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Theorem dvcmulf 22076
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 } )  oF  x.  F ) )  =  ( ( S  X.  { A }
)  oF  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 5763 . . . . 5  |-  ( A  e.  CC  ->  ( X  X.  { A }
) : X --> { A } )
42, 3syl 16 . . . 4  |-  ( ph  ->  ( X  X.  { A } ) : X --> { A } )
52snssd 4165 . . . 4  |-  ( ph  ->  { A }  C_  CC )
6 fss 5730 . . . 4  |-  ( ( ( X  X.  { A } ) : X --> { A }  /\  { A }  C_  CC )  ->  ( X  X.  { A } ) : X --> CC )
74, 5, 6syl2anc 661 . . 3  |-  ( ph  ->  ( X  X.  { A } ) : X --> CC )
8 dvcmul.f . . 3  |-  ( ph  ->  F : X --> CC )
9 c0ex 9579 . . . . . 6  |-  0  e.  _V
109fconst 5762 . . . . 5  |-  ( X  X.  { 0 } ) : X --> { 0 }
11 recnprss 22036 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
121, 11syl 16 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
13 fconstg 5763 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( S  X.  { A }
) : S --> { A } )
142, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  ( S  X.  { A } ) : S --> { A } )
15 fss 5730 . . . . . . . . 9  |-  ( ( ( S  X.  { A } ) : S --> { A }  /\  { A }  C_  CC )  ->  ( S  X.  { A } ) : S --> CC )
1614, 5, 15syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( S  X.  { A } ) : S --> CC )
17 ssid 3516 . . . . . . . . 9  |-  S  C_  S
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  S  C_  S )
19 dvcmulf.df . . . . . . . . 9  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
20 dvbsss 22034 . . . . . . . . . 10  |-  dom  ( S  _D  F )  C_  S
2120a1i 11 . . . . . . . . 9  |-  ( ph  ->  dom  ( S  _D  F )  C_  S
)
2219, 21eqsstr3d 3532 . . . . . . . 8  |-  ( ph  ->  X  C_  S )
23 eqid 2460 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
24 eqid 2460 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  S )  =  ( ( TopOpen ` fld )t  S )
2523, 24dvres 22043 . . . . . . . 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 1224 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  |`  X ) )  =  ( ( S  _D  ( S  X.  { A }
) )  |`  (
( int `  (
( TopOpen ` fld )t  S ) ) `  X ) ) )
27 resmpt 5314 . . . . . . . . . 10  |-  ( X 
C_  S  ->  (
( x  e.  S  |->  A )  |`  X )  =  ( x  e.  X  |->  A ) )
2822, 27syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  S  |->  A )  |`  X )  =  ( x  e.  X  |->  A ) )
29 fconstmpt 5035 . . . . . . . . . 10  |-  ( S  X.  { A }
)  =  ( x  e.  S  |->  A )
3029reseq1i 5260 . . . . . . . . 9  |-  ( ( S  X.  { A } )  |`  X )  =  ( ( x  e.  S  |->  A )  |`  X )
31 fconstmpt 5035 . . . . . . . . 9  |-  ( X  X.  { A }
)  =  ( x  e.  X  |->  A )
3228, 30, 313eqtr4g 2526 . . . . . . . 8  |-  ( ph  ->  ( ( S  X.  { A } )  |`  X )  =  ( X  X.  { A } ) )
3332oveq2d 6291 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  |`  X ) )  =  ( S  _D  ( X  X.  { A } ) ) )
34 resmpt 5314 . . . . . . . . 9  |-  ( X 
C_  S  ->  (
( x  e.  S  |->  0 )  |`  X )  =  ( x  e.  X  |->  0 ) )
3522, 34syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  S  |->  0 )  |`  X )  =  ( x  e.  X  |->  0 ) )
36 fconstg 5763 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  ( CC  X.  { A }
) : CC --> { A } )
372, 36syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  X.  { A } ) : CC --> { A } )
38 fss 5730 . . . . . . . . . . . . 13  |-  ( ( ( CC  X.  { A } ) : CC --> { A }  /\  { A }  C_  CC )  ->  ( CC  X.  { A } ) : CC --> CC )
3937, 5, 38syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  ( CC  X.  { A } ) : CC --> CC )
40 ssid 3516 . . . . . . . . . . . . 13  |-  CC  C_  CC
4140a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  CC  C_  CC )
42 dvconst 22048 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  ( CC  _D  ( CC  X.  { A } ) )  =  ( CC  X.  { 0 } ) )
432, 42syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  ( CC  X.  { A }
