| Step |
Hyp |
Ref |
Expression |
| 1 |
|
psrring.s |
|- S = ( I mPwSer R ) |
| 2 |
|
psrring.i |
|- ( ph -> I e. V ) |
| 3 |
|
psrring.r |
|- ( ph -> R e. Ring ) |
| 4 |
|
psr1cl.d |
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
| 5 |
|
psr1cl.z |
|- .0. = ( 0g ` R ) |
| 6 |
|
psr1cl.o |
|- .1. = ( 1r ` R ) |
| 7 |
|
psr1cl.u |
|- U = ( x e. D |-> if ( x = ( I X. { 0 } ) , .1. , .0. ) ) |
| 8 |
|
psr1cl.b |
|- B = ( Base ` S ) |
| 9 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
| 10 |
9 6
|
ringidcl |
|- ( R e. Ring -> .1. e. ( Base ` R ) ) |
| 11 |
9 5
|
ring0cl |
|- ( R e. Ring -> .0. e. ( Base ` R ) ) |
| 12 |
10 11
|
ifcld |
|- ( R e. Ring -> if ( x = ( I X. { 0 } ) , .1. , .0. ) e. ( Base ` R ) ) |
| 13 |
3 12
|
syl |
|- ( ph -> if ( x = ( I X. { 0 } ) , .1. , .0. ) e. ( Base ` R ) ) |
| 14 |
13
|
adantr |
|- ( ( ph /\ x e. D ) -> if ( x = ( I X. { 0 } ) , .1. , .0. ) e. ( Base ` R ) ) |
| 15 |
14 7
|
fmptd |
|- ( ph -> U : D --> ( Base ` R ) ) |
| 16 |
|
fvex |
|- ( Base ` R ) e. _V |
| 17 |
|
ovex |
|- ( NN0 ^m I ) e. _V |
| 18 |
4 17
|
rabex2 |
|- D e. _V |
| 19 |
16 18
|
elmap |
|- ( U e. ( ( Base ` R ) ^m D ) <-> U : D --> ( Base ` R ) ) |
| 20 |
15 19
|
sylibr |
|- ( ph -> U e. ( ( Base ` R ) ^m D ) ) |
| 21 |
1 9 4 8 2
|
psrbas |
|- ( ph -> B = ( ( Base ` R ) ^m D ) ) |
| 22 |
20 21
|
eleqtrrd |
|- ( ph -> U e. B ) |