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Theorem dchrmulcl 24040
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 24039 . 2  |-  ( ph  ->  ( X  .x.  Y
)  =  ( X  oF  x.  Y
) )
8 mulcl 9622 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
98adantl 467 . . . 4  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
10 eqid 2429 . . . . 5  |-  ( Base `  Z )  =  (
Base `  Z )
111, 2, 3, 10, 5dchrf 24033 . . . 4  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
121, 2, 3, 10, 6dchrf 24033 . . . 4  |-  ( ph  ->  Y : ( Base `  Z ) --> CC )
13 fvex 5891 . . . . 5  |-  ( Base `  Z )  e.  _V
1413a1i 11 . . . 4  |-  ( ph  ->  ( Base `  Z
)  e.  _V )
15 inidm 3677 . . . 4  |-  ( (
Base `  Z )  i^i  ( Base `  Z
) )  =  (
Base `  Z )
169, 11, 12, 14, 14, 15off 6560 . . 3  |-  ( ph  ->  ( X  oF  x.  Y ) : ( Base `  Z
) --> CC )
17 eqid 2429 . . . . . . . 8  |-  (Unit `  Z )  =  (Unit `  Z )
1810, 17unitcl 17822 . . . . . . 7  |-  ( x  e.  (Unit `  Z
)  ->  x  e.  ( Base `  Z )
)
1910, 17unitcl 17822 . . . . . . 7  |-  ( y  e.  (Unit `  Z
)  ->  y  e.  ( Base `  Z )
)
2018, 19anim12i 568 . . . . . 6  |-  ( ( x  e.  (Unit `  Z )  /\  y  e.  (Unit `  Z )
)  ->  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )
211, 3dchrrcl 24031 . . . . . . . . . . . . . 14  |-  ( X  e.  D  ->  N  e.  NN )
225, 21syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  NN )
231, 2, 10, 17, 22, 3dchrelbas2 24028 . . . . . . . . . . . 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 213 . . . . . . . . . . 11  |-  ( ph  ->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ( Base `  Z ) ( ( X `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) ) )
2524simpld 460 . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
26 eqid 2429 . . . . . . . . . . . . 13  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
2726, 10mgpbas 17664 . . . . . . . . . . . 12  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
28 eqid 2429 . . . . . . . . . . . . 13  |-  ( .r
`  Z )  =  ( .r `  Z
)
2926, 28mgpplusg 17662 . . . . . . . . . . . 12  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
30 eqid 2429 . . . . . . . . . . . . 13  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
31 cnfldmul 18911 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
3230, 31mgpplusg 17662 . . . . . . . . . . . 12  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3327, 29, 32mhmlin 16540 . . . . . . . . . . 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 1206 . . . . . . . . . 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 473 . . . . . . . . 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 24028 . . . . . . . . . . . 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 213 . . . . . . . . . . 11  |-  ( ph  ->  ( Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ( Base `  Z ) ( ( Y `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) ) )
3837simpld 460 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
3927, 29, 32mhmlin 16540 . . . . . . . . . . 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 1206 . . . . . . . . . 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 473 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  ( x
( .r `  Z
) y ) )  =  ( ( Y `
 x )  x.  ( Y `  y
) ) )
4235, 41oveq12d 6323 . . . . . . . 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 6037 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( X `  x )  e.  CC )
4443adantrr 721 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( X `  x )  e.  CC )
45 simpr 462 . . . . . . . . . 10  |-  ( ( x  e.  ( Base `  Z )  /\  y  e.  ( Base `  Z
) )  ->  y  e.  ( Base `  Z
) )
