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Theorem dchrmulcl 22588
Description: Closure of the group operation on Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
Hypotheses
Ref Expression
dchrmhm.g  |-  G  =  (DChr `  N )
dchrmhm.z  |-  Z  =  (ℤ/n `  N )
dchrmhm.b  |-  D  =  ( Base `  G
)
dchrmul.t  |-  .x.  =  ( +g  `  G )
dchrmul.x  |-  ( ph  ->  X  e.  D )
dchrmul.y  |-  ( ph  ->  Y  e.  D )
Assertion
Ref Expression
dchrmulcl  |-  ( ph  ->  ( X  .x.  Y
)  e.  D )

Proof of Theorem dchrmulcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dchrmhm.g . . 3  |-  G  =  (DChr `  N )
2 dchrmhm.z . . 3  |-  Z  =  (ℤ/n `  N )
3 dchrmhm.b . . 3  |-  D  =  ( Base `  G
)
4 dchrmul.t . . 3  |-  .x.  =  ( +g  `  G )
5 dchrmul.x . . 3  |-  ( ph  ->  X  e.  D )
6 dchrmul.y . . 3  |-  ( ph  ->  Y  e.  D )
71, 2, 3, 4, 5, 6dchrmul 22587 . 2  |-  ( ph  ->  ( X  .x.  Y
)  =  ( X  oF  x.  Y
) )
8 mulcl 9366 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
98adantl 466 . . . 4  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
10 eqid 2443 . . . . 5  |-  ( Base `  Z )  =  (
Base `  Z )
111, 2, 3, 10, 5dchrf 22581 . . . 4  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
121, 2, 3, 10, 6dchrf 22581 . . . 4  |-  ( ph  ->  Y : ( Base `  Z ) --> CC )
13 fvex 5701 . . . . 5  |-  ( Base `  Z )  e.  _V
1413a1i 11 . . . 4  |-  ( ph  ->  ( Base `  Z
)  e.  _V )
15 inidm 3559 . . . 4  |-  ( (
Base `  Z )  i^i  ( Base `  Z
) )  =  (
Base `  Z )
169, 11, 12, 14, 14, 15off 6334 . . 3  |-  ( ph  ->  ( X  oF  x.  Y ) : ( Base `  Z
) --> CC )
17 eqid 2443 . . . . . . . 8  |-  (Unit `  Z )  =  (Unit `  Z )
1810, 17unitcl 16751 . . . . . . 7  |-  ( x  e.  (Unit `  Z
)  ->  x  e.  ( Base `  Z )
)
1910, 17unitcl 16751 . . . . . . 7  |-  ( y  e.  (Unit `  Z
)  ->  y  e.  ( Base `  Z )
)
2018, 19anim12i 566 . . . . . 6  |-  ( ( x  e.  (Unit `  Z )  /\  y  e.  (Unit `  Z )
)  ->  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )
211, 3dchrrcl 22579 . . . . . . . . . . . . . 14  |-  ( X  e.  D  ->  N  e.  NN )
225, 21syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  NN )
231, 2, 10, 17, 22, 3dchrelbas2 22576 . . . . . . . . . . . 12  |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  A. x  e.  ( Base `  Z
) ( ( X `
 x )  =/=  0  ->  x  e.  (Unit `  Z ) ) ) ) )
245, 23mpbid 210 . . . . . . . . . . 11  |-  ( ph  ->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ( Base `  Z ) ( ( X `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) ) )
2524simpld 459 . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
26 eqid 2443 . . . . . . . . . . . . 13  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
2726, 10mgpbas 16597 . . . . . . . . . . . 12  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
28 eqid 2443 . . . . . . . . . . . . 13  |-  ( .r
`  Z )  =  ( .r `  Z
)
2926, 28mgpplusg 16595 . . . . . . . . . . . 12  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
30 eqid 2443 . . . . . . . . . . . . 13  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
31 cnfldmul 17824 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
3230, 31mgpplusg 16595 . . . . . . . . . . . 12  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3327, 29, 32mhmlin 15471 . . . . . . . . . . 11  |-  ( ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  x  e.  ( Base `  Z )  /\  y  e.  ( Base `  Z ) )  ->  ( X `  ( x ( .r
`  Z ) y ) )  =  ( ( X `  x
)  x.  ( X `
 y ) ) )
34333expb 1188 . . . . . . . . . 10  |-  ( ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( X `  ( x
( .r `  Z
) y ) )  =  ( ( X `
 x )  x.  ( X `  y
) ) )
3525, 34sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( X `  ( x
( .r `  Z
) y ) )  =  ( ( X `
 x )  x.  ( X `  y
) ) )
361, 2, 10, 17, 22, 3dchrelbas2 22576 . . . . . . . . . . . 12  |-  ( ph  ->  ( Y  e.  D  <->  ( Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  A. x  e.  ( Base `  Z
) ( ( Y `
 x )  =/=  0  ->  x  e.  (Unit `  Z ) ) ) ) )
376, 36mpbid 210 . . . . . . . . . . 11  |-  ( ph  ->  ( Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ( Base `  Z ) ( ( Y `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) ) )
3837simpld 459 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
3927, 29, 32mhmlin 15471 . . . . . . . . . . 11  |-  ( ( Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  x  e.  ( Base `  Z )  /\  y  e.  ( Base `  Z ) )  ->  ( Y `  ( x ( .r
`  Z ) y ) )  =  ( ( Y `  x
)  x.  ( Y `
 y ) ) )
40393expb 1188 . . . . . . . . . 10  |-  ( ( Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  ( x
( .r `  Z
) y ) )  =  ( ( Y `
 x )  x.  ( Y `  y
) ) )
4138, 40sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  ( x
( .r `  Z
) y ) )  =  ( ( Y `
 x )  x.  ( Y `  y
) ) )
4235, 41oveq12d 6109 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X `  (
x ( .r `  Z ) y ) )  x.  ( Y `
 ( x ( .r `  Z ) y ) ) )  =  ( ( ( X `  x )  x.  ( X `  y ) )  x.  ( ( Y `  x )  x.  ( Y `  y )
) ) )
4311ffvelrnda 5843 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( X `  x )  e.  CC )
4443adantrr 716 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( X `  x )  e.  CC )
45 simpr 461 . . . . . . . . . 10  |-  ( ( x  e.  ( Base `  Z )  /\  y  e.  ( Base `  Z
) )  ->  y  e.  ( Base `  Z
) )
46 ffvelrn 5841 . . . . . . . . . 10  |-  ( ( X : ( Base `  Z ) --> CC  /\  y  e.  ( Base `  Z ) )  -> 
( X `  y
)  e.  CC )
4711, 45, 46syl2an 477 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( X `  y )  e.  CC )
4812ffvelrnda 5843 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( Y `  x )  e.  CC )
4948adantrr 716 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  x )  e.  CC )
50 ffvelrn 5841 . . . . . . . . . 10  |-  ( ( Y : ( Base `  Z ) --> CC  /\  y  e.  ( Base `  Z ) )  -> 
( Y `  y
)  e.  CC )
5112, 45, 50syl2an 477 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  y )  e.  CC )
5244, 47, 49, 51mul4d 9581 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( ( X `  x )  x.  ( X `  y )
)  x.  ( ( Y `  x )  x.  ( Y `  y ) ) )  =  ( ( ( X `  x )  x.  ( Y `  x ) )  x.  ( ( X `  y )  x.  ( Y `  y )
) ) )
5342, 52eqtrd 2475 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X `  (
x ( .r `  Z ) y ) )  x.  ( Y `
 ( x ( .r `  Z ) y ) ) )  =  ( ( ( X `  x )  x.  ( Y `  x ) )  x.  ( ( X `  y )  x.  ( Y `  y )
) ) )
54 ffn 5559 . . . . . . . . . 10  |-  ( X : ( Base `  Z
) --> CC  ->  X  Fn  ( Base `  Z
) )
5511, 54syl 16 . . . . . . . . 9  |-  ( ph  ->  X  Fn  ( Base `  Z ) )
5655adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  X  Fn  ( Base `  Z
) )
57 ffn 5559 . . . . . . . . . 10  |-  ( Y : ( Base `  Z
) --> CC  ->  Y  Fn  ( Base `  Z
) )
5812, 57syl 16 . . . . . . . . 9  |-  ( ph  ->  Y  Fn  ( Base `  Z ) )
5958adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  Y  Fn  ( Base `  Z
) )
6013a1i 11 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Base `  Z )  e. 
_V )
6122nnnn0d 10636 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
622zncrng 17977 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
63 crngrng 16655 . . . . . . . . . 10  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
6461, 62, 633syl 20 . . . . . . . . 9  |-  ( ph  ->  Z  e.  Ring )
6510, 28rngcl 16658 . . . . . . . . . 10  |-  ( ( Z  e.  Ring  /\  x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
)  ->  ( x
( .r `  Z
) y )  e.  ( Base `  Z
) )
66653expb 1188 . . . . . . . . 9  |-  ( ( Z  e.  Ring  /\  (
x  e.  ( Base `  Z )  /\  y  e.  ( Base `  Z
) ) )  -> 
( x ( .r
`  Z ) y )  e.  ( Base `  Z ) )
6764, 66sylan 471 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
x ( .r `  Z ) y )  e.  ( Base `  Z
) )
68 fnfvof 6333 . . . . . . . 8  |-  ( ( ( X  Fn  ( Base `  Z )  /\  Y  Fn  ( Base `  Z ) )  /\  ( ( Base `  Z
)  e.  _V  /\  ( x ( .r
`  Z ) y )  e.  ( Base `  Z ) ) )  ->  ( ( X  oF  x.  Y
) `  ( x
( .r `  Z
) y ) )  =  ( ( X `
 ( x ( .r `  Z ) y ) )  x.  ( Y `  (
x ( .r `  Z ) y ) ) ) )
6956, 59, 60, 67, 68syl22anc 1219 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X  oF  x.  Y ) `  ( x ( .r
`  Z ) y ) )  =  ( ( X `  (
x ( .r `  Z ) y ) )  x.  ( Y `
 ( x ( .r `  Z ) y ) ) ) )
7055adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  X  Fn  ( Base `  Z )
)
7158adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  Y  Fn  ( Base `  Z )
)
7213a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( Base `  Z )  e.  _V )
73 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  x  e.  ( Base `  Z )
)
74 fnfvof 6333 . . . . . . . . . 10  |-  ( ( ( X  Fn  ( Base `  Z )  /\  Y  Fn  ( Base `  Z ) )  /\  ( ( Base `  Z
)  e.  _V  /\  x  e.  ( Base `  Z ) ) )  ->  ( ( X  oF  x.  Y
) `  x )  =  ( ( X `
 x )  x.  ( Y `  x
) ) )
7570, 71, 72, 73, 74syl22anc 1219 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( ( X  oF  x.  Y
) `  x )  =  ( ( X `
 x )  x.  ( Y `  x
) ) )
7675adantrr 716 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X  oF  x.  Y ) `  x )  =  ( ( X `  x
)  x.  ( Y `
 x ) ) )
77 simprr 756 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  y  e.  ( Base `  Z
) )
78 fnfvof 6333 . . . . . . . . 9  |-  ( ( ( X  Fn  ( Base `  Z )  /\  Y  Fn  ( Base `  Z ) )  /\  ( ( Base `  Z
)  e.  _V  /\  y  e.  ( Base `  Z ) ) )  ->  ( ( X  oF  x.  Y
) `  y )  =  ( ( X `
 y )  x.  ( Y `  y
) ) )
7956, 59, 60, 77, 78syl22anc 1219 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X  oF  x.  Y ) `  y )  =  ( ( X `  y
)  x.  ( Y `
 y ) ) )
8076, 79oveq12d 6109 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( ( X  oF  x.  Y ) `  x )  x.  (
( X  oF  x.  Y ) `  y ) )  =  ( ( ( X `
 x )  x.  ( Y `  x
) )  x.  (
( X `  y
)  x.  ( Y `
 y ) ) ) )
