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Theorem dchrmulcl 20504
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 20503 . 2  |-  ( ph  ->  ( X  .x.  Y
)  =  ( X  o F  x.  Y
) )
8 mulcl 8837 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
98adantl 452 . . . 4  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
10 eqid 2296 . . . . 5  |-  ( Base `  Z )  =  (
Base `  Z )
111, 2, 3, 10, 5dchrf 20497 . . . 4  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
121, 2, 3, 10, 6dchrf 20497 . . . 4  |-  ( ph  ->  Y : ( Base `  Z ) --> CC )
13 fvex 5555 . . . . 5  |-  ( Base `  Z )  e.  _V
1413a1i 10 . . . 4  |-  ( ph  ->  ( Base `  Z
)  e.  _V )
15 inidm 3391 . . . 4  |-  ( (
Base `  Z )  i^i  ( Base `  Z
) )  =  (
Base `  Z )
169, 11, 12, 14, 14, 15off 6109 . . 3  |-  ( ph  ->  ( X  o F  x.  Y ) : ( Base `  Z
) --> CC )
17 eqid 2296 . . . . . . . 8  |-  (Unit `  Z )  =  (Unit `  Z )
1810, 17unitcl 15457 . . . . . . 7  |-  ( x  e.  (Unit `  Z
)  ->  x  e.  ( Base `  Z )
)
1910, 17unitcl 15457 . . . . . . 7  |-  ( y  e.  (Unit `  Z
)  ->  y  e.  ( Base `  Z )
)
2018, 19anim12i 549 . . . . . 6  |-  ( ( x  e.  (Unit `  Z )  /\  y  e.  (Unit `  Z )
)  ->  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )
211, 3dchrrcl 20495 . . . . . . . . . . . . . 14  |-  ( X  e.  D  ->  N  e.  NN )
225, 21syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  NN )
231, 2, 10, 17, 22, 3dchrelbas2 20492 . . . . . . . . . . . 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 201 . . . . . . . . . . 11  |-  ( ph  ->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ( Base `  Z ) ( ( X `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) ) )
2524simpld 445 . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
26 eqid 2296 . . . . . . . . . . . . 13  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
2726, 10mgpbas 15347 . . . . . . . . . . . 12  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
28 eqid 2296 . . . . . . . . . . . . 13  |-  ( .r
`  Z )  =  ( .r `  Z
)
2926, 28mgpplusg 15345 . . . . . . . . . . . 12  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
30 eqid 2296 . . . . . . . . . . . . 13  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
31 cnfldmul 16401 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
3230, 31mgpplusg 15345 . . . . . . . . . . . 12  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3327, 29, 32mhmlin 14438 . . . . . . . . . . 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 1152 . . . . . . . . . 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 457 . . . . . . . . 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 20492 . . . . . . . . . . . 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 201 . . . . . . . . . . 11  |-  ( ph  ->  ( Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ( Base `  Z ) ( ( Y `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) ) )
3837simpld 445 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
3927, 29, 32mhmlin 14438 . . . . . . . . . . 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 1152 . . . . . . . . . 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 457 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  ( x
( .r `  Z
) y ) )  =  ( ( Y `
 x )  x.  ( Y `  y
) ) )
4235, 41oveq12d 5892 . . . . . . . 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 )
) ) )
43 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( X : ( Base `  Z ) --> CC  /\  x  e.  ( Base `  Z ) )  -> 
( X `  x
)  e.  CC )
4411, 43sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( X `  x )  e.  CC )
4544adantrr 697 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( X `  x )  e.  CC )
46 simpr 447 . . . . . . . . . 10  |-  ( ( x  e.  ( Base `  Z )  /\  y  e.  ( Base `  Z
) )  ->  y  e.  ( Base `  Z
) )
47 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( X : ( Base `  Z ) --> CC  /\  y  e.  ( Base `  Z ) )  -> 
( X `  y
)  e.  CC )
4811, 46, 47syl2an 463 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( X `  y )  e.  CC )
49 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( Y : ( Base `  Z ) --> CC  /\  x  e.  ( Base `  Z ) )  -> 
( Y `  x
)  e.  CC )
5012, 49sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( Y `  x )  e.  CC )
5150adantrr 697 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  x )  e.  CC )
52 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( Y : ( Base `  Z ) --> CC  /\  y  e.  ( Base `  Z ) )  -> 
( Y `  y
)  e.  CC )
5312, 46, 52syl2an 463 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  y )  e.  CC )
5445, 48, 51, 53mul4d 9040 . . . . . . . 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 )
) ) )
5542, 54eqtrd 2328 . . . . . . 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 )
) ) )
56 ffn 5405 . . . . . . . . . 10  |-  ( X : ( Base `  Z
) --> CC  ->  X  Fn  ( Base `  Z
) )
5711, 56syl 15 . . . . . . . . 9  |-  ( ph  ->  X  Fn  ( Base `  Z ) )
5857adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  X  Fn  ( Base `  Z
) )
59 ffn 5405 . . . . . . . . . 10  |-  ( Y : ( Base `  Z
) --> CC  ->  Y  Fn  ( Base `  Z
) )
6012, 59syl 15 . . . . . . . . 9  |-  ( ph  ->  Y  Fn  ( Base `  Z ) )
6160adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  Y  Fn  ( Base `  Z
) )
6213a1i 10 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Base `  Z )  e. 
