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Theorem dvcmulf 19294
Description: The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
Hypotheses
Ref Expression
dvcmul.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
dvcmul.f  |-  ( ph  ->  F : X --> CC )
dvcmul.a  |-  ( ph  ->  A  e.  CC )
dvcmulf.df  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
Assertion
Ref Expression
dvcmulf  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  o F  x.  F ) )  =  ( ( S  X.  { A }
)  o F  x.  ( S  _D  F
) ) )

Proof of Theorem dvcmulf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dvcmul.s . . 3  |-  ( ph  ->  S  e.  { RR ,  CC } )
2 dvcmul.a . . . . 5  |-  ( ph  ->  A  e.  CC )
3 fconstg 5428 . . . . 5  |-  ( A  e.  CC  ->  ( X  X.  { A }
) : X --> { A } )
42, 3syl 15 . . . 4  |-  ( ph  ->  ( X  X.  { A } ) : X --> { A } )
52snssd 3760 . . . 4  |-  ( ph  ->  { A }  C_  CC )
6 fss 5397 . . . 4  |-  ( ( ( X  X.  { A } ) : X --> { A }  /\  { A }  C_  CC )  ->  ( X  X.  { A } ) : X --> CC )
74, 5, 6syl2anc 642 . . 3  |-  ( ph  ->  ( X  X.  { A } ) : X --> CC )
8 dvcmul.f . . 3  |-  ( ph  ->  F : X --> CC )
9 c0ex 8832 . . . . . 6  |-  0  e.  _V
109fconst 5427 . . . . 5  |-  ( X  X.  { 0 } ) : X --> { 0 }
11 recnprss 19254 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
121, 11syl 15 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
13 fconstg 5428 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( S  X.  { A }
) : S --> { A } )
142, 13syl 15 . . . . . . . . 9  |-  ( ph  ->  ( S  X.  { A } ) : S --> { A } )
15 fss 5397 . . . . . . . . 9  |-  ( ( ( S  X.  { A } ) : S --> { A }  /\  { A }  C_  CC )  ->  ( S  X.  { A } ) : S --> CC )
1614, 5, 15syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( S  X.  { A } ) : S --> CC )
17 ssid 3197 . . . . . . . . 9  |-  S  C_  S
1817a1i 10 . . . . . . . 8  |-  ( ph  ->  S  C_  S )
19 dvcmulf.df . . . . . . . . 9  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
20 dvbsss 19252 . . . . . . . . . 10  |-  dom  ( S  _D  F )  C_  S
2120a1i 10 . . . . . . . . 9  |-  ( ph  ->  dom  ( S  _D  F )  C_  S
)
2219, 21eqsstr3d 3213 . . . . . . . 8  |-  ( ph  ->  X  C_  S )
23 eqid 2283 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
24 eqid 2283 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  S )  =  ( ( TopOpen ` fld )t  S )
2523, 24dvres 19261 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  ( S  X.  { A } ) : S --> CC )  /\  ( S  C_  S  /\  X  C_  S ) )  -> 
( S  _D  (
( S  X.  { A } )  |`  X ) )  =  ( ( S  _D  ( S  X.  { A }
) )  |`  (
( int `  (
( TopOpen ` fld )t  S ) ) `  X ) ) )
2612, 16, 18, 22, 25syl22anc 1183 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  |`  X ) )  =  ( ( S  _D  ( S  X.  { A }
) )  |`  (
( int `  (
( TopOpen ` fld )t  S ) ) `  X ) ) )
27 resmpt 5000 . . . . . . . . . 10  |-  ( X 
C_  S  ->  (
( x  e.  S  |->  A )  |`  X )  =  ( x  e.  X  |->  A ) )
2822, 27syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  S  |->  A )  |`  X )  =  ( x  e.  X  |->  A ) )
29 fconstmpt 4732 . . . . . . . . . 10  |-  ( S  X.  { A }
)  =  ( x  e.  S  |->  A )
3029reseq1i 4951 . . . . . . . . 9  |-  ( ( S  X.  { A } )  |`  X )  =  ( ( x  e.  S  |->  A )  |`  X )
31 fconstmpt 4732 . . . . . . . . 9  |-  ( X  X.  { A }
)  =  ( x  e.  X  |->  A )
3228, 30, 313eqtr4g 2340 . . . . . . . 8  |-  ( ph  ->  ( ( S  X.  { A } )  |`  X )  =  ( X  X.  { A } ) )
3332oveq2d 5874 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  |`  X ) )  =  ( S  _D  ( X  X.  { A } ) ) )