) )  =  ( CC  X.  { 0 } ) )
4443dmeqd 5196 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( CC  _D  ( CC  X.  { A } ) )  =  dom  ( CC  X.  { 0 } ) )
459fconst 5762 . . . . . . . . . . . . . . 15  |-  ( CC 
X.  { 0 } ) : CC --> { 0 }
4645fdmi 5727 . . . . . . . . . . . . . 14  |-  dom  ( CC  X.  { 0 } )  =  CC
4744, 46syl6eq 2517 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  ( CC  _D  ( CC  X.  { A } ) )  =  CC )
4812, 47sseqtr4d 3534 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  dom  ( CC 
_D  ( CC  X.  { A } ) ) )
49 dvres3 22045 . . . . . . . . . . . 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 1224 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  (
( CC  X.  { A } )  |`  S ) )  =  ( ( CC  _D  ( CC 
X.  { A }
) )  |`  S ) )
51 xpssres 5299 . . . . . . . . . . . . 13  |-  ( S 
C_  CC  ->  ( ( CC  X.  { A } )  |`  S )  =  ( S  X.  { A } ) )
5212, 51syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  X.  { A } )  |`  S )  =  ( S  X.  { A } ) )
5352oveq2d 6291 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  (
( CC  X.  { A } )  |`  S ) )  =  ( S  _D  ( S  X.  { A } ) ) )
5443reseq1d 5263 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S )  =  ( ( CC  X.  {
0 } )  |`  S ) )
55 xpssres 5299 . . . . . . . . . . . . 13  |-  ( S 
C_  CC  ->  ( ( CC  X.  { 0 } )  |`  S )  =  ( S  X.  { 0 } ) )
5612, 55syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  X.  { 0 } )  |`  S )  =  ( S  X.  { 0 } ) )
5754, 56eqtrd 2501 . . . . . . . . . . 11  |-  ( ph  ->  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S )  =  ( S  X.  { 0 } ) )
5850, 53, 573eqtr3d 2509 . . . . . . . . . 10  |-  ( ph  ->  ( S  _D  ( S  X.  { A }
) )  =  ( S  X.  { 0 } ) )
59 fconstmpt 5035 . . . . . . . . . 10  |-  ( S  X.  { 0 } )  =  ( x  e.  S  |->  0 )
6058, 59syl6eq 2517 . . . . . . . . 9  |-  ( ph  ->  ( S  _D  ( S  X.  { A }
) )  =  ( x  e.  S  |->  0 ) )
6123cnfldtopon 21018 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
62 resttopon 19421 . . . . . . . . . . . . 13  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  S  C_  CC )  ->  (
( TopOpen ` fld )t  S )  e.  (TopOn `  S ) )
6361, 12, 62sylancr 663 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( TopOpen ` fld )t  S )  e.  (TopOn `  S ) )
64 topontop 19187 . . . . . . . . . . . 12  |-  ( ( ( TopOpen ` fld )t  S )  e.  (TopOn `  S )  ->  (
( TopOpen ` fld )t  S )  e.  Top )
6563, 64syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( TopOpen ` fld )t  S )  e.  Top )
66 toponuni 19188 . . . . . . . . . . . . 13  |-  ( ( ( TopOpen ` fld )t  S )  e.  (TopOn `  S )  ->  S  =  U. ( ( TopOpen ` fld )t  S
) )
6763, 66syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  S  =  U. (
( TopOpen ` fld )t  S ) )
6822, 67sseqtrd 3533 . . . . . . . . . . 11  |-  ( ph  ->  X  C_  U. (
( TopOpen ` fld )t  S ) )
69 eqid 2460 . . . . . . . . . . . 12  |-  U. (
( TopOpen ` fld )t  S )  =  U. ( ( TopOpen ` fld )t  S )
7069ntrss2 19317 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X )  C_  X
)
7212, 8, 22, 24, 23dvbssntr 22032 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( S  _D  F )  C_  (
( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )
7319, 72eqsstr3d 3532 . . . . . . . . . 10  |-  ( ph  ->  X  C_  ( ( int `  ( ( TopOpen ` fld )t  S
) ) `  X
) )
7471, 73eqssd 3514 . . . . . . . . 9  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X )  =  X )
7560, 74reseq12d 5265 . . . . . . . 8  |-  ( ph  ->  ( ( S  _D  ( S  X.  { A } ) )  |`  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )  =  ( ( x  e.  S  |->  0 )  |`  X ) )
76 fconstmpt 5035 . . . . . . . . 9  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
7776a1i 11 . . . . . . . 8  |-  ( ph  ->  ( X  X.  {
0 } )  =  ( x  e.  X  |->  0 ) )
