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Theorem dvcmulf 21535
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 5695 . . . . 5  |-  ( A  e.  CC  ->  ( X  X.  { A }
) : X --> { A } )
42, 3syl 16 . . . 4  |-  ( ph  ->  ( X  X.  { A } ) : X --> { A } )
52snssd 4116 . . . 4  |-  ( ph  ->  { A }  C_  CC )
6 fss 5665 . . . 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 9481 . . . . . 6  |-  0  e.  _V
109fconst 5694 . . . . 5  |-  ( X  X.  { 0 } ) : X --> { 0 }
11 recnprss 21495 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
121, 11syl 16 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
13 fconstg 5695 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( S  X.  { A }
) : S --> { A } )
142, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  ( S  X.  { A } ) : S --> { A } )
15 fss 5665 . . . . . . . . 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 3473 . . . . . . . . 9  |-  S  C_  S
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  S  C_  S )
19 dvcmulf.df . . . . . . . . 9  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
20 dvbsss 21493 . . . . . . . . . 10  |-  dom  ( S  _D  F )  C_  S
2120a1i 11 . . . . . . . . 9  |-  ( ph  ->  dom  ( S  _D  F )  C_  S
)
2219, 21eqsstr3d 3489 . . . . . . . 8  |-  ( ph  ->  X  C_  S )
23 eqid 2451 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
24 eqid 2451 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  S )  =  ( ( TopOpen ` fld )t  S )
2523, 24dvres 21502 . . . . . . . 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 1220 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  |`  X ) )  =  ( ( S  _D  ( S  X.  { A }
) )  |`  (
( int `  (
( TopOpen ` fld )t  S ) ) `  X ) ) )
27 resmpt 5254 . . . . . . . . . 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 4980 . . . . . . . . . 10  |-  ( S  X.  { A }
)  =  ( x  e.  S  |->  A )
3029reseq1i 5204 . . . . . . . . 9  |-  ( ( S  X.  { A } )  |`  X )  =  ( ( x  e.  S  |->  A )  |`  X )
31 fconstmpt 4980 . . . . . . . . 9  |-  ( X  X.  { A }
)  =  ( x  e.  X  |->  A )
3228, 30, 313eqtr4g 2517 . . . . . . . 8  |-  ( ph  ->  ( ( S  X.  { A } )  |`  X )  =  ( X  X.  { A } ) )
3332oveq2d 6206 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  |`  X ) )  =  ( S  _D  ( X  X.  { A } ) ) )
34 resmpt 5254 . . . . . . . . 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 5695 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  ( CC  X.  { A }
) : CC --> { A } )
372, 36syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  X.  { A } ) : CC --> { A } )
38 fss 5665 . . . . . . . . . . . . 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 3473 . . . . . . . . . . . . 13  |-  CC  C_  CC
4140a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  CC  C_  CC )
42 dvconst 21507 . . . . . . . . . . . . . . . 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 5140 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( CC  _D  ( CC  X.  { A } ) )  =  dom  ( CC  X.  { 0 } ) )
459fconst 5694 . . . . . . . . . . . . . . 15  |-  ( CC 
X.  { 0 } ) : CC --> { 0 }
4645fdmi 5662 . . . . . . . . . . . . . 14  |-  dom  ( CC  X.  { 0 } )  =  CC
4744, 46syl6eq 2508 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  ( CC  _D  ( CC  X.  { A } ) )  =  CC )
4812, 47sseqtr4d 3491 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  dom  ( CC 
_D  ( CC  X.  { A } ) ) )
49 dvres3 21504 . . . . . . . . . . . 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 1220 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  (
( CC  X.  { A } )  |`  S ) )  =  ( ( CC  _D  ( CC 
X.  { A }
) )  |`  S ) )
51 xpssres 5242 . . . . . . . . . . . . 13  |-  ( S 
C_  CC  ->  ( ( CC  X.  { A } )  |`  S )  =  ( S  X.  { A } ) )
5212, 51syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  X.  { A } )  |`  S )  =  ( S  X.  { A } ) )
5352oveq2d 6206 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  (
( CC  X.  { A } )  |`  S ) )  =  ( S  _D  ( S  X.  { A } ) ) )
5443reseq1d 5207 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S )  =  ( ( CC  X.  {
0 } )  |`  S ) )
