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Theorem ftc1anclem3 28469
Description: Lemma for ftc1anc 28475- 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 21154 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
21ffvelrnda 5843 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( F `  x )  e.  RR )
3 i1ff 21154 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
43ffvelrnda 5843 . . . . . . 7  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( G `  x )  e.  RR )
5 absreim 12782 . . . . . . 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 827 . . . . 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 9412 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( F `  x )  e.  CC )
98sqvald 12005 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( F `
 x ) ^
2 )  =  ( ( F `  x
)  x.  ( F `
 x ) ) )
104recnd 9412 . . . . . . . . 9  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( G `  x )  e.  CC )
1110sqvald 12005 . . . . . . . 8  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( G `
 x ) ^
2 )  =  ( ( G `  x
)  x.  ( G `
 x ) ) )
129, 11oveqan12d 6110 . . . . . . 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 827 . . . . . 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 5695 . . . . 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 2475 . . . 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 4378 . . 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 9341 . . . . . . 7  |-  _i  e.  CC
18 mulcl 9366 . . . . . . 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 9364 . . . . . 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 827 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  (
( F `  x
)  +  ( _i  x.  ( G `  x ) ) )  e.  CC )
23 reex 9373 . . . . . 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 6116 . . . . . 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 5744 . . . . . 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 4882 . . . . . . . 8  |-  ( RR 
X.  { _i }
)  =  ( x  e.  RR  |->  _i )
3332a1i 11 . . . . . . 7  |-  ( G  e.  dom  S.1  ->  ( RR  X.  { _i } )  =  ( x  e.  RR  |->  _i ) )
343feqmptd 5744 . . . . . . 7  |-  ( G  e.  dom  S.1  ->  G  =  ( x  e.  RR  |->  ( G `  x ) ) )
3530, 31, 4, 33, 34offval2 6336 . . . . . 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 6336 . . . 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 12825 . . . . . 6  |-  abs : CC
--> RR
3938a1i 11 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  abs : CC --> RR )
4039feqmptd 5744 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  abs  =  (
y  e.  CC  |->  ( abs `  y ) ) )
41 fveq2 5691 . . . 4  |-  ( y  =  ( ( F `
 x )  +  ( _i  x.  ( G `  x )
) )  ->  ( abs `  y )  =  ( abs `  (
( F `  x
)  +  ( _i  x.  ( G `  x ) ) ) ) )
4222, 37, 40, 41fmptco 5876 . . 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 9406 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( F `
 x )  x.  ( F `  x
) )  e.  CC )
4410, 10mulcld 9406 . . . . . 6  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( G `
 x )  x.  ( G `  x
) )  e.  CC )
45 addcl 9364 . . . . . 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 827 . . . 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 6336 . . . . . 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 6336 . . . . . 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 6336 . . . 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 sqrf 12851 . . . . . 6  |-  sqr : CC
--> CC
5756a1i 11 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  sqr : CC --> CC )
5857feqmptd 5744 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  sqr  =  (
y  e.  CC  |->  ( sqr `  y ) ) )
59 fveq2 5691 . . . 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 5876 . . 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 2485 . 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 11392 . . . . . . 7  |-  ( x  e.  ( 0 [,) +oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
63 resqrcl 12743 . . . . . . 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 5744 . . . . . . . 8  |-  ( sqr
: CC --> CC  ->  sqr  =  ( x  e.  CC  |->  ( sqr `  x
) ) )
6856, 67ax-mp 5 . . . . . . 7  |-  sqr  =  ( x  e.  CC  |->  ( sqr `  x ) )
6968reseq1i 5106 . . . . . 6  |-  ( sqr  |`  ( 0 [,) +oo ) )  =  ( ( x  e.  CC  |->  ( sqr `  x ) )  |`  ( 0 [,) +oo ) )
7062simplbi 460 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) +oo )  ->  x  e.  RR )
7170ssriv 3360 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR
72 ax-resscn 9339 . . . . . . . 8  |-  RR  C_  CC
7371, 72sstri 3365 . . . . . . 7  |-  ( 0 [,) +oo )  C_  CC
74 resmpt 5156 . . . . . . 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 2463 . . . . 5  |-  ( sqr  |`  ( 0 [,) +oo ) )  =  ( x  e.  ( 0 [,) +oo )  |->  ( sqr `  x ) )
7765, 76fmptd 5867 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( sqr  |`  (
0 [,) +oo )
) : ( 0 [,) +oo ) --> RR )
78 ge0addcl 11397 . . . . . 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 6100 . . . . . . . . 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 5546 . . . . . . 7  |-  ( z  =  F  ->  (
( z  oF  x.  z ) : RR --> ( 0 [,) +oo )  <->  ( F  oF  x.  F ) : RR --> ( 0 [,) +oo ) ) )
83 i1ff 21154 . . . . . . . . . . . 12  |-  ( z  e.  dom  S.1  ->  z : RR --> RR )