34 resmpt 5000 . . . . . . . . 9  |-  ( X 
C_  S  ->  (
( x  e.  S  |->  0 )  |`  X )  =  ( x  e.  X  |->  0 ) )
3522, 34syl 15 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  S  |->  0 )  |`  X )  =  ( x  e.  X  |->  0 ) )
36 fconstg 5428 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  ( CC  X.  { A }
) : CC --> { A } )
372, 36syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  X.  { A } ) : CC --> { A } )
38 fss 5397 . . . . . . . . . . . . 13  |-  ( ( ( CC  X.  { A } ) : CC --> { A }  /\  { A }  C_  CC )  ->  ( CC  X.  { A } ) : CC --> CC )
3937, 5, 38syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( CC  X.  { A } ) : CC --> CC )
40 ssid 3197 . . . . . . . . . . . . 13  |-  CC  C_  CC
4140a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  CC  C_  CC )
42 dvconst 19266 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  ( CC  _D  ( CC  X.  { A } ) )  =  ( CC  X.  { 0 } ) )
432, 42syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  ( CC  X.  { A }
) )  =  ( CC  X.  { 0 } ) )
4443dmeqd 4881 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( CC  _D  ( CC  X.  { A } ) )  =  dom  ( CC  X.  { 0 } ) )
459fconst 5427 . . . . . . . . . . . . . . 15  |-  ( CC 
X.  { 0 } ) : CC --> { 0 }
4645fdmi 5394 . . . . . . . . . . . . . 14  |-  dom  ( CC  X.  { 0 } )  =  CC
4744, 46syl6eq 2331 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  ( CC  _D  ( CC  X.  { A } ) )  =  CC )
4812, 47sseqtr4d 3215 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  dom  ( CC 
_D  ( CC  X.  { A } ) ) )
49 dvres3 19263 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  ( CC  X.  { A } ) : CC --> CC )  /\  ( CC  C_  CC  /\  S  C_ 
dom  ( CC  _D  ( CC  X.  { A } ) ) ) )  ->  ( S  _D  ( ( CC  X.  { A } )  |`  S ) )  =  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S ) )
501, 39, 41, 48, 49syl22anc 1183 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  (
( CC  X.  { A } )  |`  S ) )  =  ( ( CC  _D  ( CC 
X.  { A }
) )  |`  S ) )
51 xpssres 4989 . . . . . . . . . . . . 13  |-  ( S 
C_  CC  ->  ( ( CC  X.  { A } )  |`  S )  =  ( S  X.  { A } ) )
5212, 51syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  X.  { A } )  |`  S )  =  ( S  X.  { A } ) )
5352oveq2d 5874 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  (
( CC  X.  { A } )  |`  S ) )  =  ( S  _D  ( S  X.  { A } ) ) )
5443reseq1d 4954 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S )  =  ( ( CC  X.  {
0 } )  |`  S ) )
55 xpssres 4989 . . . . . . . . . . . . 13  |-  ( S 
C_  CC  ->  ( ( CC  X.  { 0 } )  |`  S )  =  ( S  X.  { 0 } ) )
5612, 55syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  X.  { 0 } )  |`  S )  =  ( S  X.  { 0 } ) )
5754, 56eqtrd 2315 . . . . . . . . . . 11  |-  ( ph  ->  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S )  =  ( S  X.  { 0 } ) )
5850, 53, 573eqtr3d 2323 . . . . . . . . . 10  |-  ( ph  ->  ( S  _D  ( S  X.  { A }
) )  =  ( S  X.  { 0 } ) )
59 fconstmpt 4732 . . . . . . . . . 10  |-  ( S  X.  { 0 } )  =  ( x  e.  S  |->  0 )
6058, 59syl6eq 2331 . . . . . . . . 9  |-  ( ph  ->  ( S  _D  ( S  X.  { A }
) )  =  ( x  e.  S  |->  0 ) )
6123cnfldtopon 18292 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
62 resttopon 16892 . . . . . . . . . . . . 13  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  S  C_  CC )  ->  (
( TopOpen ` fld )t  S )  e.  (TopOn `  S ) )
6361, 12, 62sylancr 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( TopOpen ` fld )t  S )  e.  (TopOn `  S ) )
