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Theorem dvcmulf 22841
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 5730 . . . . 5  |-  ( A  e.  CC  ->  ( X  X.  { A }
) : X --> { A } )
42, 3syl 17 . . . 4  |-  ( ph  ->  ( X  X.  { A } ) : X --> { A } )
52snssd 4088 . . . 4  |-  ( ph  ->  { A }  C_  CC )
64, 5fssd 5698 . . 3  |-  ( ph  ->  ( X  X.  { A } ) : X --> CC )
7 dvcmul.f . . 3  |-  ( ph  ->  F : X --> CC )
8 c0ex 9588 . . . . . 6  |-  0  e.  _V
98fconst 5729 . . . . 5  |-  ( X  X.  { 0 } ) : X --> { 0 }
10 recnprss 22801 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
111, 10syl 17 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
12 fconstg 5730 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( S  X.  { A }
) : S --> { A } )
132, 12syl 17 . . . . . . . . 9  |-  ( ph  ->  ( S  X.  { A } ) : S --> { A } )
1413, 5fssd 5698 . . . . . . . 8  |-  ( ph  ->  ( S  X.  { A } ) : S --> CC )
15 ssid 3426 . . . . . . . . 9  |-  S  C_  S
1615a1i 11 . . . . . . . 8  |-  ( ph  ->  S  C_  S )
17 dvcmulf.df . . . . . . . . 9  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
18 dvbsss 22799 . . . . . . . . . 10  |-  dom  ( S  _D  F )  C_  S
1918a1i 11 . . . . . . . . 9  |-  ( ph  ->  dom  ( S  _D  F )  C_  S
)
2017, 19eqsstr3d 3442 . . . . . . . 8  |-  ( ph  ->  X  C_  S )
21 eqid 2428 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
22 eqid 2428 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  S )  =  ( ( TopOpen ` fld )t  S )
2321, 22dvres 22808 . . . . . . . 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 ) ) )
2411, 14, 16, 20, 23syl22anc 1265 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  |`  X ) )  =  ( ( S  _D  ( S  X.  { A }
) )  |`  (
( int `  (
( TopOpen ` fld )t  S ) ) `  X ) ) )
2520resmptd 5118 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  S  |->  A )  |`  X )  =  ( x  e.  X  |->  A ) )
26 fconstmpt 4840 . . . . . . . . . 10  |-  ( S  X.  { A }
)  =  ( x  e.  S  |->  A )
2726reseq1i 5063 . . . . . . . . 9  |-  ( ( S  X.  { A } )  |`  X )  =  ( ( x  e.  S  |->  A )  |`  X )
28 fconstmpt 4840 . . . . . . . . 9  |-  ( X  X.  { A }
)  =  ( x  e.  X  |->  A )
2925, 27, 283eqtr4g 2487 . . . . . . . 8  |-  ( ph  ->  ( ( S  X.  { A } )  |`  X )  =  ( X  X.  { A } ) )
3029oveq2d 6265 . . . . . . 7  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  |`  X ) )  =  ( S  _D  ( X  X.  { A } ) ) )
3120resmptd 5118 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  S  |->  0 )  |`  X )  =  ( x  e.  X  |->  0 ) )
32 fconstg 5730 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  ( CC  X.  { A }
) : CC --> { A } )
332, 32syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  X.  { A } ) : CC --> { A } )
3433, 5fssd 5698 . . . . . . . . . . . 12  |-  ( ph  ->  ( CC  X.  { A } ) : CC --> CC )
35 ssid 3426 . . . . . . . . . . . . 13  |-  CC  C_  CC
3635a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  CC  C_  CC )