46 ffvelrn 6035 . . . . . . . . . 10  |-  ( ( X : ( Base `  Z ) --> CC  /\  y  e.  ( Base `  Z ) )  -> 
( X `  y
)  e.  CC )
4711, 45, 46syl2an 479 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( X `  y )  e.  CC )
4812ffvelrnda 6037 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( Y `  x )  e.  CC )
4948adantrr 721 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  x )  e.  CC )
50 ffvelrn 6035 . . . . . . . . . 10  |-  ( ( Y : ( Base `  Z ) --> CC  /\  y  e.  ( Base `  Z ) )  -> 
( Y `  y
)  e.  CC )
5112, 45, 50syl2an 479 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  ( Y `  y )  e.  CC )
5244, 47, 49, 51mul4d 9844 . . . . . . . 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 2470 . . . . . . 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 5746 . . . . . . . . . 10  |-  ( X : ( Base `  Z
) --> CC  ->  X  Fn  ( Base `  Z
) )
5511, 54syl 17 . . . . . . . . 9  |-  ( ph  ->  X  Fn  ( Base `  Z ) )
5655adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  X  Fn  ( Base `  Z
) )
57 ffn 5746 . . . . . . . . . 10  |-  ( Y : ( Base `  Z
) --> CC  ->  Y  Fn  ( Base `  Z
) )
5812, 57syl 17 . . . . . . . . 9  |-  ( ph  ->  Y  Fn  ( Base `  Z ) )
5958adantr 466 . . . . . . . 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 10925 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
622zncrng 19046 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
63 crngring 17726 . . . . . . . . . 10  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
6461, 62, 633syl 18 . . . . . . . . 9  |-  ( ph  ->  Z  e.  Ring )
6510, 28ringcl 17729 . . . . . . . . . 10  |-  ( ( Z  e.  Ring  /\  x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
)  ->  ( x
( .r `  Z
) y )  e.  ( Base `  Z
) )
66653expb 1206 . . . . . . . . 9  |-  ( ( Z  e.  Ring  /\  (
x  e.  ( Base `  Z )  /\  y  e.  ( Base `  Z
) ) )  -> 
( x ( .r
`  Z ) y )  e.  ( Base `  Z ) )
6764, 66sylan 473 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
x ( .r `  Z ) y )  e.  ( Base `  Z
) )
68 fnfvof 6559 . . . . . . . 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 1265 . . . . . . 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 466 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  X  Fn  ( Base `  Z )
)
7158adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  Y  Fn  ( Base `  Z )
)
7213a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( Base `  Z )  e.  _V )
73 simpr 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  x  e.  ( Base `  Z )
)
74 fnfvof 6559 . . . . . . . . . 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 1265 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( ( X  oF  x.  Y
) `  x )  =  ( ( X `
 x )  x.  ( Y `  x
) ) )
7675adantrr 721 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X  oF  x.  Y ) `  x )  =  ( ( X `  x
)  x.  ( Y `
 x ) ) )
77 simprr 764 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  y  e.  ( Base `  Z
) )
78 fnfvof 6559 . . . . . . . . 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 1265 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  Z
)  /\  y  e.  ( Base `  Z )
) )  ->  (
( X  oF  x.  Y ) `  y )  =  ( ( X `  y
)  x.  ( Y `
 y ) ) )
8076, 79oveq12d 6323 . . . . . . 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 2480 . . . . . 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 476 . . . . 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 2853 . . . 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 2429 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
8510, 84ringidcl 17736 . . . . . . 7  |-  ( Z  e.  Ring  ->  ( 1r
`  Z )  e.  ( Base `  Z
) )
8664, 85syl 17 . . . . . 6  |-  ( ph  ->  ( 1r `  Z
)  e.  ( Base `  Z ) )
