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Theorem ftc1anclem3 30276
Description: Lemma for ftc1anc 30282- 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 22209 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
21ffvelrnda 6032 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( F `  x )  e.  RR )
3 i1ff 22209 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
43ffvelrnda 6032 . . . . . . 7  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( G `  x )  e.  RR )
5 absreim 13138 . . . . . . 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 831 . . . . 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 9639 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( F `  x )  e.  CC )
98sqvald 12310 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( F `
 x ) ^
2 )  =  ( ( F `  x
)  x.  ( F `
 x ) ) )
104recnd 9639 . . . . . . . . 9  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( G `  x )  e.  CC )
1110sqvald 12310 . . . . . . . 8  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( G `
 x ) ^
2 )  =  ( ( G `  x
)  x.  ( G `
 x ) ) )
129, 11oveqan12d 6315 . . . . . . 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 831 . . . . . 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 5876 . . . . 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 2498 . . . 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 4543 . . 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 9568 . . . . . . 7  |-  _i  e.  CC
18 mulcl 9593 . . . . . . 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 9591 . . . . . 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 831 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  (
( F `  x
)  +  ( _i  x.  ( G `  x ) ) )  e.  CC )
23 reex 9600 . . . . . 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 6324 . . . . . 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 5926 . . . . . 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 5052 . . . . . . . 8  |-  ( RR 
X.  { _i }
)  =  ( x  e.  RR  |->  _i )
3332a1i 11 . . . . . . 7  |-  ( G  e.  dom  S.1  ->  ( RR  X.  { _i } )  =  ( x  e.  RR  |->  _i ) )
343feqmptd 5926 . . . . . . 7  |-  ( G  e.  dom  S.1  ->  G  =  ( x  e.  RR  |->  ( G `  x ) ) )
3530, 31, 4, 33, 34offval2 6555 . . . . . 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 6555 . . . 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 13182 . . . . . 6  |-  abs : CC
--> RR
3938a1i 11 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  abs : CC --> RR )
4039feqmptd 5926 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  abs  =  (
y  e.  CC  |->  ( abs `  y ) ) )
41 fveq2 5872 . . . 4  |-  ( y  =  ( ( F `
 x )  +  ( _i  x.  ( G `  x )
) )  ->  ( abs `  y )  =  ( abs `  (
( F `  x
)  +  ( _i  x.  ( G `  x ) ) ) ) )
4222, 37, 40, 41fmptco 6065 . . 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 9633 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( F `
 x )  x.  ( F `  x
) )  e.  CC )
4410, 10mulcld 9633 . . . . . 6  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( G `
 x )  x.  ( G `  x
) )  e.  CC )
45 addcl 9591 . . . . . 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 831 . . . 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 6555 . . . . . 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 6555 . . . . . 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 6555 . . . 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 13208 . . . . . 6  |-  sqr : CC
--> CC
5756a1i 11 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  sqr : CC --> CC )
5857feqmptd 5926 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  sqr  =  (
y  e.  CC  |->  ( sqr `  y ) ) )
59 fveq2 5872 . . . 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 6065 . . 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 2508 . 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 11652 . . . . . . 7  |-  ( x  e.  ( 0 [,) +oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
63 resqrtcl 13099 . . . . . . 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 5926 . . . . . . . 8  |-  ( sqr
: CC --> CC  ->  sqr  =  ( x  e.  CC  |->  ( sqr `  x
) ) )
6856, 67ax-mp 5 . . . . . . 7  |-  sqr  =  ( x  e.  CC  |->  ( sqr `  x ) )
6968reseq1i 5279 . . . . . 6  |-  ( sqr  |`  ( 0 [,) +oo ) )  =  ( ( x  e.  CC  |->  ( sqr `  x ) )  |`  ( 0 [,) +oo ) )
70 rge0ssre 11653 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR
71 ax-resscn 9566 . . . . . . . 8  |-  RR  C_  CC
7270, 71sstri 3508 . . . . . . 7  |-  ( 0 [,) +oo )  C_  CC
73 resmpt 5333 . . . . . . 7  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( (
x  e.  CC  |->  ( sqr `  x ) )  |`  ( 0 [,) +oo ) )  =  ( x  e.  ( 0 [,) +oo )  |->  ( sqr `  x
) ) )
7472, 73ax-mp 5 . . . . . 6  |-  ( ( x  e.  CC  |->  ( sqr `  x ) )  |`  ( 0 [,) +oo ) )  =  ( x  e.  ( 0 [,) +oo )  |->  ( sqr `  x
) )
7569, 74eqtri 2486 . . . . 5  |-  ( sqr  |`  ( 0 [,) +oo ) )  =  ( x  e.  ( 0 [,) +oo )  |->  ( sqr `  x ) )
7665, 75fmptd 6056 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( sqr  |`  (
0 [,) +oo )
) : ( 0 [,) +oo ) --> RR )
77 ge0addcl 11657 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  +  y )  e.  ( 0 [,) +oo )
)
7877adantl 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 ) )
79 oveq12 6305 . . . . . . . . 9  |-  ( ( z  =  F  /\  z  =  F )  ->  ( z  oF  x.  z )  =  ( F  oF  x.  F ) )
8079anidms 645 . . . . . . . 8  |-  ( z  =  F  ->  (
z  oF  x.  z )  =  ( F  oF  x.  F ) )
8180feq1d 5723 . . . . . . 7  |-  ( z  =  F  ->  (
( z  oF  x.  z ) : RR --> ( 0 [,) +oo )  <->  ( F  oF  x.  F ) : RR --> ( 0 [,) +oo ) ) )
82 i1ff 22209 . . . . . . . . . . . 12  |-  ( z  e.  dom  S.1  ->  z : RR --> RR )
8382ffvelrnda 6032 . . . . . . . . . . 11  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  ( z `  x )  e.  RR )
8483, 83remulcld 9641 . . . . . . . . . 10  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( z `
 x )  x.  ( z `  x
) )  e.  RR )
8583msqge0d 10142 . . . . . . . . . 10  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  0  <_  (
( z `  x
)  x.  ( z `
 x ) ) )
86 elrege0 11652 . . . . . . . . . 10  |-  ( ( ( z `  x
)  x.  ( z `
 x ) )  e.  ( 0 [,) +oo )  <->  ( ( ( z `  x )  x.  ( z `  x ) )  e.  RR  /\  0  <_ 
( ( z `  x )  x.  (
z `  x )
) ) )
8784, 85, 86sylanbrc 664 . . . . . . . . 9  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( z `
 x )  x.  ( z `  x
) )  e.  ( 0 [,) +oo )
)
88 eqid 2457 . . . . . . . . 9  |-  ( x  e.  RR  |->  ( ( z `  x )  x.  ( z `  x ) ) )  =  ( x  e.  RR  |->  ( ( z `
 x )  x.  ( z `  x
) ) )
8987, 88fmptd 6056 . . . . . . . 8  |-  ( z  e.  dom  S.1  ->  ( x  e.  RR  |->  ( ( z `  x
)  x.  ( z `
 x ) ) ) : RR --> ( 0 [,) +oo ) )
9023a1i 11 . . . . . . . . . 10  |-  ( z  e.  dom  S.1  ->  RR  e.  _V )
9182feqmptd 5926 . . . . . . . . . 10  |-  ( z  e.  dom  S.1  ->  z  =  ( x  e.  RR  |->  ( z `  x ) ) )
9290, 83, 83, 91, 91offval2 6555 . . . . . . . . 9  |-  ( z  e.  dom  S.1  ->  ( z  oF  x.  z )  =  ( x  e.  RR  |->  ( ( z `  x
)  x.  ( z `
 x ) ) ) )
9392feq1d 5723 . . . . . . . 8  |-  ( z  e.  dom  S.1  ->  ( ( z  oF  x.  z ) : RR --> ( 0 [,) +oo )  <->  ( x  e.  RR  |->  ( ( z `
 x )  x.  ( z `  x
) ) ) : RR --> ( 0 [,) +oo ) ) )
9489, 93mpbird 232 . . . . . . 7  |-  ( z  e.  dom  S.1  ->  ( z  oF  x.  z ) : RR --> ( 0 [,) +oo ) )
9581, 94vtoclga 3173 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( F  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
