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Theorem dvcmulf 21399
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 5592 . . . . 5  |-  ( A  e.  CC  ->  ( X  X.  { A }
) : X --> { A } )
42, 3syl 16 . . . 4  |-  ( ph  ->  ( X  X.  { A } ) : X --> { A } )
52snssd 4013 . . . 4  |-  ( ph  ->  { A }  C_  CC )
6 fss 5562 . . . 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 9372 . . . . . 6  |-  0  e.  _V
109fconst 5591 . . . . 5  |-  ( X  X.  { 0 } ) : X --> { 0 }
11 recnprss 21359 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
121, 11syl 16 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
13 fconstg 5592 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( S  X.  { A }
) : S --> { A } )
142, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  ( S  X.  { A } ) : S --> { A } )
15 fss 5562 . . . . . . . . 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 3370 . . . . . . . . 9  |-  S  C_  S
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  S  C_  S )
19 dvcmulf.df . . . . . . . . 9  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
20 dvbsss 21357 . . . . . . . . . 10  |-  dom  ( S  _D  F )  C_  S
2120a1i 11 . . . . . . . . 9  |-  ( ph  ->  dom  ( S  _D  F )  C_  S
)
2219, 21eqsstr3d 3386 . . . . . . . 8  |-  ( ph  ->  X  C_  S )
23 eqid 2438 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
24 eqid 2438 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  S )  =  ( ( TopOpen ` fld )t  S )
2523, 24dvres 21366 . . . . . . . 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 1219 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  |`  X ) )  =  ( ( S  _D  ( S  X.  { A }
) )  |`  (
( int `  (
( TopOpen ` fld )t  S ) ) `  X ) ) )
27 resmpt 5151 . . . . . . . . . 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 4877 . . . . . . . . . 10  |-  ( S  X.  { A }
)  =  ( x  e.  S  |->  A )
3029reseq1i 5101 . . . . . . . . 9  |-  ( ( S  X.  { A } )  |`  X )  =  ( ( x  e.  S  |->  A )  |`  X )
31 fconstmpt 4877 . . . . . . . . 9  |-  ( X  X.  { A }
)  =  ( x  e.  X  |->  A )
3228, 30, 313eqtr4g 2495 . . . . . . . 8  |-  ( ph  ->  ( ( S  X.  { A } )  |`  X )  =  ( X  X.  { A } ) )
3332oveq2d 6102 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  |`  X ) )  =  ( S  _D  ( X  X.  { A } ) ) )
34 resmpt 5151 . . . . . . . . 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 5592 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  ( CC  X.  { A }
) : CC --> { A } )
372, 36syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  X.  { A } ) : CC --> { A } )
38 fss 5562 . . . . . . . . . . . . 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 3370 . . . . . . . . . . . . 13  |-  CC  C_  CC
4140a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  CC  C_  CC )
42 dvconst 21371 . . . . . . . . . . . . . . . 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 5037 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( CC  _D  ( CC  X.  { A } ) )  =  dom  ( CC  X.  { 0 } ) )
459fconst 5591 . . . . . . . . . . . . . . 15  |-  ( CC 
X.  { 0 } ) : CC --> { 0 }
4645fdmi 5559 . . . . . . . . . . . . . 14  |-  dom  ( CC  X.  { 0 } )  =  CC
4744, 46syl6eq 2486 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  ( CC  _D  ( CC  X.  { A } ) )  =  CC )
4812, 47sseqtr4d 3388 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  dom  ( CC 
_D  ( CC  X.  { A } ) ) )
49 dvres3 21368 . . . . . . . . . . . 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 1219 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  (
( CC  X.  { A } )  |`  S ) )  =  ( ( CC  _D  ( CC 
X.  { A }
) )  |`  S ) )
51 xpssres 5139 . . . . . . . . . . . . 13  |-  ( S 
C_  CC  ->  ( ( CC  X.  { A } )  |`  S )  =  ( S  X.  { A } ) )
5212, 51syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  X.  { A } )  |`  S )  =  ( S  X.  { A } ) )
5352oveq2d 6102 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  (
