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Theorem dvcmulf 19310
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 5444 . . . . 5  |-  ( A  e.  CC  ->  ( X  X.  { A }
) : X --> { A } )
42, 3syl 15 . . . 4  |-  ( ph  ->  ( X  X.  { A } ) : X --> { A } )
52snssd 3776 . . . 4  |-  ( ph  ->  { A }  C_  CC )
6 fss 5413 . . . 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 8848 . . . . . 6  |-  0  e.  _V
109fconst 5443 . . . . 5  |-  ( X  X.  { 0 } ) : X --> { 0 }
11 recnprss 19270 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
121, 11syl 15 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
13 fconstg 5444 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( S  X.  { A }
) : S --> { A } )
142, 13syl 15 . . . . . . . . 9  |-  ( ph  ->  ( S  X.  { A } ) : S --> { A } )
15 fss 5413 . . . . . . . . 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 3210 . . . . . . . . 9  |-  S  C_  S
1817a1i 10 . . . . . . . 8  |-  ( ph  ->  S  C_  S )
19 dvcmulf.df . . . . . . . . 9  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
20 dvbsss 19268 . . . . . . . . . 10  |-  dom  ( S  _D  F )  C_  S
2120a1i 10 . . . . . . . . 9  |-  ( ph  ->  dom  ( S  _D  F )  C_  S
)
2219, 21eqsstr3d 3226 . . . . . . . 8  |-  ( ph  ->  X  C_  S )
23 eqid 2296 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
24 eqid 2296 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  S )  =  ( ( TopOpen ` fld )t  S )
2523, 24dvres 19277 . . . . . . . 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 5016 . . . . . . . . . 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 4748 . . . . . . . . . 10  |-  ( S  X.  { A }
)  =  ( x  e.  S  |->  A )
3029reseq1i 4967 . . . . . . . . 9  |-  ( ( S  X.  { A } )  |`  X )  =  ( ( x  e.  S  |->  A )  |`  X )
31 fconstmpt 4748 . . . . . . . . 9  |-  ( X  X.  { A }
)  =  ( x  e.  X  |->  A )
3228, 30, 313eqtr4g 2353 . . . . . . . 8  |-  ( ph  ->  ( ( S  X.  { A } )  |`  X )  =  ( X  X.  { A } ) )
3332oveq2d 5890 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  |`  X ) )  =  ( S  _D  ( X  X.  { A } ) ) )
34 resmpt 5016 . . . . . . . . 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 5444 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  ( CC  X.  { A }
) : CC --> { A } )
372, 36syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  X.  { A } ) : CC --> { A } )
38 fss 5413 . . . . . . . . . . . . 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 3210 . . . . . . . . . . . . 13  |-  CC  C_  CC
4140a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  CC  C_  CC )
42 dvconst 19282 . . . . . . . . . . . . . . . 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 4897 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( CC  _D  ( CC  X.  { A } ) )  =  dom  ( CC  X.  { 0 } ) )
459fconst 5443 . . . . . . . . . . . . . . 15  |-  ( CC 
X.  { 0 } ) : CC --> { 0 }
4645fdmi 5410 . . . . . . . . . . . . . 14  |-  dom  ( CC  X.  { 0 } )  =  CC
4744, 46syl6eq 2344 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  ( CC  _D  ( CC  X.  { A } ) )  =  CC )
4812, 47sseqtr4d 3228 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  dom  ( CC 
_D  ( CC  X.  { A } ) ) )
49 dvres3 19279 . . . . . . . . . . . 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 5005 . . . . . . . . . . . . 13  |-  ( S 
C_  CC  ->  ( ( CC  X.  { A } )  |`  S )  =  ( S  X.  { A } ) )
5212, 51syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  X.  { A } )  |`  S )  =  ( S  X.  { A } ) )
5352oveq2d 5890 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  (
