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Metamath Proof Explorer

Theorem dchrbas

Description: Base set of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016)

Ref Expression
Hypotheses dchrval.g 
|- G = ( DChr ` N )
dchrval.z 
|- Z = ( Z/nZ ` N )
dchrval.b 
|- B = ( Base ` Z )
dchrval.u 
|- U = ( Unit ` Z )
dchrval.n 
|- ( ph -> N e. NN )
dchrbas.b 
|- D = ( Base ` G )
Assertion dchrbas 
|- ( ph -> D = { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } )

Proof

Step Hyp Ref Expression
1 dchrval.g 
 |-  G = ( DChr ` N )
2 dchrval.z 
 |-  Z = ( Z/nZ ` N )
3 dchrval.b 
 |-  B = ( Base ` Z )
4 dchrval.u 
 |-  U = ( Unit ` Z )
5 dchrval.n 
 |-  ( ph -> N e. NN )
6 dchrbas.b 
 |-  D = ( Base ` G )
7 eqidd 
 |-  ( ph -> { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } = { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } )
8 1 2 3 4 5 7 dchrval 
 |-  ( ph -> G = { <. ( Base ` ndx ) , { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } >. , <. ( +g ` ndx ) , ( oF x. |` ( { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } X. { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } ) ) >. } )
9 8 fveq2d 
 |-  ( ph -> ( Base ` G ) = ( Base ` { <. ( Base ` ndx ) , { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } >. , <. ( +g ` ndx ) , ( oF x. |` ( { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } X. { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } ) ) >. } ) )
10 ovex 
 |-  ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) e. _V
11 10 rabex 
 |-  { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } e. _V
12 eqid 
 |-  { <. ( Base ` ndx ) , { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } >. , <. ( +g ` ndx ) , ( oF x. |` ( { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } X. { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } ) ) >. } = { <. ( Base ` ndx ) , { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } >. , <. ( +g ` ndx ) , ( oF x. |` ( { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } X. { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } ) ) >. }
13 12 grpbase 
 |-  ( { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } e. _V -> { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } = ( Base ` { <. ( Base ` ndx ) , { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } >. , <. ( +g ` ndx ) , ( oF x. |` ( { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } X. { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } ) ) >. } ) )
14 11 13 ax-mp 
 |-  { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } = ( Base ` { <. ( Base ` ndx ) , { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } >. , <. ( +g ` ndx ) , ( oF x. |` ( { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } X. { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } ) ) >. } )
15 9 6 14 3eqtr4g 
 |-  ( ph -> D = { x e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } )