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) (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 ) |
||
| 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 | mulrid | |- .r = Slot ( .r ` ndx ) |
|
| 5 | plendxnmulrndx | |- ( le ` ndx ) =/= ( .r ` ndx ) |
|
| 6 | 5 | necomi | |- ( .r ` ndx ) =/= ( le ` ndx ) |
| 7 | 1 2 3 4 6 | znbaslem | |- ( N e. NN0 -> ( .r ` U ) = ( .r ` Y ) ) |