( CC  X.  { A } )  |`  S ) )  =  ( S  _D  ( S  X.  { A } ) ) )
5443reseq1d 5104 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S )  =  ( ( CC  X.  {
0 } )  |`  S ) )
55 xpssres 5139 . . . . . . . . . . . . 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 2470 . . . . . . . . . . 11  |-  ( ph  ->  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S )  =  ( S  X.  { 0 } ) )
5850, 53, 573eqtr3d 2478 . . . . . . . . . 10  |-  ( ph  ->  ( S  _D  ( S  X.  { A }
) )  =  ( S  X.  { 0 } ) )
59 fconstmpt 4877 . . . . . . . . . 10  |-  ( S  X.  { 0 } )  =  ( x  e.  S  |->  0 )
6058, 59syl6eq 2486 . . . . . . . . 9  |-  ( ph  ->  ( S  _D  ( S  X.  { A }
) )  =  ( x  e.  S  |->  0 ) )
6123cnfldtopon 20342 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
62 resttopon 18745 . . . . . . . . . . . . 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 18511 . . . . . . . . . . . 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 18512 . . . . . . . . . . . . 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 3387 . . . . . . . . . . 11  |-  ( ph  ->  X  C_  U. (
( TopOpen ` fld )t  S ) )
69 eqid 2438 . . . . . . . . . . . 12  |-  U. (
( TopOpen ` fld )t  S )  =  U. ( ( TopOpen ` fld )t  S )
7069ntrss2 18641 . . . . . . . . . . 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 21355 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( S  _D  F )  C_  (
( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )
7319, 72eqsstr3d 3386 . . . . . . . . . 10  |-  ( ph  ->  X  C_  ( ( int `  ( ( TopOpen ` fld )t  S
) ) `  X
) )
7471, 73eqssd 3368 . . . . . . . . 9  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X )  =  X )
7560, 74reseq12d 5106 . . . . . . . 8  |-  ( ph  ->  ( ( S  _D  ( S  X.  { A } ) )  |`  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )  =  ( ( x  e.  S  |->  0 )  |`  X ) )
76 fconstmpt 4877 . . . . . . . . 9  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
7776a1i 11 . . . . . . . 8  |-  ( ph  ->  ( X  X.  {
0 } )  =  ( x  e.  X  |->  0 ) )
7835, 75, 773eqtr4d 2480 . . . . . . 7  |-  ( ph  ->  ( ( S  _D  ( S  X.  { A } ) )  |`  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )  =  ( X  X.  {
0 } ) )
7926, 33, 783eqtr3d 2478 . . . . . 6  |-  ( ph  ->  ( S  _D  ( X  X.  { A }
) )  =  ( X  X.  { 0 } ) )
8079feq1d 5541 . . . . 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 5558 . . . 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 21397 . 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 3564 . . . . . 6  |-  ( X 
C_  S  <->  ( S  i^i  X )  =  X )
8622, 85sylib 196 . . . . 5  |-  ( ph  ->  ( S  i^i  X
)  =  X )
8786mpteq1d 4368 . . . 4  |-  ( ph  ->  ( x  e.  ( S  i^i  X ) 
|->  ( A  x.  ( F `  x )
) )  =  ( x  e.  X  |->  ( A  x.  ( F `
 x ) ) ) )
88 ffn 5554 . . . . . 6  |-  ( ( S  X.  { A } ) : S --> { A }  ->  ( S  X.  { A }
)  Fn  S )
8914, 88syl 16 . . . . 5  |-  ( ph  ->  ( S  X.  { A } )  Fn  S
)
90 ffn 5554 . . . . . 6  |-  ( F : X --> CC  ->  F  Fn  X )
918, 90syl 16 . . . . 5  |-  ( ph  ->  F  Fn  X )
921, 22ssexd 4434 . . . . 5  |-  ( ph  ->  X  e.  _V )
93 eqid 2438 . . . . 5  |-  ( S  i^i  X )  =  ( S  i^i  X
)
94 fvconst2g 5926 . . . . . 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 2439 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  =  ( F `  x ) )
9789, 91, 1, 92, 93, 95, 96offval 6322 . . . 4  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  F )  =  ( x  e.  ( S  i^i  X
)  |->  ( A  x.  ( F `  x ) ) ) )
98 ffn 5554 . . . . . 6  |-  ( ( X  X.  { A } ) : X --> { A }  ->  ( X  X.  { A }
)  Fn  X )
994, 98syl 16 . . . . 5  |-  ( ph  ->  ( X  X.  { A } )  Fn  X
)
100 inidm 3554 . . . . 5  |-  ( X  i^i  X )  =  X
101 fvconst2g 5926 . . . . . 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 6322 . . . 4  |-  ( ph  ->  ( ( X  X.  { A } )  oF  x.  F )  =  ( x  e.  X  |->  ( A  x.  ( F `  x ) ) ) )