64 topontop 16664 . . . . . . . . . . . 12  |-  ( ( ( TopOpen ` fld )t  S )  e.  (TopOn `  S )  ->  (
( TopOpen ` fld )t  S )  e.  Top )
6563, 64syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( ( TopOpen ` fld )t  S )  e.  Top )
66 toponuni 16665 . . . . . . . . . . . . 13  |-  ( ( ( TopOpen ` fld )t  S )  e.  (TopOn `  S )  ->  S  =  U. ( ( TopOpen ` fld )t  S
) )
6763, 66syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  S  =  U. (
( TopOpen ` fld )t  S ) )
6822, 67sseqtrd 3214 . . . . . . . . . . 11  |-  ( ph  ->  X  C_  U. (
( TopOpen ` fld )t  S ) )
69 eqid 2283 . . . . . . . . . . . 12  |-  U. (
( TopOpen ` fld )t  S )  =  U. ( ( TopOpen ` fld )t  S )
7069ntrss2 16794 . . . . . . . . . . 11  |-  ( ( ( ( TopOpen ` fld )t  S )  e.  Top  /\  X  C_  U. (
( TopOpen ` fld )t  S ) )  -> 
( ( int `  (
( TopOpen ` fld )t  S ) ) `  X )  C_  X
)
7165, 68, 70syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X )  C_  X
)
7212, 8, 22, 24, 23dvbssntr 19250 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( S  _D  F )  C_  (
( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )
7319, 72eqsstr3d 3213 . . . . . . . . . 10  |-  ( ph  ->  X  C_  ( ( int `  ( ( TopOpen ` fld )t  S
) ) `  X
) )
7471, 73eqssd 3196 . . . . . . . . 9  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X )  =  X )
7560, 74reseq12d 4956 . . . . . . . 8  |-  ( ph  ->  ( ( S  _D  ( S  X.  { A } ) )  |`  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )  =  ( ( x  e.  S  |->  0 )  |`  X ) )
76 fconstmpt 4732 . . . . . . . . 9  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
7776a1i 10 . . . . . . . 8  |-  ( ph  ->  ( X  X.  {
0 } )  =  ( x  e.  X  |->  0 ) )
7835, 75, 773eqtr4d 2325 . . . . . . 7  |-  ( ph  ->  ( ( S  _D  ( S  X.  { A } ) )  |`  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )  =  ( X  X.  {
0 } ) )
7926, 33, 783eqtr3d 2323 . . . . . 6  |-  ( ph  ->  ( S  _D  ( X  X.  { A }
) )  =  ( X  X.  { 0 } ) )
8079feq1d 5379 . . . . 5  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) ) : X --> { 0 }  <-> 
( X  X.  {
0 } ) : X --> { 0 } ) )
8110, 80mpbiri 224 . . . 4  |-  ( ph  ->  ( S  _D  ( X  X.  { A }
) ) : X --> { 0 } )
82 fdm 5393 . . . 4  |-  ( ( S  _D  ( X  X.  { A }
) ) : X --> { 0 }  ->  dom  ( S  _D  ( X  X.  { A }
) )  =  X )
8381, 82syl 15 . . 3  |-  ( ph  ->  dom  ( S  _D  ( X  X.  { A } ) )  =  X )
841, 7, 8, 83, 19dvmulf 19292 . 2  |-  ( ph  ->  ( S  _D  (
( X  X.  { A } )  o F  x.  F ) )  =  ( ( ( S  _D  ( X  X.  { A }
) )  o F  x.  F )  o F  +  ( ( S  _D  F )  o F  x.  ( X  X.  { A }
) ) ) )
85 sseqin2 3388 . . . . . 6  |-  ( X 
C_  S  <->  ( S  i^i  X )  =  X )
8622, 85sylib 188 . . . . 5  |-  ( ph  ->  ( S  i^i  X
)  =  X )
87 mpteq1 4100 . . . . 5  |-  ( ( S  i^i  X )  =  X  ->  (
x  e.  ( S  i^i  X )  |->  ( A  x.  ( F `
 x ) ) )  =  ( x  e.  X  |->  ( A  x.  ( F `  x ) ) ) )
8886, 87syl 15 . . . 4  |-  ( ph  ->  ( x  e.  ( S  i^i  X ) 
|->  ( A  x.  ( F `  x )
) )  =  ( x  e.  X  |->  ( A  x.  ( F `
 x ) ) ) )
89 ffn 5389 . . . . . 6  |-  ( ( S  X.  { A } ) : S --> { A }  ->  ( S  X.  { A }
)  Fn  S )
9014, 89syl 15 . . . . 5  |-  ( ph  ->  ( S  X.  { A } )  Fn  S
)
91 ffn 5389 . . . . . 6  |-  ( F : X --> CC  ->  F  Fn  X )
928, 91syl 15 . . . . 5  |-  ( ph  ->  F  Fn  X )
93 ssexg 4160 . . . . . 6  |-  ( ( X  C_  S  /\  S  e.  { RR ,  CC } )  ->  X  e.  _V )
9422, 1, 93syl2anc 642 . . . . 5  |-  ( ph  ->  X  e.  _V )
