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Theorem ftc1anclem3 31983
Description: Lemma for ftc1anc 31989- 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 22632 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
21ffvelrnda 6037 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( F `  x )  e.  RR )
3 i1ff 22632 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
43ffvelrnda 6037 . . . . . . 7  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( G `  x )  e.  RR )
5 absreim 13356 . . . . . . 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 479 . . . . . 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 838 . . . . 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 9676 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( F `  x )  e.  CC )
98sqvald 12419 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( F `
 x ) ^
2 )  =  ( ( F `  x
)  x.  ( F `
 x ) ) )
104recnd 9676 . . . . . . . . 9  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( G `  x )  e.  CC )
1110sqvald 12419 . . . . . . . 8  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( G `
 x ) ^
2 )  =  ( ( G `  x
)  x.  ( G `
 x ) ) )
129, 11oveqan12d 6324 . . . . . . 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 838 . . . . . 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 5885 . . . . 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 2463 . . . 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 4510 . . 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 9605 . . . . . . 7  |-  _i  e.  CC
18 mulcl 9630 . . . . . . 7  |-  ( ( _i  e.  CC  /\  ( G `  x )  e.  CC )  -> 
( _i  x.  ( G `  x )
)  e.  CC )
1917, 10, 18sylancr 667 . . . . . 6  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( _i  x.  ( G `  x ) )  e.  CC )
20 addcl 9628 . . . . . 6  |-  ( ( ( F `  x
)  e.  CC  /\  ( _i  x.  ( G `  x )
)  e.  CC )  ->  ( ( F `
 x )  +  ( _i  x.  ( G `  x )
) )  e.  CC )
218, 19, 20syl2an 479 . . . . 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 838 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  (
( F `  x
)  +  ( _i  x.  ( G `  x ) ) )  e.  CC )
23 reex 9637 . . . . . 6  |-  RR  e.  _V
2423a1i 11 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  RR  e.  _V )
252adantlr 719 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  ( F `  x )  e.  RR )
26 ovex 6333 . . . . . 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 5934 . . . . . 6  |-  ( F  e.  dom  S.1  ->  F  =  ( x  e.  RR  |->  ( F `  x ) ) )
2928adantr 466 . . . . 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 4897 . . . . . . . 8  |-  ( RR 
X.  { _i }
)  =  ( x  e.  RR  |->  _i )
3332a1i 11 . . . . . . 7  |-  ( G  e.  dom  S.1  ->  ( RR  X.  { _i } )  =  ( x  e.  RR  |->  _i ) )
343feqmptd 5934 . . . . . . 7  |-  ( G  e.  dom  S.1  ->  G  =  ( x  e.  RR  |->  ( G `  x ) ) )
3530, 31, 4, 33, 34offval2 6562 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ( ( RR  X.  {
_i } )  oF  x.  G )  =  ( x  e.  RR  |->  ( _i  x.  ( G `  x ) ) ) )
3635adantl 467 . . . . 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 6562 . . . 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 13400 . . . . . 6  |-  abs : CC
--> RR
3938a1i 11 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  abs : CC --> RR )
4039feqmptd 5934 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  abs  =  (
y  e.  CC  |->  ( abs `  y ) ) )
41 fveq2 5881 . . . 4  |-  ( y  =  ( ( F `
 x )  +  ( _i  x.  ( G `  x )
) )  ->  ( abs `  y )  =  ( abs `  (
( F `  x
)  +  ( _i  x.  ( G `  x ) ) ) ) )
4222, 37, 40, 41fmptco 6071 . . 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 9670 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( F `
 x )  x.  ( F `  x
) )  e.  CC )
4410, 10mulcld 9670 . . . . . 6  |-  ( ( G  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( G `
 x )  x.  ( G `  x
) )  e.  CC )
