Description: Lemma for znbas . (Contributed by Mario Carneiro, 14-Jun-2015) (Revised by Mario Carneiro, 14-Aug-2015) (Revised by AV, 13-Jun-2019) (Revised by AV, 9-Sep-2021) (Revised by AV, 3-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | znval2.s | |- S = ( RSpan ` ZZring ) |
|
| znval2.u | |- U = ( ZZring /s ( ZZring ~QG ( S ` { N } ) ) ) |
||
| znval2.y | |- Y = ( Z/nZ ` N ) |
||
| znbaslem.e | |- E = Slot ( E ` ndx ) |
||
| znbaslem.n | |- ( E ` ndx ) =/= ( le ` ndx ) |
||
| Assertion | znbaslem | |- ( N e. NN0 -> ( E ` U ) = ( E ` Y ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | znval2.s | |- S = ( RSpan ` ZZring ) |
|
| 2 | znval2.u | |- U = ( ZZring /s ( ZZring ~QG ( S ` { N } ) ) ) |
|
| 3 | znval2.y | |- Y = ( Z/nZ ` N ) |
|
| 4 | znbaslem.e | |- E = Slot ( E ` ndx ) |
|
| 5 | znbaslem.n | |- ( E ` ndx ) =/= ( le ` ndx ) |
|
| 6 | 4 5 | setsnid | |- ( E ` U ) = ( E ` ( U sSet <. ( le ` ndx ) , ( le ` Y ) >. ) ) |
| 7 | eqid | |- ( le ` Y ) = ( le ` Y ) |
|
| 8 | 1 2 3 7 | znval2 | |- ( N e. NN0 -> Y = ( U sSet <. ( le ` ndx ) , ( le ` Y ) >. ) ) |
| 9 | 8 | fveq2d | |- ( N e. NN0 -> ( E ` Y ) = ( E ` ( U sSet <. ( le ` ndx ) , ( le ` Y ) >. ) ) ) |
| 10 | 6 9 | eqtr4id | |- ( N e. NN0 -> ( E ` U ) = ( E ` Y ) ) |