( CC  X.  { A } )  |`  S ) )  =  ( S  _D  ( S  X.  { A } ) ) )
5443reseq1d 4970 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S )  =  ( ( CC  X.  {
0 } )  |`  S ) )
55 xpssres 5005 . . . . . . . . . . . . 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 2328 . . . . . . . . . . 11  |-  ( ph  ->  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S )  =  ( S  X.  { 0 } ) )
5850, 53, 573eqtr3d 2336 . . . . . . . . . 10  |-  ( ph  ->  ( S  _D  ( S  X.  { A }
) )  =  ( S  X.  { 0 } ) )
59 fconstmpt 4748 . . . . . . . . . 10  |-  ( S  X.  { 0 } )  =  ( x  e.  S  |->  0 )
6058, 59syl6eq 2344 . . . . . . . . 9  |-  ( ph  ->  ( S  _D  ( S  X.  { A }
) )  =  ( x  e.  S  |->  0 ) )
6123cnfldtopon 18308 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
62 resttopon 16908 . . . . . . . . . . . . 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 16680 . . . . . . . . . . . 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 16681 . . . . . . . . . . . . 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 3227 . . . . . . . . . . 11  |-  ( ph  ->  X  C_  U. (
( TopOpen ` fld )t  S ) )
69 eqid 2296 . . . . . . . . . . . 12  |-  U. (
( TopOpen ` fld )t  S )  =  U. ( ( TopOpen ` fld )t  S )
7069ntrss2 16810 . . . . . . . . . . 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 19266 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( S  _D  F )  C_  (
( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )
7319, 72eqsstr3d 3226 . . . . . . . . . 10  |-  ( ph  ->  X  C_  ( ( int `  ( ( TopOpen ` fld )t  S
) ) `  X
) )
7471, 73eqssd 3209 . . . . . . . . 9  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X )  =  X )
7560, 74reseq12d 4972 . . . . . . . 8  |-  ( ph  ->  ( ( S  _D  ( S  X.  { A } ) )  |`  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )  =  ( ( x  e.  S  |->  0 )  |`  X ) )
76 fconstmpt 4748 . . . . . . . . 9  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
7776a1i 10 . . . . . . . 8  |-  ( ph  ->  ( X  X.  {
0 } )  =  ( x  e.  X  |->  0 ) )
7835, 75, 773eqtr4d 2338 . . . . . . 7  |-  ( ph  ->  ( ( S  _D  ( S  X.  { A } ) )  |`  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )  =  ( X  X.  {
0 } ) )
7926, 33, 783eqtr3d 2336 . . . . . 6  |-  ( ph  ->  ( S  _D  ( X  X.  { A }
) )  =  ( X  X.  { 0 } ) )
8079feq1d 5395 . . . . 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 5409 . . . 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 19308 . 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 3401 . . . . . 6  |-  ( X 
C_  S  <->  ( S  i^i  X )  =  X )
8622, 85sylib 188 . . . . 5  |-  ( ph  ->  ( S  i^i  X
)  =  X )
87 mpteq1 4116 . . . . 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 5405 . . . . . 6  |-  ( ( S  X.  { A } ) : S --> { A }  ->  ( S  X.  { A }
)  Fn  S )
9014, 89syl 15 . . . . 5  |-  ( ph  ->  ( S  X.  { A } )  Fn  S
)
91 ffn 5405 . . . . . 6  |-  ( F : X --> CC  ->  F  Fn  X )
928, 91syl 15 . . . . 5  |-  ( ph  ->  F  Fn  X )
93 ssexg 4176 . . . . . 6  |-  ( ( X  C_  S  /\  S  e.  { RR ,  CC } )  ->  X  e.  _V )
9422, 1, 93syl2anc 642 . . . . 5  |-  ( ph  ->  X  e.  _V )
95 eqid 2296 . . . . 5  |-  ( S  i^i  X )  =  ( S  i^i  X
)
96 fvconst2g 5743 . . . . . 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 2297 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  =  ( F `  x ) )
9990, 92, 1, 94, 95, 97, 98offval 6101 . . . 4  |-  ( ph  ->  ( ( S  X.  { A } )  o F  x.  F )  =  ( x  e.  ( S  i^i  X
)  |->  ( A  x.  ( F `  x ) ) ) )
100 ffn 5405 . . . . . 6  |-  ( ( X  X.  { A } ) : X --> { A }  ->  ( X  X.  { A }