45 addcl 9628 . . . . . 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 479 . . . . 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 838 . . . 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 719 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  x  e.  RR )  ->  (
( F `  x
)  x.  ( F `
 x ) )  e.  CC )
4944adantll 718 . . . . 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 6562 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( F  oF  x.  F )  =  ( x  e.  RR  |->  ( ( F `  x
)  x.  ( F `
 x ) ) ) )
5251adantr 466 . . . . 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 6562 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ( G  oF  x.  G )  =  ( x  e.  RR  |->  ( ( G `  x
)  x.  ( G `
 x ) ) ) )
5453adantl 467 . . . . 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 6562 . . . 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 13426 . . . . . 6  |-  sqr : CC
--> CC
5756a1i 11 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  sqr : CC --> CC )
5857feqmptd 5934 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  sqr  =  (
y  e.  CC  |->  ( sqr `  y ) ) )
59 fveq2 5881 . . . 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 6071 . . 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 2473 . 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 11745 . . . . . . 7  |-  ( x  e.  ( 0 [,) +oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
63 resqrtcl 13317 . . . . . . 7  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( sqr `  x
)  e.  RR )
6462, 63sylbi 198 . . . . . 6  |-  ( x  e.  ( 0 [,) +oo )  ->  ( sqr `  x )  e.  RR )
6564adantl 467 . . . . 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 5934 . . . . . . . 8  |-  ( sqr
: CC --> CC  ->  sqr  =  ( x  e.  CC  |->  ( sqr `  x
) ) )
6856, 67ax-mp 5 . . . . . . 7  |-  sqr  =  ( x  e.  CC  |->  ( sqr `  x ) )
6968reseq1i 5120 . . . . . 6  |-  ( sqr  |`  ( 0 [,) +oo ) )  =  ( ( x  e.  CC  |->  ( sqr `  x ) )  |`  ( 0 [,) +oo ) )
70 rge0ssre 11747 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR
71 ax-resscn 9603 . . . . . . . 8  |-  RR  C_  CC
7270, 71sstri 3473 . . . . . . 7  |-  ( 0 [,) +oo )  C_  CC
73 resmpt 5173 . . . . . . 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 2451 . . . . 5  |-  ( sqr  |`  ( 0 [,) +oo ) )  =  ( x  e.  ( 0 [,) +oo )  |->  ( sqr `  x ) )
7665, 75fmptd 6061 . . . 4  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( sqr  |`  (
0 [,) +oo )
) : ( 0 [,) +oo ) --> RR )
77 ge0addcl 11751 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  +  y )  e.  ( 0 [,) +oo )
)
7877adantl 467 . . . . 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 6314 . . . . . . . . 9  |-  ( ( z  =  F  /\  z  =  F )  ->  ( z  oF  x.  z )  =  ( F  oF  x.  F ) )
8079anidms 649 . . . . . . . 8  |-  ( z  =  F  ->  (
z  oF  x.  z )  =  ( F  oF  x.  F ) )
8180feq1d 5732 . . . . . . 7  |-  ( z  =  F  ->  (
( z  oF  x.  z ) : RR --> ( 0 [,) +oo )  <->  ( F  oF  x.  F ) : RR --> ( 0 [,) +oo ) ) )
82 i1ff 22632 . . . . . . . . . . . 12  |-  ( z  e.  dom  S.1  ->  z : RR --> RR )
8382ffvelrnda 6037 . . . . . . . . . . 11  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  ( z `  x )  e.  RR )
8483, 83remulcld 9678 . . . . . . . . . 10  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( z `
 x )  x.  ( z `  x
) )  e.  RR )
8583msqge0d 10189 . . . . . . . . . 10  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  0  <_  (
( z `  x
)  x.  ( z `
 x ) ) )
86 elrege0 11745 . . . . . . . . . 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 668 . . . . . . . . 9  |-  ( ( z  e.  dom  S.1  /\  x  e.  RR )  ->  ( ( z `
 x )  x.  ( z `  x
) )  e.  ( 0 [,) +oo )
)
88 eqid 2422 . . . . . . . . 9  |-  ( x  e.  RR  |->  ( ( z `  x )  x.  ( z `  x ) ) )  =  ( x  e.  RR  |->  ( ( z `
 x )  x.  ( z `  x
) ) )
8987, 88fmptd 6061 . . . . . . . 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 5934 . . . . . . . . . 10  |-  ( z  e.  dom  S.1  ->  z  =  ( x  e.  RR  |->  ( z `  x ) ) )