10487, 97, 1033eqtr4d 2480 . . 3  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  F )  =  ( ( X  X.  { A }
)  oF  x.  F ) )
105104oveq2d 6102 . 2  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  oF  x.  F ) )  =  ( S  _D  ( ( X  X.  { A } )  oF  x.  F ) ) )
10686mpteq1d 4368 . . 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 21361 . . . . . . 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 5542 . . . . . 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 5554 . . . . 5  |-  ( ( S  _D  F ) : X --> CC  ->  ( S  _D  F )  Fn  X )
112110, 111syl 16 . . . 4  |-  ( ph  ->  ( S  _D  F
)  Fn  X )
113 eqidd 2439 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  =  ( ( S  _D  F ) `  x ) )
11489, 112, 1, 92, 93, 95, 113offval 6322 . . 3  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  ( S  _D  F ) )  =  ( x  e.  ( S  i^i  X
)  |->  ( A  x.  ( ( S  _D  F ) `  x
) ) ) )
115 0cnd 9371 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  0  e.  CC )
116 ovex 6111 . . . . . 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 6101 . . . . . . 7  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F )  =  ( ( X  X.  { 0 } )  oF  x.  F ) )
119 0cnd 9371 . . . . . . . 8  |-  ( ph  ->  0  e.  CC )
120 mul02 9539 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
121120adantl 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
12292, 8, 119, 119, 121caofid2 6346 . . . . . . 7  |-  ( ph  ->  ( ( X  X.  { 0 } )  oF  x.  F
)  =  ( X  X.  { 0 } ) )
123118, 122eqtrd 2470 . . . . . 6  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F )  =  ( X  X.  { 0 } ) )
124123, 76syl6eq 2486 . . . . 5  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F )  =  ( x  e.  X  |->  0 ) )
125 fvex 5696 . . . . . . 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 5739 . . . . . 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 6331 . . . . 5  |-  ( ph  ->  ( ( S  _D  F )  oF  x.  ( X  X.  { A } ) )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `
 x )  x.  A ) ) )
13192, 115, 117, 124, 130offval2 6331 . . . 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 5838 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  CC )
133132, 127mulcld 9398 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  e.  CC )
134133addid2d 9562 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
0  +  ( ( ( S  _D  F
) `  x )  x.  A ) )  =  ( ( ( S  _D  F ) `  x )  x.  A
) )
135132, 127mulcomd 9399 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  =  ( A  x.  ( ( S  _D  F ) `  x
) ) )
136134, 135eqtrd 2470 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
0  +  ( ( ( S  _D  F
) `  x )  x.  A ) )  =  ( A  x.  (
( S  _D  F
) `  x )
) )
137136mpteq2dva 4373 . . . 4  |-  ( ph  ->  ( x  e.  X  |->  ( 0  +  ( ( ( S  _D  F ) `  x
)  x.  A ) ) )  =  ( x  e.  X  |->  ( A  x.  ( ( S  _D  F ) `
 x ) ) ) )
138131, 137eqtrd 2470 . . 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 2480 . 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 2480 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 1369    e. wcel 1756   _Vcvv 2967    i^i cin 3322    C_ wss 3323   {csn 3872   {cpr 3874   U.cuni 4086    e. cmpt 4345    X. cxp 4833   dom cdm 4835    |` cres 4837    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086    oFcof 6313   CCcc 9272   RRcr 9273   0cc0 9274    + caddc 9277    x. cmul 9279   ↾t crest 14351   TopOpenctopn 14352  ℂfldccnfld 17798   Topctop 18478  TopOnctopon 18479   intcnt 18601    _D cdv 21318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-icc 11299  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-lp 18720  df-perf 18721  df-cn 18811  df-cnp 18812  df-haus 18899  df-tx 19115  df-hmeo 19308  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-tms 19877  df-cncf 20434  df-limc 21321  df-dv 21322
This theorem is referenced by: (None)
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