8153, 69, 803eqtr4d 2485 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X  oF  x.  Y ) `  ( x ( .r
`  Z ) y ) )  =  ( ( ( X  oF  x.  Y ) `  x )  x.  (
( X  oF  x.  Y ) `  y ) ) )
8220, 81sylan2 474 . . . . 5  |-  ( (
ph  /\  ( x  e.  (Unit `  Z )  /\  y  e.  (Unit `  Z ) ) )  ->  ( ( X  oF  x.  Y
) `  ( x
( .r `  Z
) y ) )  =  ( ( ( X  oF  x.  Y ) `  x
)  x.  ( ( X  oF  x.  Y ) `  y
) ) )
8382ralrimivva 2808 . . . 4  |-  ( ph  ->  A. x  e.  (Unit `  Z ) A. y  e.  (Unit `  Z )
( ( X  oF  x.  Y ) `  ( x ( .r
`  Z ) y ) )  =  ( ( ( X  oF  x.  Y ) `  x )  x.  (
( X  oF  x.  Y ) `  y ) ) )
84 eqid 2443 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
8510, 84rngidcl 16665 . . . . . . 7  |-  ( Z  e.  Ring  ->  ( 1r
`  Z )  e.  ( Base `  Z
) )
8664, 85syl 16 . . . . . 6  |-  ( ph  ->  ( 1r `  Z
)  e.  ( Base `  Z ) )
87 fnfvof 6333 . . . . . 6  |-  ( ( ( X  Fn  ( Base `  Z )  /\  Y  Fn  ( Base `  Z ) )  /\  ( ( Base `  Z
)  e.  _V  /\  ( 1r `  Z )  e.  ( Base `  Z
) ) )  -> 
( ( X  oF  x.  Y ) `  ( 1r `  Z
) )  =  ( ( X `  ( 1r `  Z ) )  x.  ( Y `  ( 1r `  Z ) ) ) )
8855, 58, 14, 86, 87syl22anc 1219 . . . . 5  |-  ( ph  ->  ( ( X  oF  x.  Y ) `  ( 1r `  Z
) )  =  ( ( X `  ( 1r `  Z ) )  x.  ( Y `  ( 1r `  Z ) ) ) )
8926, 84rngidval 16605 . . . . . . . . 9  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
90 cnfld1 17841 . . . . . . . . . 10  |-  1  =  ( 1r ` fld )
9130, 90rngidval 16605 . . . . . . . . 9  |-  1  =  ( 0g `  (mulGrp ` fld ) )
9289, 91mhm0 15472 . . . . . . . 8  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( X `  ( 1r `  Z ) )  =  1 )
9325, 92syl 16 . . . . . . 7  |-  ( ph  ->  ( X `  ( 1r `  Z ) )  =  1 )
9489, 91mhm0 15472 . . . . . . . 8  |-  ( Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( Y `  ( 1r `  Z ) )  =  1 )
9538, 94syl 16 . . . . . . 7  |-  ( ph  ->  ( Y `  ( 1r `  Z ) )  =  1 )
9693, 95oveq12d 6109 . . . . . 6  |-  ( ph  ->  ( ( X `  ( 1r `  Z ) )  x.  ( Y `
 ( 1r `  Z ) ) )  =  ( 1  x.  1 ) )
97 1t1e1 10469 . . . . . 6  |-  ( 1  x.  1 )  =  1
9896, 97syl6eq 2491 . . . . 5  |-  ( ph  ->  ( ( X `  ( 1r `  Z ) )  x.  ( Y `
 ( 1r `  Z ) ) )  =  1 )
9988, 98eqtrd 2475 . . . 4  |-  ( ph  ->  ( ( X  oF  x.  Y ) `  ( 1r `  Z
) )  =  1 )
10075neeq1d 2621 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X  oF  x.  Y ) `  x )  =/=  0  <->  ( ( X `  x
)  x.  ( Y `
 x ) )  =/=  0 ) )
10143, 48mulne0bd 9987 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X `  x
)  =/=  0  /\  ( Y `  x
)  =/=  0 )  <-> 
( ( X `  x )  x.  ( Y `  x )
)  =/=  0 ) )
102100, 101bitr4d 256 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X  oF  x.  Y ) `  x )  =/=  0  <->  ( ( X `  x
)  =/=  0  /\  ( Y `  x
)  =/=  0 ) ) )
10324simprd 463 . . . . . . . 8  |-  ( ph  ->  A. x  e.  (