37 dvconst 22813 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  ( CC  _D  ( CC  X.  { A } ) )  =  ( CC  X.  { 0 } ) )
382, 37syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  ( CC  X.  { A }
) )  =  ( CC  X.  { 0 } ) )
3938dmeqd 4999 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( CC  _D  ( CC  X.  { A } ) )  =  dom  ( CC  X.  { 0 } ) )
408fconst 5729 . . . . . . . . . . . . . . 15  |-  ( CC 
X.  { 0 } ) : CC --> { 0 }
4140fdmi 5694 . . . . . . . . . . . . . 14  |-  dom  ( CC  X.  { 0 } )  =  CC
4239, 41syl6eq 2478 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  ( CC  _D  ( CC  X.  { A } ) )  =  CC )
4311, 42sseqtr4d 3444 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  dom  ( CC 
_D  ( CC  X.  { A } ) ) )
44 dvres3 22810 . . . . . . . . . . . 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 ) )
451, 34, 36, 43, 44syl22anc 1265 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  (
( CC  X.  { A } )  |`  S ) )  =  ( ( CC  _D  ( CC 
X.  { A }
) )  |`  S ) )
46 xpssres 5101 . . . . . . . . . . . . 13  |-  ( S 
C_  CC  ->  ( ( CC  X.  { A } )  |`  S )  =  ( S  X.  { A } ) )
4711, 46syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  X.  { A } )  |`  S )  =  ( S  X.  { A } ) )
4847oveq2d 6265 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  (
( CC  X.  { A } )  |`  S ) )  =  ( S  _D  ( S  X.  { A } ) ) )
4938reseq1d 5066 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S )  =  ( ( CC  X.  {
0 } )  |`  S ) )
50 xpssres 5101 . . . . . . . . . . . . 13  |-  ( S 
C_  CC  ->  ( ( CC  X.  { 0 } )  |`  S )  =  ( S  X.  { 0 } ) )
5111, 50syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  X.  { 0 } )  |`  S )  =  ( S  X.  { 0 } ) )
5249, 51eqtrd 2462 . . . . . . . . . . 11  |-  ( ph  ->  ( ( CC  _D  ( CC  X.  { A } ) )  |`  S )  =  ( S  X.  { 0 } ) )
5345, 48, 523eqtr3d 2470 . . . . . . . . . 10  |-  ( ph  ->  ( S  _D  ( S  X.  { A }
) )  =  ( S  X.  { 0 } ) )
54 fconstmpt 4840 . . . . . . . . . 10  |-  ( S  X.  { 0 } )  =  ( x  e.  S  |->  0 )
5553, 54syl6eq 2478 . . . . . . . . 9  |-  ( ph  ->  ( S  _D  ( S  X.  { A }
) )  =  ( x  e.  S  |->  0 ) )
5621cnfldtopon 21745 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
57 resttopon 20119 . . . . . . . . . . . . 13  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  S  C_  CC )  ->  (
( TopOpen ` fld )t  S )  e.  (TopOn `  S ) )
5856, 11, 57sylancr 667 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( TopOpen ` fld )t  S )  e.  (TopOn `  S ) )
59 topontop 19883 . . . . . . . . . . . 12  |-  ( ( ( TopOpen ` fld )t  S )  e.  (TopOn `  S )  ->  (
( TopOpen ` fld )t  S )  e.  Top )
6058, 59syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( ( TopOpen ` fld )t  S )  e.  Top )
61 toponuni 19884 . . . . . . . . . . . . 13  |-  ( ( ( TopOpen ` fld )t  S )  e.  (TopOn `  S )  ->  S  =  U. ( ( TopOpen ` fld )t  S
) )
6258, 61syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  S  =  U. (
