.  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) )  =  ( x  e.  RR  |->  ( sqr `  ( ( ( F `  x
)  x.  ( F `
 x ) )  +  ( ( G `
 x )  x.  ( G `  x
) ) ) ) ) )
6116, 42, 603eqtr4d 2518 . 2  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( abs  o.  ( F  oF  +  ( ( RR 
X.  { _i }
)  oF  x.  G ) ) )  =  ( sqr  o.  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) ) )
62 elrege0 11626 . . . . . . 7  |-  ( x  e.  ( 0 [,) +oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
63 resqrtcl 13049 . . . . . . 7  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( sqr `  x
)  e.  RR )
6462, 63sylbi 195 . . . . . 6  |-  ( x  e.  ( 0 [,) +oo )  ->  ( sqr `  x )  e.  RR )
6564adantl 466 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  ( 0 [,) +oo ) )  ->  ( sqr `  x )  e.  RR )
66 id 22 . . . . . . . . 9  |-  ( sqr
: CC --> CC  ->  sqr
: CC --> CC )
6766feqmptd 5919 . . . . . . . 8  |-  ( sqr
: CC --> CC  ->  sqr  =  ( x  e.  CC  |->  ( sqr `  x
) ) )
6856, 67ax-mp 5 . . . . . . 7  |-  sqr  =  ( x  e.  CC  |->  ( sqr `  x ) )
6968reseq1i 5268 . . . . . 6  |-  ( sqr  |`  ( 0 [,) +oo ) )  =  ( ( x  e.  CC  |->  ( sqr `  x ) )  |`  ( 0 [,) +oo ) )
7062simplbi 460 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) +oo )  ->  x  e.  RR )
7170ssriv 3508 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR
72 ax-resscn 9548 . . . . . . . 8  |-  RR  C_  CC
7371, 72sstri 3513 . . . . . . 7  |-  ( 0 [,) +oo )  C_  CC
74 resmpt 5322 . . . . . . 7  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( (
x  e.  CC  |->  ( sqr `  x ) )  |`  ( 0 [,) +oo ) )  =  ( x  e.  ( 0 [,) +oo )  |->  ( sqr `  x
) ) )
7573, 74ax-mp 5 . . . . . 6  |-  ( ( x  e.  CC  |->  ( sqr `  x ) )  |`  ( 0 [,) +oo ) )  =  ( x  e.  ( 0 [,) +oo )  |->  ( sqr `  x
) )
7669, 75eqtri 2496 . . . . 5  |-  ( sqr  |`  ( 0 [,) +oo ) )  =  ( x  e.  ( 0 [,) +oo )  |->  ( sqr `  x ) )
7765, 76fmptd 6044 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( sqr  |`  (
0 [,) +oo )
) : ( 0 [,) +oo ) --> RR )
78 ge0addcl 11631 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  +  y )  e.  ( 0 [,) +oo )
)
7978adantl 466 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) ) )  -> 
( x  +  y )  e.  ( 0 [,) +oo ) )
80 oveq12 6292 . . . . . . . . 9  |-  ( ( z  =  F  /\  z  =  F )  ->  ( z  oF  x.  z )  =  ( F  oF  x.  F ) )
8180anidms 645 . . . . . . . 8  |-  ( z  =  F  ->  (
z  oF  x.  z )  =  ( F  oF  x.  F ) )
8281feq1d 5716 . . . . . . 7  |-  ( z  =  F  ->  (
( z  oF  x.  z ) : RR --> ( 0 [,) +oo )  <->  ( F  oF  x.  F ) : RR --> ( 0 [,) +oo ) ) )
83 i1ff 21834 . . . . . . . . . . . 12  |-  ( z  e.  dom  S.1  ->  z : RR --> RR )
8483ffvelrnda 6020 . . . . . . . . . . 11  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  ( z `  x )  e.  RR )
8584, 84remulcld 9623 . . . . . . . . . 10  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( z `
 x )  x.  ( z `  x
) )  e.  RR )
8684msqge0d 10120 . . . . . . . . . 10  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  0  <_  (
( z `  x
)  x.  ( z `
 x ) ) )
87 elrege0 11626 . . . . . . . . . 10  |-  ( ( ( z `  x
)  x.  ( z `
 x ) )  e.  ( 0 [,) +oo )  <->  ( ( ( z `  x )  x.  ( z `  x ) )  e.  RR  /\  0  <_ 
( ( z `  x )  x.  (
z `  x )
) ) )
8885, 86, 87sylanbrc 664 . . . . . . . . 9  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( z `
 x )  x.  ( z `  x
) )  e.  ( 0 [,) +oo )
)
89 eqid 2467 . . . . . . . . 9  |-  ( x  e.  RR  |->  ( ( z `  x )  x.  ( z `  x ) ) )  =  ( x  e.  RR  |->  ( ( z `
 x )  x.  ( z `  x
) ) )
9088, 89fmptd 6044 . . . . . . . 8  |-  ( z  e.  dom  S.1  ->  ( x  e.  RR  |->  ( ( z `  x
)  x.  ( z `
 x ) ) ) : RR --> ( 0 [,) +oo ) )
9123a1i 11 . . . . . . . . . 10  |-  ( z  e.  dom  S.1  ->  RR  e.  _V )