7835, 75, 773eqtr4d 2511 . . . . . . 7  |-  ( ph  ->  ( ( S  _D  ( S  X.  { A } ) )  |`  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )  =  ( X  X.  {
0 } ) )
7926, 33, 783eqtr3d 2509 . . . . . 6  |-  ( ph  ->  ( S  _D  ( X  X.  { A }
) )  =  ( X  X.  { 0 } ) )
8079feq1d 5708 . . . . 5  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) ) : X --> { 0 }  <-> 
( X  X.  {
0 } ) : X --> { 0 } ) )
8110, 80mpbiri 233 . . . 4  |-  ( ph  ->  ( S  _D  ( X  X.  { A }
) ) : X --> { 0 } )
82 fdm 5726 . . . 4  |-  ( ( S  _D  ( X  X.  { A }
) ) : X --> { 0 }  ->  dom  ( S  _D  ( X  X.  { A }
) )  =  X )
8381, 82syl 16 . . 3  |-  ( ph  ->  dom  ( S  _D  ( X  X.  { A } ) )  =  X )
841, 7, 8, 83, 19dvmulf 22074 . 2  |-  ( ph  ->  ( S  _D  (
( X  X.  { A } )  oF  x.  F ) )  =  ( ( ( S  _D  ( X  X.  { A }
) )  oF  x.  F )  oF  +  ( ( S  _D  F )  oF  x.  ( X  X.  { A }
) ) ) )
85 sseqin2 3710 . . . . . 6  |-  ( X 
C_  S  <->  ( S  i^i  X )  =  X )
8622, 85sylib 196 . . . . 5  |-  ( ph  ->  ( S  i^i  X
)  =  X )
8786mpteq1d 4521 . . . 4  |-  ( ph  ->  ( x  e.  ( S  i^i  X ) 
|->  ( A  x.  ( F `  x )
) )  =  ( x  e.  X  |->  ( A  x.  ( F `
 x ) ) ) )
88 ffn 5722 . . . . . 6  |-  ( ( S  X.  { A } ) : S --> { A }  ->  ( S  X.  { A }
)  Fn  S )
8914, 88syl 16 . . . . 5  |-  ( ph  ->  ( S  X.  { A } )  Fn  S
)
90 ffn 5722 . . . . . 6  |-  ( F : X --> CC  ->  F  Fn  X )
918, 90syl 16 . . . . 5  |-  ( ph  ->  F  Fn  X )
921, 22ssexd 4587 . . . . 5  |-  ( ph  ->  X  e.  _V )
93 eqid 2460 . . . . 5  |-  ( S  i^i  X )  =  ( S  i^i  X
)
94 fvconst2g 6105 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  S )  ->  ( ( S  X.  { A } ) `  x )  =  A )
952, 94sylan 471 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  (
( S  X.  { A } ) `  x
)  =  A )
96 eqidd 2461 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  =  ( F `  x ) )
9789, 91, 1, 92, 93, 95, 96offval 6522 . . . 4  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  F )  =  ( x  e.  ( S  i^i  X
)  |->  ( A  x.  ( F `  x ) ) ) )
98 ffn 5722 . . . . . 6  |-  ( ( X  X.  { A } ) : X --> { A }  ->  ( X  X.  { A }
)  Fn  X )
994, 98syl 16 . . . . 5  |-  ( ph  ->  ( X  X.  { A } )  Fn  X
)
100 inidm 3700 . . . . 5  |-  ( X  i^i  X )  =  X
101 fvconst2g 6105 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  X )  ->  ( ( X  X.  { A } ) `  x )  =  A )
1022, 101sylan 471 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( X  X.  { A } ) `  x
)  =  A )
10399, 91, 92, 92, 100, 102, 96offval 6522 . . . 4  |-  ( ph  ->  ( ( X  X.  { A } )  oF  x.  F )  =  ( x  e.  X  |->  ( A  x.  ( F `  x ) ) ) )
10487, 97, 1033eqtr4d 2511 . . 3  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  F )  =  ( ( X  X.  { A }
)  oF  x.  F ) )
105104oveq2d 6291 . 2  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  oF  x.  F ) )  =  ( S  _D  ( ( X  X.  { A } )  oF  x.  F ) ) )
10686mpteq1d 4521 . . 3  |-  ( ph  ->  ( x  e.  ( S  i^i  X ) 
|->  ( A  x.  (
( S  _D  F
) `  x )
) )  =  ( x  e.  X  |->  ( A  x.  ( ( S  _D  F ) `
 x ) ) ) )
107 dvfg 22038 . . . . . . 7  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  F ) : dom  ( S  _D  F
) --> CC )
1081, 107syl 16 . . . . . 6  |-  ( ph  ->  ( S  _D  F
) : dom  ( S  _D  F ) --> CC )
10919feq2d 5709 . . . . . 6  |-  ( ph  ->  ( ( S  _D  F ) : dom  ( S  _D  F
) --> CC  <->  ( S  _D  F ) : X --> CC ) )
110108, 109mpbid 210 . . . . 5  |-  ( ph  ->  ( S  _D  F
) : X --> CC )
111 ffn 5722 . . . . 5  |-  ( ( S  _D  F ) : X --> CC  ->  ( S  _D  F )  Fn  X )
112110, 111syl 16 . . . 4  |-  ( ph  ->  ( S  _D  F
)  Fn  X )
113 eqidd 2461 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  =  ( ( S  _D  F ) `  x ) )
11489, 112, 1, 92, 93, 95, 113offval 6522 . . 3  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  ( S  _D  F ) )  =  ( x  e.  ( S  i^i  X
)  |->  ( A  x.  ( ( S  _D  F ) `  x