9290, 83, 83, 91, 91offval2 6562 . . . . . . . . 9  |-  ( z  e.  dom  S.1  ->  ( z  oF  x.  z )  =  ( x  e.  RR  |->  ( ( z `  x
)  x.  ( z `
 x ) ) ) )
9392feq1d 5732 . . . . . . . 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 235 . . . . . . 7  |-  ( z  e.  dom  S.1  ->  ( z  oF  x.  z ) : RR --> ( 0 [,) +oo ) )
9581, 94vtoclga 3145 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( F  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
9695adantr 466 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( F  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
97 oveq12 6314 . . . . . . . . 9  |-  ( ( z  =  G  /\  z  =  G )  ->  ( z  oF  x.  z )  =  ( G  oF  x.  G ) )
9897anidms 649 . . . . . . . 8  |-  ( z  =  G  ->  (
z  oF  x.  z )  =  ( G  oF  x.  G ) )
9998feq1d 5732 . . . . . . 7  |-  ( z  =  G  ->  (
( z  oF  x.  z ) : RR --> ( 0 [,) +oo )  <->  ( G  oF  x.  G ) : RR --> ( 0 [,) +oo ) ) )
10099, 94vtoclga 3145 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ( G  oF  x.  G ) : RR --> ( 0 [,) +oo ) )
101100adantl 467 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( G  oF  x.  G ) : RR --> ( 0 [,) +oo ) )
102 inidm 3671 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
10378, 96, 101, 24, 24, 102off 6560 . . . 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 5757 . . . 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 665 . . 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 5360 . . . 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 5746 . . . . . . . 8  |-  ( sqr
: CC --> CC  ->  sqr 
Fn  CC )
10856, 107ax-mp 5 . . . . . . 7  |-  sqr  Fn  CC
109 readdcl 9629 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
110109adantl 467 . . . . . . . . . 10  |-  ( ( ( F  e.  dom  S.1 
/\  G  e.  dom  S.1 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
111 remulcl 9631 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
112111adantl 467 . . . . . . . . . . . 12  |-  ( ( F  e.  dom  S.1  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( x  x.  y )  e.  RR )
113112, 1, 1, 50, 50, 102off 6560 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( F  oF  x.  F ) : RR --> RR )
114113adantr 466 . . . . . . . . . 10  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( F  oF  x.  F ) : RR --> RR )
115111adantl 467 . . . . . . . . . . . 12  |-  ( ( G  e.  dom  S.1  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( x  x.  y )  e.  RR )
116115, 3, 3, 30, 30, 102off 6560 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( G  oF  x.  G ) : RR --> RR )
117116adantl 467 . . . . . . . . . 10  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( G  oF  x.  G ) : RR --> RR )
118110, 114, 117, 24, 24, 102off 6560 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) : RR --> RR )
119 frn 5752 . . . . . . . . 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 17 . . . . . . . 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 3476 . . . . . . 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 5707 . . . . . . 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 667 . . . . . 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 22652 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  ( F  oF  x.  F )  e.  dom  S.1 )
126125adantr 466 . . . . . . . 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 22652 . . . . . . . . 9  |-  ( G  e.  dom  S.1  ->  ( G  oF  x.  G )  e.  dom  S.1 )
129128adantl 467 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( G  oF  x.  G )  e.  dom  S.1 )
130126, 129i1fadd 22651 . . . . . . 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 22633 . . . . . . 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 17 . . . . . 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 7858 . . . . . 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 665 . . . . 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 7866 . . . . 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 17 . . . 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 2511 . . 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 5039 . . . . . . 7  |-  `' ( sqr  o.  ( ( F  oF  x.  F )  oF  +  ( G  oF  x.  G )