87 fnfvof 6559 . . . . . 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 1265 . . . . 5  |-  ( ph  ->  ( ( X  oF  x.  Y ) `  ( 1r `  Z
) )  =  ( ( X `  ( 1r `  Z ) )  x.  ( Y `  ( 1r `  Z ) ) ) )
8926, 84ringidval 17672 . . . . . . . . 9  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
90 cnfld1 18928 . . . . . . . . . 10  |-  1  =  ( 1r ` fld )
9130, 90ringidval 17672 . . . . . . . . 9  |-  1  =  ( 0g `  (mulGrp ` fld ) )
9289, 91mhm0 16541 . . . . . . . 8  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( X `  ( 1r `  Z ) )  =  1 )
9325, 92syl 17 . . . . . . 7  |-  ( ph  ->  ( X `  ( 1r `  Z ) )  =  1 )
9489, 91mhm0 16541 . . . . . . . 8  |-  ( Y  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( Y `  ( 1r `  Z ) )  =  1 )
9538, 94syl 17 . . . . . . 7  |-  ( ph  ->  ( Y `  ( 1r `  Z ) )  =  1 )
9693, 95oveq12d 6323 . . . . . 6  |-  ( ph  ->  ( ( X `  ( 1r `  Z ) )  x.  ( Y `
 ( 1r `  Z ) ) )  =  ( 1  x.  1 ) )
97 1t1e1 10757 . . . . . 6  |-  ( 1  x.  1 )  =  1
9896, 97syl6eq 2486 . . . . 5  |-  ( ph  ->  ( ( X `  ( 1r `  Z ) )  x.  ( Y `
 ( 1r `  Z ) ) )  =  1 )
9988, 98eqtrd 2470 . . . 4  |-  ( ph  ->  ( ( X  oF  x.  Y ) `  ( 1r `  Z
) )  =  1 )
10075neeq1d 2708 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X  oF  x.  Y ) `  x )  =/=  0  <->  ( ( X `  x
)  x.  ( Y `
 x ) )  =/=  0 ) )
10143, 48mulne0bd 10262 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X `  x
)  =/=  0  /\  ( Y `  x
)  =/=  0 )  <-> 
( ( X `  x )  x.  ( Y `  x )
)  =/=  0 ) )
102100, 101bitr4d 259 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X  oF  x.  Y ) `  x )  =/=  0  <->  ( ( X `  x
)  =/=  0  /\  ( Y `  x
)  =/=  0 ) ) )
10324simprd 464 . . . . . . . 8  |-  ( ph  ->  A. x  e.  (
Base `  Z )
( ( X `  x )  =/=  0  ->  x  e.  (Unit `  Z ) ) )
104103r19.21bi 2801 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( ( X `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) )
105104adantrd 469 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X `  x
)  =/=  0  /\  ( Y `  x
)  =/=  0 )  ->  x  e.  (Unit `  Z ) ) )
106102, 105sylbid 218 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  Z )
)  ->  ( (
( X  oF  x.  Y ) `  x )  =/=  0  ->  x  e.  (Unit `  Z ) ) )
107106ralrimiva 2846 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  Z )
( ( ( X  oF  x.  Y
) `  x )  =/=  0  ->  x  e.  (Unit `  Z )
) )
10883, 99, 1073jca 1185 . . 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 24029 . . 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 930 . 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   _Vcvv 3087    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    oFcof 6543   CCcc 9536   0cc0 9538   1c1 9539    x. cmul 9543   NNcn 10609   NN0cn0 10869   Basecbs 15084   +g cplusg 15152   .rcmulr 15153   MndHom cmhm 16531  mulGrpcmgp 17658   1rcur 17670   Ringcrg 17715   CRingccrg 17716  Unitcui 17802  ℂfldccnfld 18905  ℤ/nczn 19005  DChrcdchr 24023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-addf 9617  ax-mulf 9618
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-ec 7373  df-qs 7377  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-0g 15299  df-imas 15365  df-qus 15366  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-nsg 16766  df-eqg 16767  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-cring 17718  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-subrg 17941  df-lmod 18028  df-lss 18091  df-lsp 18130  df-sra 18330  df-rgmod 18331  df-lidl 18332  df-rsp 18333  df-2idl 18391  df-cnfld 18906  df-zring 18974  df-zn 19009  df-dchr 24024
This theorem is referenced by:  dchrabl  24045  dchrinv  24052
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