95 eqid 2283 . . . . 5  |-  ( S  i^i  X )  =  ( S  i^i  X
)
96 fvconst2g 5727 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  S )  ->  ( ( S  X.  { A } ) `  x )  =  A )
972, 96sylan 457 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  (
( S  X.  { A } ) `  x
)  =  A )
98 eqidd 2284 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  =  ( F `  x ) )
9990, 92, 1, 94, 95, 97, 98offval 6085 . . . 4  |-  ( ph  ->  ( ( S  X.  { A } )  o F  x.  F )  =  ( x  e.  ( S  i^i  X
)  |->  ( A  x.  ( F `  x ) ) ) )
100 ffn 5389 . . . . . 6  |-  ( ( X  X.  { A } ) : X --> { A }  ->  ( X  X.  { A }
)  Fn  X )
1014, 100syl 15 . . . . 5  |-  ( ph  ->  ( X  X.  { A } )  Fn  X
)
102 inidm 3378 . . . . 5  |-  ( X  i^i  X )  =  X
103 fvconst2g 5727 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  X )  ->  ( ( X  X.  { A } ) `  x )  =  A )
1042, 103sylan 457 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( X  X.  { A } ) `  x
)  =  A )
105101, 92, 94, 94, 102, 104, 98offval 6085 . . . 4  |-  ( ph  ->  ( ( X  X.  { A } )  o F  x.  F )  =  ( x  e.  X  |->  ( A  x.  ( F `  x ) ) ) )
10688, 99, 1053eqtr4d 2325 . . 3  |-  ( ph  ->  ( ( S  X.  { A } )  o F  x.  F )  =  ( ( X  X.  { A }
)  o F  x.  F ) )
107106oveq2d 5874 . 2  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  o F  x.  F ) )  =  ( S  _D  ( ( X  X.  { A } )  o F  x.  F ) ) )
108 mpteq1 4100 . . . 4  |-  ( ( S  i^i  X )  =  X  ->  (
x  e.  ( S  i^i  X )  |->  ( A  x.  ( ( S  _D  F ) `
 x ) ) )  =  ( x  e.  X  |->  ( A  x.  ( ( S  _D  F ) `  x ) ) ) )
10986, 108syl 15 . . 3  |-  ( ph  ->  ( x  e.  ( S  i^i  X ) 
|->  ( A  x.  (
( S  _D  F
) `  x )
) )  =  ( x  e.  X  |->  ( A  x.  ( ( S  _D  F ) `
 x ) ) ) )
110 dvfg 19256 . . . . . . 7  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  F ) : dom  ( S  _D  F
) --> CC )
1111, 110syl 15 . . . . . 6  |-  ( ph  ->  ( S  _D  F
) : dom  ( S  _D  F ) --> CC )
11219feq2d 5380 . . . . . 6  |-  ( ph  ->  ( ( S  _D  F ) : dom  ( S  _D  F
) --> CC  <->  ( S  _D  F ) : X --> CC ) )
113111, 112mpbid 201 . . . . 5  |-  ( ph  ->  ( S  _D  F
) : X --> CC )
114 ffn 5389 . . . . 5  |-  ( ( S  _D  F ) : X --> CC  ->  ( S  _D  F )  Fn  X )
115113, 114syl 15 . . . 4  |-  ( ph  ->  ( S  _D  F
)  Fn  X )
116 eqidd 2284 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  =  ( ( S  _D  F ) `  x ) )
11790, 115, 1, 94, 95, 97, 116offval 6085 . . 3  |-  ( ph  ->  ( ( S  X.  { A } )  o F  x.  ( S  _D  F ) )  =  ( x  e.  ( S  i^i  X
)  |->  ( A  x.  ( ( S  _D  F ) `  x
) ) ) )
118 0cn 8831 . . . . . 6  |-  0  e.  CC
119118a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  0  e.  CC )
120 ovex 5883 . . . . . 6  |-  ( ( ( S  _D  F
) `  x )  x.  A )  e.  _V
121120a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  e.  _V )
12279oveq1d 5873 . . . . . . 7  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  o F  x.  F )  =  ( ( X  X.  { 0 } )  o F  x.  F ) )
123118a1i 10 . . . . . . . 8  |-  ( ph  ->  0  e.  CC )
124 mul02 8990 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
125124adantl 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
12694, 8, 123, 123, 125caofid2 6108 . . . . . . 7  |-  ( ph  ->  ( ( X  X.  { 0 } )  o F  x.  F
)  =  ( X  X.  { 0 } ) )
127122, 126eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  o F  x.  F )  =  ( X  X.  { 0 } ) )
128127, 76syl6eq 2331 . . . . 5  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  o F  x.  F )  =  ( x  e.  X  |->  0 ) )
129 fvex 5539 . . . . . . 7  |-  ( ( S  _D  F ) `
 x )  e. 
