Description: The multiplicative structure of Z/nZ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015) (Revised by AV, 13-Jun-2019)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | znval2.s | |- S = ( RSpan ` ZZring ) |
|
| znval2.u | |- U = ( ZZring /s ( ZZring ~QG ( S ` { N } ) ) ) |
||
| znval2.y | |- Y = ( Z/nZ ` N ) |
||
| Assertion | znmul | |- ( N e. NN0 -> ( .r ` U ) = ( .r ` 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 | df-mulr | |- .r = Slot 3 |
|
| 5 | 3nn | |- 3 e. NN |
|
| 6 | 3lt10 | |- 3 < ; 1 0 |
|
| 7 | 1 2 3 4 5 6 | znbaslem | |- ( N e. NN0 -> ( .r ` U ) = ( .r ` Y ) ) |