_V )
6322nnnn0d 10034 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
642zncrng 16514 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
65 crngrng 15367 . . . . . . . . . 10  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
6663, 64, 653syl 18 . . . . . . . . 9  |-  ( ph  ->  Z  e.  Ring )
6710, 28rngcl 15370 . . . . . . . . . 10  |-  ( ( Z  e.  Ring  /\  x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
)  ->  ( x
( .r `  Z
) y )  e.  ( Base `  Z
) )
68673expb 1152 . . . . . . . . 9  |-  ( ( Z  e.  Ring  /\  (
x  e.  ( Base `  Z )  /\  y  e.  ( Base `  Z
) ) )  -> 
( x ( .r
`  Z ) y )  e.  ( Base `  Z ) )
6966, 68sylan 457 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
x ( .r `  Z ) y )  e.  ( Base `  Z
) )
70 fnfvof 6106 . . . . . . . 8  |-  ( ( ( X  Fn  ( Base `  Z )  /\  Y  Fn  ( Base `  Z ) )  /\  ( ( Base `  Z
)  e.  _V  /\  ( x ( .r
`  Z ) y )  e.  ( Base `  Z ) ) )  ->  ( ( X  o F  x.  Y
) `  ( x
( .r `  Z
) y ) )  =  ( ( X `
 ( x ( .r `  Z ) y ) )  x.  ( Y `  (
x ( .r `  Z ) y ) ) ) )
7158, 61, 62, 69, 70syl22anc 1183 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X  o F  x.  Y ) `  ( x ( .r
`  Z ) y ) )  =  ( ( X `  (
x ( .r `  Z ) y ) )  x.  ( Y `
 ( x ( .r `  Z ) y ) ) ) )
7257adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  X  Fn  ( Base `  Z )
)
7360adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  Y  Fn  ( Base `  Z )
)
7413a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( Base `  Z )  e.  _V )
75 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  x  e.  ( Base `  Z )
)
76 fnfvof 6106 . . . . . . . . . 10  |-  ( ( ( X  Fn  ( Base `  Z )  /\  Y  Fn  ( Base `  Z ) )  /\  ( ( Base `  Z
)  e.  _V  /\  x  e.  ( Base `  Z ) ) )  ->  ( ( X  o F  x.  Y
) `  x )  =  ( ( X `
 x )  x.  ( Y `  x
) ) )
7772, 73, 74, 75, 76syl22anc 1183 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( ( X  o F  x.  Y
) `  x )  =  ( ( X `
 x )  x.  ( Y `  x
) ) )
7877adantrr 697 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X  o F  x.  Y ) `  x )  =  ( ( X `  x
)  x.  ( Y `
 x ) ) )
79 simprr 733 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  y  e.  ( Base `  Z
) )
80 fnfvof 6106 . . . . . . . . 9  |-  ( ( ( X  Fn  ( Base `  Z )  /\  Y  Fn  ( Base `  Z ) )  /\  ( ( Base `  Z
)  e.  _V  /\  y  e.  ( Base `  Z ) ) )  ->  ( ( X  o F  x.  Y
) `  y )  =  ( ( X `
 y )  x.  ( Y `  y
) ) )
8158, 61, 62, 79, 80syl22anc 1183 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X  o F  x.  Y ) `  y )  =  ( ( X `  y
)  x.  ( Y `
 y ) ) )
8278, 81oveq12d 5892 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( ( X  o F  x.  Y ) `  x )  x.  (