9695adantr 465 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( F  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
97 oveq12 6305 . . . . . . . . 9  |-  ( ( z  =  G  /\  z  =  G )  ->  ( z  oF  x.  z )  =  ( G  oF  x.  G ) )
9897anidms 645 . . . . . . . 8  |-  ( z  =  G  ->  (
z  oF  x.  z )  =  ( G  oF  x.  G ) )
9998feq1d 5723 . . . . . . 7  |-  ( z  =  G  ->  (
( z  oF  x.  z ) : RR --> ( 0 [,) +oo )  <->  ( G  oF  x.  G ) : RR --> ( 0 [,) +oo ) ) )
10099, 94vtoclga 3173 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ( G  oF  x.  G ) : RR --> ( 0 [,) +oo ) )
101100adantl 466 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( G  oF  x.  G ) : RR --> ( 0 [,) +oo ) )
102 inidm 3703 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
10378, 96, 101, 24, 24, 102off 6553 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) : RR --> ( 0 [,) +oo ) )
104 fco2 5748 . . . 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 )
10576, 103, 104syl2anc 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 )
106 rnco 5519 . . . 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 ) ) )
107 ffn 5737 . . . . . . . 8  |-  ( sqr
: CC --> CC  ->  sqr 
Fn  CC )
10856, 107ax-mp 5 . . . . . . 7  |-  sqr  Fn  CC
109 readdcl 9592 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
110109adantl 466 . . . . . . . . . 10  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
111 remulcl 9594 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
112111adantl 466 . . . . . . . . . . . 12  |-  ( ( F  e.  dom  S.1  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( x  x.  y )  e.  RR )
113112, 1, 1, 50, 50, 102off 6553 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( F  oF  x.  F ) : RR --> RR )
114113adantr 465 . . . . . . . . . 10  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( F  oF  x.  F ) : RR --> RR )
115111adantl 466 . . . . . . . . . . . 12  |-  ( ( G  e.  dom  S.1  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( x  x.  y )  e.  RR )
116115, 3, 3, 30, 30, 102off 6553 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( G  oF  x.  G ) : RR --> RR )
117116adantl 466 . . . . . . . . . 10  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( G  oF  x.  G ) : RR --> RR )
118110, 114, 117, 24, 24, 102off 6553 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) : RR --> RR )
119 frn 5743 . . . . . . . . 9  |-  ( ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) : RR --> RR  ->  ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  C_  RR )
120118, 119syl 16 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  C_  RR )
121120, 71syl6ss 3511 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  C_  CC )
122 fnssres 5700 . . . . . . 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 ) ) )
123108, 121, 122sylancr 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 ) ) )
124 id 22 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  F  e.  dom  S.1 )
125124, 124i1fmul 22229 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ( F  oF  x.  F )  e.  dom  S.1 )
126125adantr 465 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( F  oF  x.  F )  e.  dom  S.1 )
127 id 22 . . . . . . . . . 10  |-  ( G  e.  dom  S.1  ->  G  e.  dom  S.1 )
128127, 127i1fmul 22229 . . . . . . . . 9  |-  ( G  e.  dom  S.1  ->  ( G  oF  x.  G )  e.  dom  S.1 )
129128adantl 466 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( G  oF  x.  G )  e.  dom  S.1 )
130126, 129i1fadd 22228 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  e. 
dom  S.1 )
131 i1frn 22210 . . . . . . 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 )
132130, 131syl 16 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ran  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  e. 
