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Theorem dchrmulcl 22531
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 22530 . 2  |-  ( ph  ->  ( X  .x.  Y
)  =  ( X  oF  x.  Y
) )
8 mulcl 9362 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
98adantl 463 . . . 4  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
10 eqid 2441 . . . . 5  |-  ( Base `  Z )  =  (
Base `  Z )
111, 2, 3, 10, 5dchrf 22524 . . . 4  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
121, 2, 3, 10, 6dchrf 22524 . . . 4  |-  ( ph  ->  Y : ( Base `  Z ) --> CC )
13 fvex 5698 . . . . 5  |-  ( Base `  Z )  e.  _V
1413a1i 11 . . . 4  |-  ( ph  ->  ( Base `  Z
)  e.  _V )
15 inidm 3556 . . . 4  |-  ( (
Base `  Z )  i^i  ( Base `  Z
) )  =  (
Base `  Z )
169, 11, 12, 14, 14, 15off 6333 . . 3  |-  ( ph  ->  ( X  oF  x.  Y ) : ( Base `  Z
) --> CC )
17 eqid 2441 . . . . . . . 8  |-  (Unit `  Z )  =  (Unit `  Z )
1810, 17unitcl 16741 . . . . . . 7  |-  ( x  e.  (Unit `  Z
)  ->  x  e.  ( Base `  Z )
)
1910, 17unitcl 16741 . . . . . . 7  |-  ( y  e.  (Unit `  Z
)  ->  y  e.  ( Base `  Z )
)
2018, 19anim12i 563 . . . . . 6  |-  ( ( x  e.  (Unit `  Z )  /\  y  e.  (Unit `  Z )
)  ->  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )
211, 3dchrrcl 22522 . . . . . . . . . . . . . 14  |-  ( X  e.  D  ->  N  e.  NN )
225, 21syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  NN )
231, 2, 10, 17, 22, 3dchrelbas2 22519 . . . . . . . . . . . 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 456 . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
26 eqid 2441 . . . . . . . . . . . . 13  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
2726, 10mgpbas 16587 . . . . . . . . . . . 12  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
28 eqid 2441 . . . . . . . . . . . . 13  |-  ( .r
`  Z )  =  ( .r `  Z
)
2926, 28mgpplusg 16585 . . . . . . . . . . . 12  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
30 eqid 2441 . . . . . . . . . . . . 13  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
31 cnfldmul 17724 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
3230, 31mgpplusg 16585 . . . . . . . . . . . 12  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3327, 29, 32mhmlin 15467 . . . . . . . . . . 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 1183 . . . . . . . . . 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 468 . . . . . . . . 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 22519 . . . . . . . . . . . 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 456 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
3927, 29, 32mhmlin 15467 . . . . . . . . . . 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 1183 . . . . . . . . . 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 468 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  ( x
( .r `  Z
) y ) )  =  ( ( Y `
 x )  x.  ( Y `  y
) ) )
4235, 41oveq12d 6108 . . . . . . . 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 5840 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( X `  x )  e.  CC )
4443adantrr 711 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( X `  x )  e.  CC )
45 simpr 458 . . . . . . . . . 10  |-  ( ( x  e.  ( Base `  Z )  /\  y  e.  ( Base `  Z
) )  ->  y  e.  ( Base `  Z
) )
46 ffvelrn 5838 . . . . . . . . . 10  |-  ( ( X : ( Base `  Z ) --> CC  /\  y  e.  ( Base `  Z ) )  -> 
( X `  y
)  e.  CC )
4711, 45, 46syl2an 474 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( X `  y )  e.  CC )
4812ffvelrnda 5840 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( Y `  x )  e.  CC )
4948adantrr 711 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  x )  e.  CC )
50 ffvelrn 5838 . . . . . . . . . 10  |-  ( ( Y : ( Base `  Z ) --> CC  /\  y  e.  ( Base `  Z ) )  -> 
( Y `  y
)  e.  CC )
5112, 45, 50syl2an 474 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  y )  e.  CC )
5244, 47, 49, 51mul4d 9577 . . . . . . . 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 2473 . . . . . . 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 5556 . . . . . . . . . 10  |-  ( X : ( Base `  Z
) --> CC  ->  X  Fn  ( Base `  Z
) )
5511, 54syl 16 . . . . . . . . 9  |-  ( ph  ->  X  Fn  ( Base `  Z ) )
5655adantr 462 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  X  Fn  ( Base `  Z
) )
57 ffn 5556 . . . . . . . . . 10  |-  ( Y : ( Base `  Z
) --> CC  ->  Y  Fn  ( Base `  Z
) )
5812, 57syl 16 . . . . . . . . 9  |-  ( ph  ->  Y  Fn  ( Base `  Z ) )
5958adantr 462 . . . . . . . 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 10632 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
622zncrng 17877 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
63 crngrng 16645 . . . . . . . . . 10  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
6461, 62, 633syl 20 . . . . . . . . 9  |-  ( ph  ->  Z  e.  Ring )
6510, 28rngcl 16648 . . . . . . . . . 10  |-  ( ( Z  e.  Ring  /\  x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
)  ->  ( x
( .r `  Z
) y )  e.  ( Base `  Z
) )