) ) ) )
115 0cnd 9578 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  0  e.  CC )
116 ovex 6300 . . . . . 6  |-  ( ( ( S  _D  F
) `  x )  x.  A )  e.  _V
117116a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  e.  _V )
11879oveq1d 6290 . . . . . . 7  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F )  =  ( ( X  X.  { 0 } )  oF  x.  F ) )
119 0cnd 9578 . . . . . . . 8  |-  ( ph  ->  0  e.  CC )
120 mul02 9746 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
121120adantl 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
12292, 8, 119, 119, 121caofid2 6546 . . . . . . 7  |-  ( ph  ->  ( ( X  X.  { 0 } )  oF  x.  F
)  =  ( X  X.  { 0 } ) )
123118, 122eqtrd 2501 . . . . . 6  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F )  =  ( X  X.  { 0 } ) )
124123, 76syl6eq 2517 . . . . 5  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F )  =  ( x  e.  X  |->  0 ) )
125 fvex 5867 . . . . . . 7  |-  ( ( S  _D  F ) `
 x )  e. 
_V
126125a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  _V )
1272adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  CC )
128110feqmptd 5911 . . . . . 6  |-  ( ph  ->  ( S  _D  F
)  =  ( x  e.  X  |->  ( ( S  _D  F ) `
 x ) ) )
12931a1i 11 . . . . . 6  |-  ( ph  ->  ( X  X.  { A } )  =  ( x  e.  X  |->  A ) )
13092, 126, 127, 128, 129offval2 6531 . . . . 5  |-  ( ph  ->  ( ( S  _D  F )  oF  x.  ( X  X.  { A } ) )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `
 x )  x.  A ) ) )
13192, 115, 117, 124, 130offval2 6531 . . . 4  |-  ( ph  ->  ( ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F
)  oF  +  ( ( S  _D  F )  oF  x.  ( X  X.  { A } ) ) )  =  ( x  e.  X  |->  ( 0  +  ( ( ( S  _D  F ) `
 x )  x.  A ) ) ) )
132110ffvelrnda 6012 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  CC )
133132, 127mulcld 9605 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  e.  CC )
134133addid2d 9769 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
0  +  ( ( ( S  _D  F
) `  x )  x.  A ) )  =  ( ( ( S  _D  F ) `  x )  x.  A
) )
135132, 127mulcomd 9606 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  =  ( A  x.  ( ( S  _D  F ) `  x
) ) )
136134, 135eqtrd 2501 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
0  +  ( ( ( S  _D  F
) `  x )  x.  A ) )  =  ( A  x.  (
( S  _D  F
) `  x )
) )
137136mpteq2dva 4526 . . . 4  |-  ( ph  ->  ( x  e.  X  |->  ( 0  +  ( ( ( S  _D  F ) `  x
)  x.  A ) ) )  =  ( x  e.  X  |->  ( A  x.  ( ( S  _D  F ) `
 x ) ) ) )
138131, 137eqtrd 2501 . . 3  |-  ( ph  ->  ( ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F
)  oF  +  ( ( S  _D  F )  oF  x.  ( X  X.  { A } ) ) )  =  ( x  e.  X  |->  ( A  x.  ( ( S  _D  F ) `  x ) ) ) )
139106, 114, 1383eqtr4d 2511 . 2  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  ( S  _D  F ) )  =  ( ( ( S  _D  ( X  X.  { A }
) )  oF  x.  F )  oF  +  ( ( S  _D  F )  oF  x.  ( X  X.  { A }
) ) ) )
14084, 105, 1393eqtr4d 2511 1  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  oF  x.  F ) )  =  ( ( S  X.  { A }
)  oF  x.  ( S  _D  F
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106    i^i cin 3468    C_ wss 3469   {csn 4020   {cpr 4022   U.cuni 4238    |-> cmpt 4498    X. cxp 4990   dom cdm 4992    |` cres 4994    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275    oFcof 6513   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484    x. cmul 9486   ↾t crest 14665   TopOpenctopn 14666  ℂfldccnfld 18184   Topctop 19154  TopOnctopon 19155   intcnt 19277    _D cdv 21995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cn 19487  df-cnp 19488  df-haus 19575  df-tx 19791  df-hmeo 19984  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-tms 20553  df-cncf 21110  df-limc 21998  df-dv 21999
This theorem is referenced by:  dvsinax  31060
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