55 xpssres 5242 . . . . . . . . . . . . 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 2492 . . . . . . . . . . 11  |-  ( ph  ->  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S )  =  ( S  X.  { 0 } ) )
5850, 53, 573eqtr3d 2500 . . . . . . . . . 10  |-  ( ph  ->  ( S  _D  ( S  X.  { A }
) )  =  ( S  X.  { 0 } ) )
59 fconstmpt 4980 . . . . . . . . . 10  |-  ( S  X.  { 0 } )  =  ( x  e.  S  |->  0 )
6058, 59syl6eq 2508 . . . . . . . . 9  |-  ( ph  ->  ( S  _D  ( S  X.  { A }
) )  =  ( x  e.  S  |->  0 ) )
6123cnfldtopon 20478 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
62 resttopon 18881 . . . . . . . . . . . . 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 18647 . . . . . . . . . . . 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 18648 . . . . . . . . . . . . 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 3490 . . . . . . . . . . 11  |-  ( ph  ->  X  C_  U. (
( TopOpen ` fld )t  S ) )
69 eqid 2451 . . . . . . . . . . . 12  |-  U. (
( TopOpen ` fld )t  S )  =  U. ( ( TopOpen ` fld )t  S )
7069ntrss2 18777 . . . . . . . . . . 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 21491 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( S  _D  F )  C_  (
( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )
7319, 72eqsstr3d 3489 . . . . . . . . . 10  |-  ( ph  ->  X  C_  ( ( int `  ( ( TopOpen ` fld )t  S
) ) `  X
) )
7471, 73eqssd 3471 . . . . . . . . 9  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X )  =  X )
7560, 74reseq12d 5209 . . . . . . . 8  |-  ( ph  ->  ( ( S  _D  ( S  X.  { A } ) )  |`  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )  =  ( ( x  e.  S  |->  0 )  |`  X ) )
76 fconstmpt 4980 . . . . . . . . 9  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
7776a1i 11 . . . . . . . 8  |-  ( ph  ->  ( X  X.  {
0 } )  =  ( x  e.  X  |->  0 ) )
7835, 75, 773eqtr4d 2502 . . . . . . 7  |-  ( ph  ->  ( ( S  _D  ( S  X.  { A } ) )  |`  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )  =  ( X  X.  {
0 } ) )
7926, 33, 783eqtr3d 2500 . . . . . 6  |-  ( ph  ->  ( S  _D  ( X  X.  { A }
) )  =  ( X  X.  { 0 } ) )
8079feq1d 5644 . . . . 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 5661 . . . 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 21533 . 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 3667 . . . . . 6  |-  ( X 
C_  S  <->  ( S  i^i  X )  =  X )
8622, 85sylib 196 . . . . 5  |-  ( ph  ->  ( S  i^i  X
)  =  X )
8786mpteq1d 4471 . . . 4  |-  ( ph  ->  ( x  e.  ( S  i^i  X ) 
|->  ( A  x.  ( F `  x )
) )  =  ( x  e.  X  |->  ( A  x.  ( F `
 x ) ) ) )
88 ffn 5657 . . . . . 6  |-  ( ( S  X.  { A } ) : S --> { A }  ->  ( S  X.  { A }
)  Fn  S )
8914, 88syl 16 . . . . 5  |-  ( ph  ->  ( S  X.  { A } )  Fn  S
)
90 ffn 5657 . . . . . 6  |-  ( F : X --> CC  ->  F  Fn  X )
918, 90syl 16 . . . . 5  |-  ( ph  ->  F  Fn  X )
921, 22ssexd 4537 . . . . 5  |-  ( ph  ->  X  e.  _V )
93 eqid 2451 . . . . 5  |-  ( S  i^i  X )  =  ( S  i^i  X
)
94 fvconst2g 6030 . . . . . 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 2452 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  =  ( F `  x ) )
9789, 91, 1, 92, 93, 95, 96offval 6427 . . . 4  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  F )  =  ( x  e.  ( S  i^i  X
)  |->  ( A  x.  ( F `  x ) ) ) )
98 ffn 5657 . . . . . 6  |-  ( ( X  X.  { A } ) : X --> { A }  ->  ( X  X.  { A }
)  Fn  X )
994, 98syl 16 . . . . 5  |-  ( ph  ->  ( X  X.  { A } )  Fn  X
)
100 inidm 3657 . . . . 5  |-  ( X  i^i  X )  =  X
101 fvconst2g 6030 . . . . . 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 6427 . . . 4  |-  ( ph  ->  ( ( X  X.  { A } )  oF  x.  F )  =  ( x  e.  X  |->  ( A  x.  ( F `  x ) ) ) )
10487, 97, 1033eqtr4d 2502 . . 3  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  F )  =  ( ( X  X.  { A }
)  oF  x.  F ) )
105104oveq2d 6206 . 2  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  oF  x.  F ) )  =  ( S  _D  ( ( X  X.  { A } )  oF  x.  F ) ) )
10686mpteq1d 4471 . . 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 21497 . . . . . . 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 5645 . . . . . 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 5657 . . . . 5  |-  ( ( S  _D  F ) : X --> CC  ->  ( S  _D  F )  Fn  X )