Base `  Z )
( ( X `  x )  =/=  0  ->  x  e.  (Unit `  Z ) ) )
104103r19.21bi 2814 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( ( X `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) )
105104adantrd 468 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X `  x
)  =/=  0  /\  ( Y `  x
)  =/=  0 )  ->  x  e.  (Unit `  Z ) ) )
106102, 105sylbid 215 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X  oF  x.  Y ) `  x )  =/=  0  ->  x  e.  (Unit `  Z ) ) )
107106ralrimiva 2799 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  Z )
( ( ( X  oF  x.  Y
) `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) )
10883, 99, 1073jca 1168 . . 3  |-  ( ph  ->  ( A. x  e.  (Unit `  Z ) A. y  e.  (Unit `  Z ) ( ( X  oF  x.  Y ) `  (
x ( .r `  Z ) y ) )  =  ( ( ( X  oF  x.  Y ) `  x )  x.  (
( X  oF  x.  Y ) `  y ) )  /\  ( ( X  oF  x.  Y ) `  ( 1r `  Z
) )  =  1  /\  A. x  e.  ( Base `  Z
) ( ( ( X  oF  x.  Y ) `  x
)  =/=  0  ->  x  e.  (Unit `  Z
) ) ) )
1091, 2, 10, 17, 22, 3dchrelbas3 22577 . . 3  |-  ( ph  ->  ( ( X  oF  x.  Y )  e.  D  <->  ( ( X  oF  x.  Y
) : ( Base `  Z ) --> CC  /\  ( A. x  e.  (Unit `  Z ) A. y  e.  (Unit `  Z )
( ( X  oF  x.  Y ) `  ( x ( .r
`  Z ) y ) )  =  ( ( ( X  oF  x.  Y ) `  x )  x.  (
( X  oF  x.  Y ) `  y ) )  /\  ( ( X  oF  x.  Y ) `  ( 1r `  Z
) )  =  1  /\  A. x  e.  ( Base `  Z
) ( ( ( X  oF  x.  Y ) `  x
)  =/=  0  ->  x  e.  (Unit `  Z
) ) ) ) ) )
11016, 108, 109mpbir2and 913 . 2  |-  ( ph  ->  ( X  oF  x.  Y )  e.  D )
1117, 110eqeltrd 2517 1  |-  ( ph  ->  ( X  .x.  Y
)  e.  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   _Vcvv 2972    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091    oFcof 6318   CCcc 9280   0cc0 9282   1c1 9283    x. cmul 9287   NNcn 10322   NN0cn0 10579   Basecbs 14174   +g cplusg 14238   .rcmulr 14239   MndHom cmhm 15462  mulGrpcmgp 16591   1rcur 16603   Ringcrg 16645   CRingccrg 16646  Unitcui 16731  ℂfldccnfld 17818  ℤ/nczn 17934  DChrcdchr 22571
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-addf 9361  ax-mulf 9362
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-ec 7103  df-qs 7107  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-0g 14380  df-imas 14446  df-divs 14447  df-mnd 15415  df-mhm 15464  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-nsg 15679  df-eqg 15680  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-cring 16648  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-subrg 16863  df-lmod 16950  df-lss 17014  df-lsp 17053  df-sra 17253  df-rgmod 17254  df-lidl 17255  df-rsp 17256  df-2idl 17314  df-cnfld 17819  df-zring 17884  df-zn 17938  df-dchr 22572
This theorem is referenced by:  dchrabl  22593  dchrinv  22600
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