( X  o F  x.  Y ) `  y ) )  =  ( ( ( X `
 x )  x.  ( Y `  x
) )  x.  (
( X `  y
)  x.  ( Y `
 y ) ) ) )
8355, 71, 823eqtr4d 2338 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X  o F  x.  Y ) `  ( x ( .r
`  Z ) y ) )  =  ( ( ( X  o F  x.  Y ) `  x )  x.  (
( X  o F  x.  Y ) `  y ) ) )
8420, 83sylan2 460 . . . . 5  |-  ( (
ph  /\  ( x  e.  (Unit `  Z )  /\  y  e.  (Unit `  Z ) ) )  ->  ( ( X  o F  x.  Y
) `  ( x
( .r `  Z
) y ) )  =  ( ( ( X  o F  x.  Y ) `  x
)  x.  ( ( X  o F  x.  Y ) `  y
) ) )
8584ralrimivva 2648 . . . 4  |-  ( ph  ->  A. x  e.  (Unit `  Z ) A. y  e.  (Unit `  Z )
( ( X  o F  x.  Y ) `  ( x ( .r
`  Z ) y ) )  =  ( ( ( X  o F  x.  Y ) `  x )  x.  (
( X  o F  x.  Y ) `  y ) ) )
86 eqid 2296 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
8710, 86rngidcl 15377 . . . . . . 7  |-  ( Z  e.  Ring  ->  ( 1r
`  Z )  e.  ( Base `  Z
) )
8866, 87syl 15 . . . . . 6  |-  ( ph  ->  ( 1r `  Z
)  e.  ( Base `  Z ) )
89 fnfvof 6106 . . . . . 6  |-  ( ( ( X  Fn  ( Base `  Z )  /\  Y  Fn  ( Base `  Z ) )  /\  ( ( Base `  Z
)  e.  _V  /\  ( 1r `  Z )  e.  ( Base `  Z
) ) )  -> 
( ( X  o F  x.  Y ) `  ( 1r `  Z
) )  =  ( ( X `  ( 1r `  Z ) )  x.  ( Y `  ( 1r `  Z ) ) ) )
9057, 60, 14, 88, 89syl22anc 1183 . . . . 5  |-  ( ph  ->  ( ( X  o F  x.  Y ) `  ( 1r `  Z
) )  =  ( ( X `  ( 1r `  Z ) )  x.  ( Y `  ( 1r `  Z ) ) ) )
9126, 86rngidval 15359 . . . . . . . . 9  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
92 cnfld1 16415 . . . . . . . . . 10  |-  1  =  ( 1r ` fld )
9330, 92rngidval 15359 . . . . . . . . 9  |-  1  =  ( 0g `  (mulGrp ` fld ) )
9491, 93mhm0 14439 . . . . . . . 8  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( X `  ( 1r `  Z ) )  =  1 )
9525, 94syl 15 . . . . . . 7  |-  ( ph  ->  ( X `  ( 1r `  Z ) )  =  1 )
9691, 93mhm0 14439 . . . . . . . 8  |-  ( Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( Y `  ( 1r `  Z ) )  =  1 )
9738, 96syl 15 . . . . . . 7  |-  ( ph  ->  ( Y `  ( 1r `  Z ) )  =  1 )
9895, 97oveq12d 5892 . . . . . 6  |-  ( ph  ->  ( ( X `  ( 1r `  Z ) )  x.  ( Y `
 ( 1r `  Z ) ) )  =  ( 1  x.  1 ) )
99 1t1e1 9886 . . . . . 6  |-  ( 1  x.  1 )  =  1
10098, 99syl6eq 2344 . . . . 5  |-  ( ph  ->  ( ( X `  ( 1r `  Z ) )  x.  ( Y `
 ( 1r `  Z ) ) )  =  1 )
10190, 100eqtrd 2328 . . . 4  |-  ( ph  ->  ( ( X  o F  x.  Y ) `  ( 1r `  Z
) )  =  1 )
10277neeq1d 2472 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X  o F  x.  Y ) `  x )  =/=  0  <->  ( ( X `  x
)  x.  ( Y `
 x ) )  =/=  0 ) )
10344, 50mulne0bd 9435 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X `  x
)  =/=  0  /\  ( Y `  x