8483ffvelrnda 5843 . . . . . . . . . . 11  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  ( z `  x )  e.  RR )
8584, 84remulcld 9414 . . . . . . . . . 10  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( z `
 x )  x.  ( z `  x
) )  e.  RR )
8684msqge0d 9908 . . . . . . . . . 10  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  0  <_  (
( z `  x
)  x.  ( z `
 x ) ) )
87 elrege0 11392 . . . . . . . . . 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 2443 . . . . . . . . 9  |-  ( x  e.  RR  |->  ( ( z `  x )  x.  ( z `  x ) ) )  =  ( x  e.  RR  |->  ( ( z `
 x )  x.  ( z `  x
) ) )
9088, 89fmptd 5867 . . . . . . . 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 5744 . . . . . . . . . 10  |-  ( z  e.  dom  S.1  ->  z  =  ( x  e.  RR  |->  ( z `  x ) ) )
9391, 84, 84, 92, 92offval2 6336 . . . . . . . . 9  |-  ( z  e.  dom  S.1  ->  ( z  oF  x.  z )  =  ( x  e.  RR  |->  ( ( z `  x
)  x.  ( z `
 x ) ) ) )
9493feq1d 5546 . . . . . . . 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 3036 . . . . . 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 6100 . . . . . . . . 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 5546 . . . . . . 7  |-  ( z  =  G  ->  (
( z  oF  x.  z ) : RR --> ( 0 [,) +oo )  <->  ( G  oF  x.  G ) : RR --> ( 0 [,) +oo ) ) )
101100, 95vtoclga 3036 . . . . . 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 3559 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
10479, 97, 102, 24, 24, 103off 6334 . . . 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 5569 . . . 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 5344 . . . 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 5559 . . . . . . . 8  |-  ( sqr
: CC --> CC  ->  sqr 
Fn  CC )
10956, 108ax-mp 5 . . . . . . 7  |-  sqr  Fn  CC
110 readdcl 9365 . . . . . . . . . . 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 9367 . . . . . . . . . . . . 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 6334 . . . . . . . . . . 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 6334 . . . . . . . . . . 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 6334 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) : RR --> RR )
120 frn 5565 . . . . . . . . 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 3368 . . . . . . 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 5524 . . . . . . 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 21174 . . . . . . . . 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 21174 . . . . . . . . 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 21173 . . . . . . 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 21155 . . . . . . 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 7589 . . . . . 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 7596 . . . . 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 2527 . . 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 5025 . . . . . . 7  |-  `' ( sqr  o.  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G )
) )  =  ( `' ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  o.  `' sqr )
140139imaeq1i 5166 . . . . . 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 5343 . . . . . 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 2463 . . . . 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 21156 . . . . . 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 2527 . . . 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 5694 . . . 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 4001 . . . . . . . 8  |-  ( x  e.  ( ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } )  ->  x  =/=  0 )
149 c0ex 9380 . . . . . . . . . . . 12  |-  0  e.  _V
150149elsnc 3901 . . . . . . . . . . 11  |-  ( 0  e.  { x }  <->  0  =  x )
151 eqcom 2445 . . . . . . . . . . 11  |-  ( 0  =  x  <->  x  = 
0 )
152150, 151bitri 249 . . . . . . . . . 10  |-  ( 0  e.  { x }  <->  x  =  0 )
153152necon3bbii 2639 . . . . . . . . 9  |-  ( -.  0  e.  { x } 
<->  x  =/=  0 )
154 sqr0 12731 . . . . . . . . . 10  |-  ( sqr `  0 )  =  0
155154eleq1i 2506 . . . . . . . . 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 5823 . . . . . . . 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 21157 . . . . 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 2527 . . 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 21159 . 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 2517 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 1369    e. wcel 1756    =/= wne 2606   _Vcvv 2972    \ cdif 3325    C_ wss 3328   {csn 3877   class class class wbr 4292    e. cmpt 4350    X. cxp 4838   `'ccnv 4839   dom cdm 4840   ran crn 4841    |` cres 4842   "cima 4843    o. ccom 4844    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091    oFcof 6318   Fincfn 7310   CCcc 9280   RRcr 9281   0cc0 9282   _ici 9284    + caddc 9285    x. cmul 9287   +oocpnf 9415    <_ cle 9419   2c2 10371   [,)cico 11302   ^cexp 11865   sqrcsqr 12722   abscabs 12723   volcvol 20947   S.1citg1 21095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-xadd 11090  df-ioo 11304  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-sum 13164  df-xmet 17810  df-met 17811  df-ovol 20948  df-vol 20949  df-mbf 21099  df-itg1 21100
This theorem is referenced by:  ftc1anclem7  28473  ftc1anclem8  28474
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