) )  =  ( `' ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) )  o.  `' sqr )
139138imaeq1i 5184 . . . . . 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 5359 . . . . . 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 2451 . . . . 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 22634 . . . . . 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 17 . . . . 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 2511 . . . 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 466 . . 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 5884 . . . 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 4126 . . . . . . . 8  |-  ( x  e.  ( ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } )  ->  x  =/=  0 )
148 c0ex 9644 . . . . . . . . . . . 12  |-  0  e.  _V
149148elsnc 4022 . . . . . . . . . . 11  |-  ( 0  e.  { x }  <->  0  =  x )
150 eqcom 2431 . . . . . . . . . . 11  |-  ( 0  =  x  <->  x  = 
0 )
151149, 150bitri 252 . . . . . . . . . 10  |-  ( 0  e.  { x }  <->  x  =  0 )
152151necon3bbii 2681 . . . . . . . . 9  |-  ( -.  0  e.  { x } 
<->  x  =/=  0 )
153 sqrt0 13305 . . . . . . . . . 10  |-  ( sqr `  0 )  =  0
154153eleq1i 2498 . . . . . . . . 9  |-  ( ( sqr `  0 )  e.  { x }  <->  0  e.  { x }
)
155152, 154xchnxbir 310 . . . . . . . 8  |-  ( -.  ( sqr `  0
)  e.  { x } 
<->  x  =/=  0 )
156147, 155sylibr 215 . . . . . . 7  |-  ( x  e.  ( ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } )  ->  -.  ( sqr `  0 )  e. 
{ x } )
157156olcd 394 . . . . . 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 490 . . . . . . 7  |-  ( -.  ( 0  e.  CC  /\  ( sqr `  0
)  e.  { x } )  <->  ( -.  0  e.  CC  \/  -.  ( sqr `  0
)  e.  { x } ) )
159 elpreima 6017 . . . . . . . 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 310 . . . . . 6  |-  ( -.  0  e.  ( `' sqr " { x } )  <->  ( -.  0  e.  CC  \/  -.  ( sqr `  0
)  e.  { x } ) )
162157, 161sylibr 215 . . . . 5  |-  ( x  e.  ( ran  ( sqr  o.  ( ( F  oF  x.  F
)  oF  +  ( G  oF  x.  G ) ) ) 
\  { 0 } )  ->  -.  0  e.  ( `' sqr " {
x } ) )
163 i1fima2 22635 . . . . 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 479 . . . 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 2511 . . 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 22637 . 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 2507 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   _Vcvv 3080    \ cdif 3433    C_ wss 3436   {csn 3998   class class class wbr 4423    |-> cmpt 4482    X. cxp 4851   `'ccnv 4852   dom cdm 4853   ran crn 4854    |` cres 4855   "cima 4856    o. ccom 4857    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    oFcof 6543   Fincfn 7580   CCcc 9544   RRcr 9545   0cc0 9546   _ici 9548    + caddc 9549    x. cmul 9551   +oocpnf 9679    <_ cle 9683   2c2 10666   [,)cico 11644   ^cexp 12278   sqrcsqrt 13296   abscabs 13297   volcvol 22413   S.1citg1 22571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-inf2 8155  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-2o 7194  df-oadd 7197  df-er 7374  df-map 7485  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-inf 7966  df-oi 8034  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-q 11272  df-rp 11310  df-xadd 11417  df-ioo 11646  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12034  df-seq 12220  df-exp 12279  df-hash 12522  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13551  df-sum 13752  df-xmet 18962  df-met 18963  df-ovol 22414  df-vol 22416  df-mbf 22575  df-itg1 22576
This theorem is referenced by:  ftc1anclem7  31987  ftc1anclem8  31988
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