9283feqmptd 5919 . . . . . . . . . 10  |-  ( z  e.  dom  S.1  ->  z  =  ( x  e.  RR  |->  ( z `  x ) ) )
9391, 84, 84, 92, 92offval2 6539 . . . . . . . . 9  |-  ( z  e.  dom  S.1  ->  ( z  oF  x.  z )  =  ( x  e.  RR  |->  ( ( z `  x
)  x.  ( z `
 x ) ) ) )
9493feq1d 5716 . . . . . . . 8  |-  ( z  e.  dom  S.1  ->  ( ( z  oF  x.  z ) : RR --> ( 0 [,) +oo )  <->  ( x  e.  RR  |->  ( ( z `
 x )  x.  ( z `  x
) ) ) : RR --> ( 0 [,) +oo ) ) )
9590, 94mpbird 232 . . . . . . 7  |-  ( z  e.  dom  S.1  ->  ( z  oF  x.  z ) : RR --> ( 0 [,) +oo ) )
9682, 95vtoclga 3177 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( F  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
9796adantr 465 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( F  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
98 oveq12 6292 . . . . . . . . 9  |-  ( ( z  =  G  /\  z  =  G )  ->  ( z  oF  x.  z )  =  ( G  oF  x.  G ) )
9998anidms 645 . . . . . . . 8  |-  ( z  =  G  ->  (
z  oF  x.  z )  =  ( G  oF  x.  G ) )
10099feq1d 5716 . . . . . . 7  |-  ( z  =  G  ->  (
( z  oF  x.  z ) : RR --> ( 0 [,) +oo )  <->  ( G  oF  x.  G ) : RR --> ( 0 [,) +oo ) ) )
101100, 95vtoclga 3177 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ( G  oF  x.  G ) : RR --> ( 0 [,) +oo ) )
102101adantl 466 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( G  oF  x.  G ) : RR --> ( 0 [,) +oo ) )
103 inidm 3707 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
10479, 97, 102, 24, 24, 103off 6537 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) : RR --> ( 0 [,) +oo ) )
105 fco2 5741 . . . 4  |-  ( ( ( sqr  |`  (
0 [,) +oo )
) : ( 0 [,) +oo ) --> RR 
/\  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) : RR --> ( 0 [,) +oo ) )  ->  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) : RR --> RR )
10677, 104, 105syl2anc 661 . . 3  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( sqr  o.  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) ) : RR --> RR )
107 rnco 5512 . . . 4  |-  ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) )  =  ran  ( sqr  |`  ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) )
108 ffn 5730 . . . . . . . 8  |-  ( sqr
: CC --> CC  ->  sqr 
Fn  CC )
10956, 108ax-mp 5 . . . . . . 7  |-  sqr  Fn  CC
110 readdcl 9574 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
111110adantl 466 . . . . . . . . . 10  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
112 remulcl 9576 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
113112adantl 466 . . . . . . . . . . . 12  |-  ( ( F  e.  dom  S.1  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( x  x.  y )  e.  RR )
114113, 1, 1, 50, 50, 103off 6537 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( F  oF  x.  F ) : RR --> RR )
115114adantr 465 . . . . . . . . . 10  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( F  oF  x.  F ) : RR --> RR )
116112adantl 466 . . . . . . . . . . . 12  |-  ( ( G  e.  dom  S.1  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( x  x.  y )  e.  RR )
117116, 3, 3, 30, 30, 103off 6537 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( G  oF  x.  G ) : RR --> RR )
118117adantl 466 . . . . . . . . . 10  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( G  oF  x.  G ) : RR --> RR )
119111, 115, 118, 24, 24, 103off 6537 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) : RR --> RR )
120 frn 5736 . . . . . . . . 9  |-  ( ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) : RR --> RR  ->  ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  C_  RR )
121119, 120syl 16 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  C_  RR )
122121, 72syl6ss 3516 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  C_  CC )
123 fnssres 5693 . . . . . . 7  |-  ( ( sqr  Fn  CC  /\  ran  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) )  C_  CC )  ->  ( sqr  |`  ran  (
( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) )  Fn 
ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) )
124109, 122, 123sylancr 663 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( sqr  |`  ran  (
( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) )  Fn 
ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) )
125 id 22 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  F  e.  dom  S.1 )
126125, 125i1fmul 21854 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ( F  oF  x.  F )  e.  dom  S.1 )
127126adantr 465 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( F  oF  x.  F )  e.  dom  S.1 )
128 id 22 . . . . . . . . . 10  |-  ( G  e.  dom  S.1  ->  G  e.  dom  S.1 )
129128, 128i1fmul 21854 . . . . . . . . 9  |-  ( G  e.  dom  S.1  ->  ( G  oF  x.  G )  e.  dom  S.1 )
130129adantl 466 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( G  oF  x.  G )  e.  dom  S.1 )
131127, 130i1fadd 21853 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  e. 
dom  S.1 )
132 i1frn 21835 . . . . . . 7  |-  ( ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) )  e.  dom  S.1 
->  ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  e. 
Fin )
133131, 132syl 16 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  e. 
Fin )
134 fnfi 7797 . . . . . 6  |-  ( ( ( sqr  |`  ran  (
( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) )  Fn 
ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  /\  ran  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) )  e.  Fin )  ->  ( sqr  |`  ran  (
( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) )  e. 
Fin )
135124, 133, 134syl2anc 661 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( sqr  |`  ran  (
( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) )  e. 
Fin )
136 rnfi 7804 . . . . 5  |-  ( ( sqr  |`  ran  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G )
) )  e.  Fin  ->  ran  ( sqr  |`  ran  (
( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) )  e. 
Fin )
137135, 136syl 16 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ran  ( sqr  |` 
ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) )  e.  Fin )
138107, 137syl5eqel 2559 . . 3  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ran  ( sqr  o.  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) )  e. 
Fin )
139 cnvco 5187 . . . . . . 7  |-  `' ( sqr  o.  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G )
) )  =  ( `' ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  o.  `' sqr )
140139imaeq1i 5333 . . . . . 6  |-  ( `' ( sqr  o.  (
( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) ) " { x } )  =  ( ( `' ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) )  o.  `' sqr ) " { x } )
141 imaco 5511 . . . . . 6  |-  ( ( `' ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  o.  `' sqr ) " {
x } )  =  ( `' ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G )
) " ( `' sqr " { x } ) )
142140, 141eqtri 2496 . . . . 5  |-  ( `' ( sqr  o.  (
( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) ) " { x } )  =  ( `' ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) " ( `' sqr " { x } ) )
143 i1fima 21836 . . . . . 6  |-  ( ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) )  e.  dom  S.1 
->  ( `' ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G )
) " ( `' sqr " { x } ) )  e. 
dom  vol )
144131, 143syl 16 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( `' ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) " ( `' sqr " { x } ) )  e. 
dom  vol )
145142, 144syl5eqel 2559 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( `' ( sqr  o.  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G )
) ) " {
x } )  e. 