_V
130129a1i 10 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  _V )
1312adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  CC )
132113feqmptd 5575 . . . . . 6  |-  ( ph  ->  ( S  _D  F
)  =  ( x  e.  X  |->  ( ( S  _D  F ) `
 x ) ) )
13331a1i 10 . . . . . 6  |-  ( ph  ->  ( X  X.  { A } )  =  ( x  e.  X  |->  A ) )
13494, 130, 131, 132, 133offval2 6095 . . . . 5  |-  ( ph  ->  ( ( S  _D  F )  o F  x.  ( X  X.  { A } ) )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `
 x )  x.  A ) ) )
13594, 119, 121, 128, 134offval2 6095 . . . 4  |-  ( ph  ->  ( ( ( S  _D  ( X  X.  { A } ) )  o F  x.  F
)  o F  +  ( ( S  _D  F )  o F  x.  ( X  X.  { A } ) ) )  =  ( x  e.  X  |->  ( 0  +  ( ( ( S  _D  F ) `
 x )  x.  A ) ) ) )
136 ffvelrn 5663 . . . . . . . . 9  |-  ( ( ( S  _D  F
) : X --> CC  /\  x  e.  X )  ->  ( ( S  _D  F ) `  x
)  e.  CC )
137113, 136sylan 457 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  CC )
138137, 131mulcld 8855 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  e.  CC )
139138addid2d 9013 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
0  +  ( ( ( S  _D  F
) `  x )  x.  A ) )  =  ( ( ( S  _D  F ) `  x )  x.  A
) )
140137, 131mulcomd 8856 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  =  ( A  x.  ( ( S  _D  F ) `  x
) ) )
141139, 140eqtrd 2315 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
0  +  ( ( ( S  _D  F
) `  x )  x.  A ) )  =  ( A  x.  (
( S  _D  F
) `  x )
) )
142141mpteq2dva 4106 . . . 4  |-  ( ph  ->  ( x  e.  X  |->  ( 0  +  ( ( ( S  _D  F ) `  x
)  x.  A ) ) )  =  ( x  e.  X  |->  ( A  x.  ( ( S  _D  F ) `
 x ) ) ) )
143135, 142eqtrd 2315 . . 3  |-  ( ph  ->  ( ( ( S  _D  ( X  X.  { A } ) )  o F  x.  F
)  o F  +  ( ( S  _D  F )  o F  x.  ( X  X.  { A } ) ) )  =  ( x  e.  X  |->  ( A  x.  ( ( S  _D  F ) `  x ) ) ) )
144109, 117, 1433eqtr4d 2325 . 2  |-  ( ph  ->  ( ( S  X.  { A } )  o F  x.  ( S  _D  F ) )  =  ( ( ( S  _D  ( X  X.  { A }
) )  o F  x.  F )  o F  +  ( ( S  _D  F )  o F  x.  ( X  X.  { A }
) ) ) )
14584, 107, 1443eqtr4d 2325 1  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  o F  x.  F ) )  =  ( ( S  X.  { A }
)  o F  x.  ( S  _D  F
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   {csn 3640   {cpr 3641   U.cuni 3827    e. cmpt 4077    X. cxp 4687   dom cdm 4689    |` cres 4691    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742   ↾t crest 13325   TopOpenctopn 13326  ℂfldccnfld 16377   Topctop 16631  TopOnctopon 16632   intcnt 16754    _D cdv 19213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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