( TopOpen ` fld )t  S ) )
6320, 62sseqtrd 3443 . . . . . . . . . . 11  |-  ( ph  ->  X  C_  U. (
( TopOpen ` fld )t  S ) )
64 eqid 2428 . . . . . . . . . . . 12  |-  U. (
( TopOpen ` fld )t  S )  =  U. ( ( TopOpen ` fld )t  S )
6564ntrss2 20014 . . . . . . . . . . 11  |-  ( ( ( ( TopOpen ` fld )t  S )  e.  Top  /\  X  C_  U. (
( TopOpen ` fld )t  S ) )  -> 
( ( int `  (
( TopOpen ` fld )t  S ) ) `  X )  C_  X
)
6660, 63, 65syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X )  C_  X
)
6711, 7, 20, 22, 21dvbssntr 22797 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( S  _D  F )  C_  (
( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )
6817, 67eqsstr3d 3442 . . . . . . . . . 10  |-  ( ph  ->  X  C_  ( ( int `  ( ( TopOpen ` fld )t  S
) ) `  X
) )
6966, 68eqssd 3424 . . . . . . . . 9  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X )  =  X )
7055, 69reseq12d 5068 . . . . . . . 8  |-  ( ph  ->  ( ( S  _D  ( S  X.  { A } ) )  |`  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )  =  ( ( x  e.  S  |->  0 )  |`  X ) )
71 fconstmpt 4840 . . . . . . . . 9  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
7271a1i 11 . . . . . . . 8  |-  ( ph  ->  ( X  X.  {
0 } )  =  ( x  e.  X  |->  0 ) )
7331, 70, 723eqtr4d 2472 . . . . . . 7  |-  ( ph  ->  ( ( S  _D  ( S  X.  { A } ) )  |`  ( ( int `  (
( TopOpen ` fld )t  S ) ) `  X ) )  =  ( X  X.  {
0 } ) )
7424, 30, 733eqtr3d 2470 . . . . . 6  |-  ( ph  ->  ( S  _D  ( X  X.  { A }
) )  =  ( X  X.  { 0 } ) )
7574feq1d 5675 . . . . 5  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) ) : X --> { 0 }  <-> 
( X  X.  {
0 } ) : X --> { 0 } ) )
769, 75mpbiri 236 . . . 4  |-  ( ph  ->  ( S  _D  ( X  X.  { A }
) ) : X --> { 0 } )
77 fdm 5693 . . . 4  |-  ( ( S  _D  ( X  X.  { A }
) ) : X --> { 0 }  ->  dom  ( S  _D  ( X  X.  { A }
) )  =  X )
7876, 77syl 17 . . 3  |-  ( ph  ->  dom  ( S  _D  ( X  X.  { A } ) )  =  X )
791, 6, 7, 78, 17dvmulf 22839 . 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 }
) ) ) )
80 sseqin2 3624 . . . . . 6  |-  ( X 
C_  S  <->  ( S  i^i  X )  =  X )
8120, 80sylib 199 . . . . 5  |-  ( ph  ->  ( S  i^i  X
)  =  X )
8281mpteq1d 4448 . . . 4  |-  ( ph  ->  ( x  e.  ( S  i^i  X ) 
|->  ( A  x.  ( F `  x )
) )  =  ( x  e.  X  |->  ( A  x.  ( F `
 x ) ) ) )
83 ffn 5689 . . . . . 6  |-  ( ( S  X.  { A } ) : S --> { A }  ->  ( S  X.  { A }
)  Fn  S )
8413, 83syl 17 . . . . 5  |-  ( ph  ->  ( S  X.  { A } )  Fn  S
)
85 ffn 5689 . . . . . 6  |-  ( F : X --> CC  ->  F  Fn  X )
867, 85syl 17 . . . . 5  |-  ( ph  ->  F  Fn  X )
871, 20ssexd 4514 . . . . 5  |-  ( ph  ->  X  e.  _V )
88 eqid 2428 . . . . 5  |-  ( S  i^i  X )  =  ( S  i^i  X
)
89 fvconst2g 6077 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  S )  ->  ( ( S  X.  { A } ) `  x )  =  A )
902, 89sylan 473 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  (
( S  X.  { A } ) `  x
)  =  A )
91 eqidd 2429 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  =  ( F `  x ) )