dom  vol )
146145adantr 465 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  ( ran  ( sqr 
o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } ) )  ->  ( `' ( sqr  o.  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) ) " { x } )  e.  dom  vol )
147142fveq2i 5868 . . . 4  |-  ( vol `  ( `' ( sqr 
o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) )
" { x }
) )  =  ( vol `  ( `' ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) " ( `' sqr " { x } ) ) )
148 eldifsni 4153 . . . . . . . 8  |-  ( x  e.  ( ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } )  ->  x  =/=  0 )
149 c0ex 9589 . . . . . . . . . . . 12  |-  0  e.  _V
150149elsnc 4051 . . . . . . . . . . 11  |-  ( 0  e.  { x }  <->  0  =  x )
151 eqcom 2476 . . . . . . . . . . 11  |-  ( 0  =  x  <->  x  = 
0 )
152150, 151bitri 249 . . . . . . . . . 10  |-  ( 0  e.  { x }  <->  x  =  0 )
153152necon3bbii 2728 . . . . . . . . 9  |-  ( -.  0  e.  { x } 
<->  x  =/=  0 )
154 sqrt0 13037 . . . . . . . . . 10  |-  ( sqr `  0 )  =  0
155154eleq1i 2544 . . . . . . . . 9  |-  ( ( sqr `  0 )  e.  { x }  <->  0  e.  { x }
)
156153, 155xchnxbir 309 . . . . . . . 8  |-  ( -.  ( sqr `  0
)  e.  { x } 
<->  x  =/=  0 )
157148, 156sylibr 212 . . . . . . 7  |-  ( x  e.  ( ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } )  ->  -.  ( sqr `  0 )  e. 
{ x } )
158157olcd 393 . . . . . 6  |-  ( x  e.  ( ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } )  ->  ( -.  0  e.  CC  \/  -.  ( sqr `  0
)  e.  { x } ) )
159 ianor 488 . . . . . . 7  |-  ( -.  ( 0  e.  CC  /\  ( sqr `  0
)  e.  { x } )  <->  ( -.  0  e.  CC  \/  -.  ( sqr `  0
)  e.  { x } ) )
160 elpreima 6000 . . . . . . . 8  |-  ( sqr 
Fn  CC  ->  ( 0  e.  ( `' sqr " { x } )  <-> 
( 0  e.  CC  /\  ( sqr `  0
)  e.  { x } ) ) )
16156, 108, 160mp2b 10 . . . . . . 7  |-  ( 0  e.  ( `' sqr " { x } )  <-> 
( 0  e.  CC  /\  ( sqr `  0
)  e.  { x } ) )
162159, 161xchnxbir 309 . . . . . 6  |-  ( -.  0  e.  ( `' sqr " { x } )  <->  ( -.  0  e.  CC  \/  -.  ( sqr `  0
)  e.  { x } ) )
163158, 162sylibr 212 . . . . 5  |-  ( x  e.  ( ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } )  ->  -.  0  e.  ( `' sqr " {
x } ) )
164 i1fima2 21837 . . . . 5  |-  ( ( ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) )  e.  dom  S.1 
/\  -.  0  e.  ( `' sqr " { x } ) )  -> 
( vol `  ( `' ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) "
( `' sqr " {
x } ) ) )  e.  RR )
165131, 163, 164syl2an 477 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  ( ran  ( sqr 
o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } ) )  ->  ( vol `  ( `' ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) " ( `' sqr " { x } ) ) )  e.  RR )
166147, 165syl5eqel 2559 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  ( ran  ( sqr 
o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } ) )  ->  ( vol `  ( `' ( sqr  o.  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G )
) ) " {
x } ) )  e.  RR )
167106, 138, 146, 166i1fd 21839 . 2  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( sqr  o.  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) )  e. 
dom  S.1 )
16861, 167eqeltrd 2555 1  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( abs  o.  ( F  oF  +  ( ( RR 
X.  { _i }
)  oF  x.  G ) ) )  e.  dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473    C_ wss 3476   {csn 4027   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002    o. ccom 5003    Fn wfn 5582   -->wf 5583   ` cfv 5587  (class class class)co 6283    oFcof 6521   Fincfn 7516   CCcc 9489   RRcr 9490   0cc0 9491   _ici 9493    + caddc 9494    x. cmul 9496   +oocpnf 9624    <_ cle 9628   2c2 10584   [,)cico 11530   ^cexp 12133   sqrcsqrt 13028   abscabs 13029   volcvol 21626   S.1citg1 21775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-q 11182  df-rp 11220  df-xadd 11318  df-ioo 11532  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-sum 13471  df-xmet 18199  df-met 18200  df-ovol 21627  df-vol 21628  df-mbf 21779  df-itg1 21780
This theorem is referenced by:  ftc1anclem7  29689  ftc1anclem8  29690
  Copyright terms: Public domain W3C validator