112110, 111syl 16 . . . 4  |-  ( ph  ->  ( S  _D  F
)  Fn  X )
113 eqidd 2452 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  =  ( ( S  _D  F ) `  x ) )
11489, 112, 1, 92, 93, 95, 113offval 6427 . . 3  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  ( S  _D  F ) )  =  ( x  e.  ( S  i^i  X
)  |->  ( A  x.  ( ( S  _D  F ) `  x
) ) ) )
115 0cnd 9480 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  0  e.  CC )
116 ovex 6215 . . . . . 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 6205 . . . . . . 7  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F )  =  ( ( X  X.  { 0 } )  oF  x.  F ) )
119 0cnd 9480 . . . . . . . 8  |-  ( ph  ->  0  e.  CC )
120 mul02 9648 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
121120adantl 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
12292, 8, 119, 119, 121caofid2 6451 . . . . . . 7  |-  ( ph  ->  ( ( X  X.  { 0 } )  oF  x.  F
)  =  ( X  X.  { 0 } ) )
123118, 122eqtrd 2492 . . . . . 6  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F )  =  ( X  X.  { 0 } ) )
124123, 76syl6eq 2508 . . . . 5  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F )  =  ( x  e.  X  |->  0 ) )
125 fvex 5799 . . . . . . 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 5843 . . . . . 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 6436 . . . . 5  |-  ( ph  ->  ( ( S  _D  F )  oF  x.  ( X  X.  { A } ) )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `
 x )  x.  A ) ) )
13192, 115, 117, 124, 130offval2 6436 . . . 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 5942 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  CC )
133132, 127mulcld 9507 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  e.  CC )
134133addid2d 9671 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
0  +  ( ( ( S  _D  F
) `  x )  x.  A ) )  =  ( ( ( S  _D  F ) `  x )  x.  A
) )
135132, 127mulcomd 9508 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  =  ( A  x.  ( ( S  _D  F ) `  x
) ) )
136134, 135eqtrd 2492 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
0  +  ( ( ( S  _D  F
) `  x )  x.  A ) )  =  ( A  x.  (
( S  _D  F
) `  x )
) )
137136mpteq2dva 4476 . . . 4  |-  ( ph  ->  ( x  e.  X  |->  ( 0  +  ( ( ( S  _D  F ) `  x
)  x.  A ) ) )  =  ( x  e.  X  |->  ( A  x.  ( ( S  _D  F ) `
 x ) ) ) )
138131, 137eqtrd 2492 . . 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 2502 . 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 2502 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 1370    e. wcel 1758   _Vcvv 3068    i^i cin 3425    C_ wss 3426   {csn 3975   {cpr 3977   U.cuni 4189    |-> cmpt 4448    X. cxp 4936   dom cdm 4938    |` cres 4940    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190    oFcof 6418   CCcc 9381   RRcr 9382   0cc0 9383    + caddc 9386    x. cmul 9388   ↾t crest 14461   TopOpenctopn 14462  ℂfldccnfld 17927   Topctop 18614  TopOnctopon 18615   intcnt 18737    _D cdv 21454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-fi 7762  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-q 11055  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-icc 11408  df-fz 11539  df-fzo 11650  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-hom 14364  df-cco 14365  df-rest 14463  df-topn 14464  df-0g 14482  df-gsum 14483  df-topgen 14484  df-pt 14485  df-prds 14488  df-xrs 14542  df-qtop 14547  df-imas 14548  df-xps 14550  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-submnd 15567  df-mulg 15650  df-cntz 15937  df-cmn 16383  df-psmet 17918  df-xmet 17919  df-met 17920  df-bl 17921  df-mopn 17922  df-fbas 17923  df-fg 17924  df-cnfld 17928  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cld 18739  df-ntr 18740  df-cls 18741  df-nei 18818  df-lp 18856  df-perf 18857  df-cn 18947  df-cnp 18948  df-haus 19035  df-tx 19251  df-hmeo 19444  df-fil 19535  df-fm 19627  df-flim 19628  df-flf 19629  df-xms 20011  df-ms 20012  df-tms 20013  df-cncf 20570  df-limc 21457  df-dv 21458
This theorem is referenced by: (None)
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