9284, 86, 1, 87, 88, 90, 91offval 6496 . . . 4  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  F )  =  ( x  e.  ( S  i^i  X
)  |->  ( A  x.  ( F `  x ) ) ) )
93 ffn 5689 . . . . . 6  |-  ( ( X  X.  { A } ) : X --> { A }  ->  ( X  X.  { A }
)  Fn  X )
944, 93syl 17 . . . . 5  |-  ( ph  ->  ( X  X.  { A } )  Fn  X
)
95 inidm 3614 . . . . 5  |-  ( X  i^i  X )  =  X
96 fvconst2g 6077 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  X )  ->  ( ( X  X.  { A } ) `  x )  =  A )
972, 96sylan 473 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( X  X.  { A } ) `  x
)  =  A )
9894, 86, 87, 87, 95, 97, 91offval 6496 . . . 4  |-  ( ph  ->  ( ( X  X.  { A } )  oF  x.  F )  =  ( x  e.  X  |->  ( A  x.  ( F `  x ) ) ) )
9982, 92, 983eqtr4d 2472 . . 3  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  F )  =  ( ( X  X.  { A }
)  oF  x.  F ) )
10099oveq2d 6265 . 2  |-  ( ph  ->  ( S  _D  (
( S  X.  { A } )  oF  x.  F ) )  =  ( S  _D  ( ( X  X.  { A } )  oF  x.  F ) ) )
10181mpteq1d 4448 . . 3  |-  ( ph  ->  ( x  e.  ( S  i^i  X ) 
|->  ( A  x.  (
( S  _D  F
) `  x )
) )  =  ( x  e.  X  |->  ( A  x.  ( ( S  _D  F ) `
 x ) ) ) )
102 dvfg 22803 . . . . . . 7  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  F ) : dom  ( S  _D  F
) --> CC )
1031, 102syl 17 . . . . . 6  |-  ( ph  ->  ( S  _D  F
) : dom  ( S  _D  F ) --> CC )
10417feq2d 5676 . . . . . 6  |-  ( ph  ->  ( ( S  _D  F ) : dom  ( S  _D  F
) --> CC  <->  ( S  _D  F ) : X --> CC ) )
105103, 104mpbid 213 . . . . 5  |-  ( ph  ->  ( S  _D  F
) : X --> CC )
106 ffn 5689 . . . . 5  |-  ( ( S  _D  F ) : X --> CC  ->  ( S  _D  F )  Fn  X )
107105, 106syl 17 . . . 4  |-  ( ph  ->  ( S  _D  F
)  Fn  X )
108 eqidd 2429 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  =  ( ( S  _D  F ) `  x ) )
10984, 107, 1, 87, 88, 90, 108offval 6496 . . 3  |-  ( ph  ->  ( ( S  X.  { A } )  oF  x.  ( S  _D  F ) )  =  ( x  e.  ( S  i^i  X
)  |->  ( A  x.  ( ( S  _D  F ) `  x
) ) ) )
110 0cnd 9587 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  0  e.  CC )
111 ovex 6277 . . . . . 6  |-  ( ( ( S  _D  F
) `  x )  x.  A )  e.  _V
112111a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  e.  _V )
11374oveq1d 6264 . . . . . . 7  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F )  =  ( ( X  X.  { 0 } )  oF  x.  F ) )
114 0cnd 9587 . . . . . . . 8  |-  ( ph  ->  0  e.  CC )
115 mul02 9762 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
116115adantl 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
11787, 7, 114, 114, 116caofid2 6520 . . . . . . 7  |-  ( ph  ->  ( ( X  X.  { 0 } )  oF  x.  F
)  =  ( X  X.  { 0 } ) )
118113, 117eqtrd 2462 . . . . . 6  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F )  =  ( X  X.  { 0 } ) )
119118, 71syl6eq 2478 . . . . 5  |-  ( ph  ->  ( ( S  _D  ( X  X.  { A } ) )  oF  x.  F )  =  ( x  e.  X  |->  0 ) )
120 fvex 5835 . . . . . . 7  |-  ( ( S  _D  F ) `
 x )  e. 