66653expb 1183 . . . . . . . . 9  |-  ( ( Z  e.  Ring  /\  (
x  e.  ( Base `  Z )  /\  y  e.  ( Base `  Z
) ) )  -> 
( x ( .r
`  Z ) y )  e.  ( Base `  Z ) )
6764, 66sylan 468 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
x ( .r `  Z ) y )  e.  ( Base `  Z
) )
68 fnfvof 6332 . . . . . . . 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 1214 . . . . . . 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 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  X  Fn  ( Base `  Z )
)
7158adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  Y  Fn  ( Base `  Z )
)
7213a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( Base `  Z )  e.  _V )
73 simpr 458 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  x  e.  ( Base `  Z )
)
74 fnfvof 6332 . . . . . . . . . 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 1214 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( ( X  oF  x.  Y
) `  x )  =  ( ( X `
 x )  x.  ( Y `  x
) ) )
7675adantrr 711 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X  oF  x.  Y ) `  x )  =  ( ( X `  x
)  x.  ( Y `
 x ) ) )
77 simprr 751 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  y  e.  ( Base `  Z
) )
78 fnfvof 6332 . . . . . . . . 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 1214 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X  oF  x.  Y ) `  y )  =  ( ( X `  y
)  x.  ( Y `
 y ) ) )
8076, 79oveq12d 6108 . . . . . . 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 2483 . . . . . 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 471 . . . . 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 2806 . . . 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 2441 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
8510, 84rngidcl 16655 . . . . . . 7  |-  ( Z  e.  Ring  ->  ( 1r
`  Z )  e.  ( Base `  Z
) )
8664, 85syl 16 . . . . . 6  |-  ( ph  ->  ( 1r `  Z
)  e.  ( Base `  Z ) )
87 fnfvof 6332 . . . . . 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 1214 . . . . 5  |-  ( ph  ->  ( ( X  oF  x.  Y ) `  ( 1r `  Z
) )  =  ( ( X `  ( 1r `  Z ) )  x.  ( Y `  ( 1r `  Z ) ) ) )
8926, 84rngidval 16595 . . . . . . . . 9  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
90 cnfld1 17741 . . . . . . . . . 10  |-  1  =  ( 1r ` fld )
9130, 90rngidval 16595 . . . . . . . . 9  |-  1  =  ( 0g `  (mulGrp ` fld ) )
9289, 91mhm0 15468 . . . . . . . 8  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( X `  ( 1r `  Z ) )  =  1 )
9325, 92syl 16 . . . . . . 7  |-  ( ph  ->  ( X `  ( 1r `  Z ) )  =  1 )
9489, 91mhm0 15468 . . . . . . . 8  |-  ( Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( Y `  ( 1r `  Z ) )  =  1 )
9538, 94syl 16 . . . . . . 7  |-  ( ph  ->  ( Y `  ( 1r `  Z ) )  =  1 )
9693, 95oveq12d 6108 . . . . . 6  |-  ( ph  ->  ( ( X `  ( 1r `  Z ) )  x.  ( Y `
 ( 1r `  Z ) ) )  =  ( 1  x.  1 ) )
97 1t1e1 10465 . . . . . 6  |-  ( 1  x.  1 )  =  1
9896, 97syl6eq 2489 . . . . 5  |-  ( ph  ->  ( ( X `  ( 1r `  Z ) )  x.  ( Y `
 ( 1r `  Z ) ) )  =  1 )
9988, 98eqtrd 2473 . . . 4  |-  ( ph  ->  ( ( X  oF  x.  Y ) `  ( 1r `  Z
) )  =  1 )
10075neeq1d 2619 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X  oF  x.  Y ) `  x )  =/=  0  <->  ( ( X `  x
)  x.  ( Y `
 x ) )  =/=  0 ) )
10143, 48mulne0bd 9983 . . . . . . 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 460 . . . . . . . 8  |-  ( ph  ->  A. x  e.  (
Base `  Z )
( ( X `  x )  =/=  0  ->  x  e.  (Unit `  Z ) ) )
104103r19.21bi 2812 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( ( X `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) )
105104adantrd 465 . . . . . 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 2797 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  Z )
( ( ( X  oF  x.  Y
) `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) )
10883, 99, 1073jca 1163 . . 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 22520 . . 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 908 . 2  |-  ( ph  ->  ( X  oF  x.  Y )  e.  D )
1117, 110eqeltrd 2515 1  |-  ( ph  ->  ( X  .x.  Y
)  e.  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   _Vcvv 2970    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   CCcc 9276   0cc0 9278   1c1 9279    x. cmul 9283   NNcn 10318   NN0cn0 10575   Basecbs 14170   +g cplusg 14234   .rcmulr 14235   MndHom cmhm 15458  mulGrpcmgp 16581   1rcur 16593   Ringcrg 16635   CRingccrg 16636  Unitcui 16721  ℂfldccnfld 17718  ℤ/nczn 17834  DChrcdchr 22514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-0g 14376  df-imas 14442  df-divs 14443  df-mnd 15411  df-mhm 15460  df-grp 15538  df-minusg 15539  df-sbg 15540  df-subg 15671  df-nsg 15672  df-eqg 15673  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-subrg 16843  df-lmod 16930  df-lss 16992  df-lsp 17031  df-sra 17231  df-rgmod 17232  df-lidl 17233  df-rsp 17234  df-2idl 17292  df-cnfld 17719  df-zring 17784  df-zn 17838  df-dchr 22515
This theorem is referenced by:  dchrabl  22536  dchrinv  22543
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