)  =/=  0 )  <-> 
( ( X `  x )  x.  ( Y `  x )
)  =/=  0 ) )
104102, 103bitr4d 247 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X  o F  x.  Y ) `  x )  =/=  0  <->  ( ( X `  x
)  =/=  0  /\  ( Y `  x
)  =/=  0 ) ) )
10524simprd 449 . . . . . . . 8  |-  ( ph  ->  A. x  e.  (
Base `  Z )
( ( X `  x )  =/=  0  ->  x  e.  (Unit `  Z ) ) )
106105r19.21bi 2654 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( ( X `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) )
107106adantrd 454 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X `  x
)  =/=  0  /\  ( Y `  x
)  =/=  0 )  ->  x  e.  (Unit `  Z ) ) )
108104, 107sylbid 206 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X  o F  x.  Y ) `  x )  =/=  0  ->  x  e.  (Unit `  Z ) ) )
109108ralrimiva 2639 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  Z )
( ( ( X  o F  x.  Y
) `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) )
11085, 101, 1093jca 1132 . . 3  |-  ( ph  ->  ( A. x  e.  (Unit `  Z ) A. y  e.  (Unit `  Z ) ( ( X  o F  x.  Y ) `  (
x ( .r `  Z ) y ) )  =  ( ( ( X  o F  x.  Y ) `  x )  x.  (
( X  o F  x.  Y ) `  y ) )  /\  ( ( X  o F  x.  Y ) `  ( 1r `  Z
) )  =  1  /\  A. x  e.  ( Base `  Z
) ( ( ( X  o F  x.  Y ) `  x
)  =/=  0  ->  x  e.  (Unit `  Z
) ) ) )
1111, 2, 10, 17, 22, 3dchrelbas3 20493 . . 3  |-  ( ph  ->  ( ( X  o F  x.  Y )  e.  D  <->  ( ( X  o F  x.  Y
) : ( Base `  Z ) --> CC  /\  ( A. x  e.  (Unit `  Z ) A. y  e.  (Unit `  Z )
( ( X  o F  x.  Y ) `  ( x ( .r
`  Z ) y ) )  =  ( ( ( X  o F  x.  Y ) `  x )  x.  (
( X  o F  x.  Y ) `  y ) )  /\  ( ( X  o F  x.  Y ) `  ( 1r `  Z
) )  =  1  /\  A. x  e.  ( Base `  Z
) ( ( ( X  o F  x.  Y ) `  x
)  =/=  0  ->  x  e.  (Unit `  Z
) ) ) ) ) )
11216, 110, 111mpbir2and 888 . 2  |-  ( ph  ->  ( X  o F  x.  Y )  e.  D )
1137, 112eqeltrd 2370 1  |-  ( ph  ->  ( X  .x.  Y
)  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758   NNcn 9762   NN0cn0 9981   Basecbs 13164   +g cplusg 13224   .rcmulr 13225   MndHom cmhm 14429  mulGrpcmgp 15341   Ringcrg 15353   CRingccrg 15354   1rcur 15355  Unitcui 15437  ℂfldccnfld 16393  ℤ/nczn 16470  DChrcdchr 20487
This theorem is referenced by:  dchrabl  20509  dchrinv  20516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-imas 13427  df-divs 13428  df-mnd 14383  df-mhm 14431  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-nsg 14635  df-eqg 14636  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-sra 15941  df-rgmod 15942  df-lidl 15943  df-rsp 15944  df-2idl 16000  df-cnfld 16394  df-zn 16474  df-dchr 20488
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