)  Fn  X )
1014, 100syl 15 . . . . 5  |-  ( ph  ->  ( X  X.  { A } )  Fn  X
)
102 inidm 3391 . . . . 5  |-  ( X  i^i  X )  =  X
103 fvconst2g 5743 . . . . . 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 6101 . . . 4  |-  ( ph  ->  ( ( X  X.  { A } )  o F  x.  F )  =  ( x  e.  X  |->  ( A  x.  ( F `  x ) ) ) )
10688, 99, 1053eqtr4d 2338 . . 3  |-  ( ph  ->  ( ( S  X.  { A } )  o F  x.  F )  =  ( ( X  X.  { A }
)  o F  x.  F ) )
107106oveq2d 5890 . 2  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  o F  x.  F ) )  =  ( S  _D  ( ( X  X.  { A } )  o F  x.  F ) ) )
108 mpteq1 4116 . . . 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 19272 . . . . . . 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 5396 . . . . . 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 5405 . . . . 5  |-  ( ( S  _D  F ) : X --> CC  ->  ( S  _D  F )  Fn  X )
115113, 114syl 15 . . . 4  |-  ( ph  ->  ( S  _D  F
)  Fn  X )
116 eqidd 2297 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  =  ( ( S  _D  F ) `  x ) )
11790, 115, 1, 94, 95, 97, 116offval 6101 . . 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 8847 . . . . . 6  |-  0  e.  CC
119118a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  0  e.  CC )
120 ovex 5899 . . . . . 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 5889 . . . . . . 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 9006 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
125124adantl 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
12694, 8, 123, 123, 125caofid2 6124 . . . . . . 7  |-  ( ph  ->  ( ( X  X.  { 0 } )  o F  x.  F
)  =  ( X  X.  { 0 } ) )
127122, 126eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  o F  x.  F )  =  ( X  X.  { 0 } ) )
128127, 76syl6eq 2344 . . . . 5  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  o F  x.  F )  =  ( x  e.  X  |->  0 ) )
129 fvex 5555 . . . . . . 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 5591 . . . . . 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 6111 . . . . 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 6111 . . . 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 5679 . . . . . . . . 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 8871 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  e.  CC )
139138addid2d 9029 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
0  +  ( ( ( S  _D  F
) `  x )  x.  A ) )  =  ( ( ( S  _D  F ) `  x )  x.  A
) )
140137, 131mulcomd 8872 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  =  ( A  x.  ( ( S  _D  F ) `  x
) ) )
141139, 140eqtrd 2328 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
0  +  ( ( ( S  _D  F
) `  x )  x.  A ) )  =  ( A  x.  (
( S  _D  F
) `  x )
) )
142141mpteq2dva 4122 . . . 4  |-  ( ph  ->  ( x  e.  X  |->  ( 0  +  ( ( ( S  _D  F ) `  x
)  x.  A ) ) )  =  ( x  e.  X  |->  ( A  x.  ( ( S  _D  F ) `
 x ) ) ) )
143135, 142eqtrd 2328 . . 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 2338 . 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 2338 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 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   {csn 3653   {cpr 3654   U.cuni 3843    e. cmpt 4093    X. cxp 4703   dom cdm 4705    |` cres 4707    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756    x. cmul 8758   ↾t crest 13341   TopOpenctopn 13342  ℂfldccnfld 16393   Topctop 16647  TopOnctopon 16648   intcnt 16770    _D cdv 19229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
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