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Theorem dchrmulcl 20488
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 20487 . 2  |-  ( ph  ->  ( X  .x.  Y
)  =  ( X  o F  x.  Y
) )
8 mulcl 8821 . . . . 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 2283 . . . . 5  |-  ( Base `  Z )  =  (
Base `  Z )
111, 2, 3, 10, 5dchrf 20481 . . . 4  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
121, 2, 3, 10, 6dchrf 20481 . . . 4  |-  ( ph  ->  Y : ( Base `  Z ) --> CC )
13 fvex 5539 . . . . 5  |-  ( Base `  Z )  e.  _V
1413a1i 10 . . . 4  |-  ( ph  ->  ( Base `  Z
)  e.  _V )
15 inidm 3378 . . . 4  |-  ( (
Base `  Z )  i^i  ( Base `  Z
) )  =  (
Base `  Z )
169, 11, 12, 14, 14, 15off 6093 . . 3  |-  ( ph  ->  ( X  o F  x.  Y ) : ( Base `  Z
) --> CC )
17 eqid 2283 . . . . . . . 8  |-  (Unit `  Z )  =  (Unit `  Z )
1810, 17unitcl 15441 . . . . . . 7  |-  ( x  e.  (Unit `  Z
)  ->  x  e.  ( Base `  Z )
)
1910, 17unitcl 15441 . . . . . . 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 20479 . . . . . . . . . . . . . 14  |-  ( X  e.  D  ->  N  e.  NN )
225, 21syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  NN )
231, 2, 10, 17, 22, 3dchrelbas2 20476 . . . . . . . . . . . 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 2283 . . . . . . . . . . . . 13  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
2726, 10mgpbas 15331 . . . . . . . . . . . 12  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
28 eqid 2283 . . . . . . . . . . . . 13  |-  ( .r
`  Z )  =  ( .r `  Z
)
2926, 28mgpplusg 15329 . . . . . . . . . . . 12  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
30 eqid 2283 . . . . . . . . . . . . 13  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
31 cnfldmul 16385 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
3230, 31mgpplusg 15329 . . . . . . . . . . . 12  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3327, 29, 32mhmlin 14422 . . . . . . . . . . 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 20476 . . . . . . . . . . . 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 14422 . . . . . . . . . . 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 5876 . . . . . . . 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 5663 . . . . . . . . . . 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 5663 . . . . . . . . . 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 5663 . . . . . . . . . . 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 5663 . . . . . . . . . 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 9024 . . . . . . . 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 2315 . . . . . . 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 5389 . . . . . . . . . 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 5389 . . . . . . . . . 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 10018 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
642zncrng 16498 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
65 crngrng 15351 . . . . . . . . . 10  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
6663, 64, 653syl 18 . . . . . . . . 9  |-  ( ph  ->  Z  e.  Ring )
6710, 28rngcl 15354 . . . . . . . . . 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 6090 . . . . . . . 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 6090 . . . . . . . . . 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 6090 . . . . . . . . 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 5876 . . . . . . 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 2325 . . . . . 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 2635 . . . 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 2283 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
8710, 86rngidcl 15361 . . . . . . 7  |-  ( Z  e.  Ring  ->  ( 1r
`  Z )  e.  ( Base `  Z
) )
8866, 87syl 15 . . . . . 6  |-  ( ph  ->  ( 1r `  Z
)  e.  ( Base `  Z ) )
89 fnfvof 6090 . . . . . 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 15343 . . . . . . . . 9  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
92 cnfld1 16399 . . . . . . . . . 10  |-  1  =  ( 1r ` fld )
9330, 92rngidval 15343 . . . . . . . . 9  |-  1  =  ( 0g `  (mulGrp ` fld ) )
9491, 93mhm0 14423 . . . . . . . 8  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( X `  ( 1r `  Z ) )  =  1 )
9525, 94syl 15 . . . . . . 7  |-  ( ph  ->  ( X `  ( 1r `  Z ) )  =  1 )
9691, 93mhm0 14423 . . . . . . . 8  |-  ( Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( Y `  ( 1r `  Z ) )  =  1 )
9738, 96syl 15 . . . . . . 7  |-  ( ph  ->  ( Y `  ( 1r `  Z ) )  =  1 )
9895, 97oveq12d 5876 . . . . . 6  |-  ( ph  ->  ( ( X `  ( 1r `  Z ) )  x.  ( Y `
 ( 1r `  Z ) ) )  =  ( 1  x.  1 ) )
99 1t1e1 9870 . . . . . 6  |-  ( 1  x.  1 )  =  1
10098, 99syl6eq 2331 . . . . 5  |-  ( ph  ->  ( ( X `  ( 1r `  Z ) )  x.  ( Y `
 ( 1r `  Z ) ) )  =  1 )
10190, 100eqtrd 2315 . . . 4  |-  ( ph  ->  ( ( X  o F  x.  Y ) `  ( 1r `  Z
) )  =  1 )
10277neeq1d 2459 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X  o F  x.  Y ) `  x )  =/=  0  <->  ( ( X `  x
)  x.  ( Y `
 x ) )  =/=  0 ) )
10344, 50mulne0bd 9419 . . . . . . 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 2641 . . . . . . 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 2626 . . . 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 20477 . . 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 2357 1  |-  ( ph  ->  ( X  .x.  Y
)  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742   NNcn 9746   NN0cn0 9965   Basecbs 13148   +g cplusg 13208   .rcmulr 13209   MndHom cmhm 14413  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338   1rcur 15339  Unitcui 15421  ℂfldccnfld 16377  ℤ/nczn 16454  DChrcdchr 20471
This theorem is referenced by:  dchrabl  20493  dchrinv  20500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-imas 13411  df-divs 13412  df-mnd 14367  df-mhm 14415  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-nsg 14619  df-eqg 14620  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-cnfld 16378  df-zn 16458  df-dchr 20472
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