_V
121120a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  _V )
1222adantr 466 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  CC )
123105feqmptd 5878 . . . . . 6  |-  ( ph  ->  ( S  _D  F
)  =  ( x  e.  X  |->  ( ( S  _D  F ) `
 x ) ) )
12428a1i 11 . . . . . 6  |-  ( ph  ->  ( X  X.  { A } )  =  ( x  e.  X  |->  A ) )
12587, 121, 122, 123, 124offval2 6506 . . . . 5  |-  ( ph  ->  ( ( S  _D  F )  oF  x.  ( X  X.  { A } ) )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `
 x )  x.  A ) ) )
12687, 110, 112, 119, 125offval2 6506 . . . 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 ) ) ) )
127105ffvelrnda 5981 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  CC )
128127, 122mulcld 9614 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  e.  CC )
129128addid2d 9785 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
0  +  ( ( ( S  _D  F
) `  x )  x.  A ) )  =  ( ( ( S  _D  F ) `  x )  x.  A
) )
130127, 122mulcomd 9615 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  A )  =  ( A  x.  ( ( S  _D  F ) `  x
) ) )
131129, 130eqtrd 2462 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
0  +  ( ( ( S  _D  F
) `  x )  x.  A ) )  =  ( A  x.  (
( S  _D  F
) `  x )
) )
132131mpteq2dva 4453 . . . 4  |-  ( ph  ->  ( x  e.  X  |->  ( 0  +  ( ( ( S  _D  F ) `  x
)  x.  A ) ) )  =  ( x  e.  X  |->  ( A  x.  ( ( S  _D  F ) `
 x ) ) ) )
133126, 132eqtrd 2462 . . 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 ) ) ) )
134101, 109, 1333eqtr4d 2472 . 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 }
) ) ) )
13579, 100, 1343eqtr4d 2472 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 370    = wceq 1437    e. wcel 1872   _Vcvv 3022    i^i cin 3378    C_ wss 3379   {csn 3941   {cpr 3943   U.cuni 4162    |-> cmpt 4425    X. cxp 4794   dom cdm 4796    |` cres 4798    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249    oFcof 6487   CCcc 9488   RRcr 9489   0cc0 9490    + caddc 9493    x. cmul 9495   ↾t crest 15262   TopOpenctopn 15263  ℂfldccnfld 18913   Topctop 19859  TopOnctopon 19860   intcnt 19974    _D cdv 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-iin 4245  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-of 6489  df-om 6651  df-1st 6751  df-2nd 6752  df-supp 6870  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-2o 7138  df-oadd 7141  df-er 7318  df-map 7429  df-pm 7430  df-ixp 7478  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-fsupp 7837  df-fi 7878  df-sup 7909  df-inf 7910  df-oi 7978  df-card 8325  df-cda 8549  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-7 10624  df-8 10625  df-9 10626  df-10 10627  df-n0 10821  df-z 10889  df-dec 11003  df-uz 11111  df-q 11216  df-rp 11254  df-xneg 11360  df-xadd 11361  df-xmul 11362  df-icc 11593  df-fz 11736  df-fzo 11867  df-seq 12164  df-exp 12223  df-hash 12466  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-struct 15066  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-ress 15071  df-plusg 15146  df-mulr 15147  df-starv 15148  df-sca 15149  df-vsca 15150  df-ip 15151  df-tset 15152  df-ple 15153  df-ds 15155  df-unif 15156  df-hom 15157  df-cco 15158  df-rest 15264  df-topn 15265  df-0g 15283  df-gsum 15284  df-topgen 15285  df-pt 15286  df-prds 15289  df-xrs 15343  df-qtop 15349  df-imas 15350  df-xps 15353  df-mre 15435  df-mrc 15436  df-acs 15438  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-submnd 16526  df-mulg 16619  df-cntz 16914  df-cmn 17375  df-psmet 18905  df-xmet 18906  df-met 18907  df-bl 18908  df-mopn 18909  df-fbas 18910  df-fg 18911  df-cnfld 18914  df-top 19863  df-bases 19864  df-topon 19865  df-topsp 19866  df-cld 19976  df-ntr 19977  df-cls 19978  df-nei 20056  df-lp 20094  df-perf 20095  df-cn 20185  df-cnp 20186  df-haus 20273  df-tx 20519  df-hmeo 20712  df-fil 20803  df-fm 20895  df-flim 20896  df-flf 20897  df-xms 21277  df-ms 21278  df-tms 21279  df-cncf 21852  df-limc 22763  df-dv 22764
This theorem is referenced by:  dvsinax  37666
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