Fin )
133 fnfi 7816 . . . . . 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 )
134123, 132, 133syl2anc 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 )
135 rnfi 7823 . . . . 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 )
136134, 135syl 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 )
137106, 136syl5eqel 2549 . . 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 )
138 cnvco 5198 . . . . . . 7  |-  `' ( sqr  o.  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G )
) )  =  ( `' ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  o.  `' sqr )
139138imaeq1i 5344 . . . . . 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 } )
140 imaco 5518 . . . . . 6  |-  ( ( `' ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  o.  `' sqr ) " {
x } )  =  ( `' ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G )
) " ( `' sqr " { x } ) )
141139, 140eqtri 2486 . . . . 5  |-  ( `' ( sqr  o.  (
( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) ) " { x } )  =  ( `' ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G
) ) " ( `' sqr " { x } ) )
142 i1fima 22211 . . . . . 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 )
143130, 142syl 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 )
144141, 143syl5eqel 2549 . . . 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 )
145144adantr 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 )
146141fveq2i 5875 . . . 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 } ) ) )
147 eldifsni 4158 . . . . . . . 8  |-  ( x  e.  ( ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } )  ->  x  =/=  0 )
148 c0ex 9607 . . . . . . . . . . . 12  |-  0  e.  _V
149148elsnc 4056 . . . . . . . . . . 11  |-  ( 0  e.  { x }  <->  0  =  x )
150 eqcom 2466 . . . . . . . . . . 11  |-  ( 0  =  x  <->  x  = 
0 )
151149, 150bitri 249 . . . . . . . . . 10  |-  ( 0  e.  { x }  <->  x  =  0 )
152151necon3bbii 2718 . . . . . . . . 9  |-  ( -.  0  e.  { x } 
<->  x  =/=  0 )
153 sqrt0 13087 . . . . . . . . . 10  |-  ( sqr `  0 )  =  0
154153eleq1i 2534 . . . . . . . . 9  |-  ( ( sqr `  0 )  e.  { x }  <->  0  e.  { x }
)
155152, 154xchnxbir 309 . . . . . . . 8  |-  ( -.  ( sqr `  0
)  e.  { x } 
<->  x  =/=  0 )
156147, 155sylibr 212 . . . . . . 7  |-  ( x  e.  ( ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } )  ->  -.  ( sqr `  0 )  e. 
{ x } )
157156olcd 393 . . . . . 6  |-  ( x  e.  ( ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } )  ->  ( -.  0  e.  CC  \/  -.  ( sqr `  0
)  e.  { x } ) )
158 ianor 488 . . . . . . 7  |-  ( -.  ( 0  e.  CC  /\  ( sqr `  0
)  e.  { x } )  <->  ( -.  0  e.  CC  \/  -.  ( sqr `  0
)  e.  { x } ) )
159 elpreima 6008 . . . . . . . 8  |-  ( sqr 
Fn  CC  ->  ( 0  e.  ( `' sqr " { x } )  <-> 
( 0  e.  CC  /\  ( sqr `  0
)  e.  { x } ) ) )
16056, 107, 159mp2b 10 . . . . . . 7  |-  ( 0  e.  ( `' sqr " { x } )  <-> 
( 0  e.  CC  /\  ( sqr `  0
)  e.  { x } ) )
161158, 160xchnxbir 309 . . . . . 6  |-  ( -.  0  e.  ( `' sqr " { x } )  <->  ( -.  0  e.  CC  \/  -.  ( sqr `  0
)  e.  { x } ) )
162157, 161sylibr 212 . . . . 5  |-  ( x  e.  ( ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } )  ->  -.  0  e.  ( `' sqr " {
x } ) )
163 i1fima2 22212 . . . . 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 )
164130, 162, 163syl2an 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 )
165146, 164syl5eqel 2549 . . 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 )
166105, 137, 145, 165i1fd 22214 . 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 )
16761, 166eqeltrd 2545 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 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    \ cdif 3468    C_ wss 3471   {csn 4032   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011    o. ccom 5012    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537   Fincfn 7535   CCcc 9507   RRcr 9508   0cc0 9509   _ici 9511    + caddc 9512    x. cmul 9514   +oocpnf 9642    <_ cle 9646   2c2 10606   [,)cico 11556   ^cexp 12169   sqrcsqrt 13078   abscabs 13079   volcvol 22001   S.1citg1 22150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xadd 11344  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-xmet 18539  df-met 18540  df-ovol 22002  df-vol 22003  df-mbf 22154  df-itg1 22155
This theorem is referenced by:  ftc1anclem7  30280  ftc1anclem8  30281
  Copyright terms: Public domain W3C validator