Description: The ring multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015) (Revised by AV, 31-Oct-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | tngbas.t | |- T = ( G toNrmGrp N ) |
|
| tngmulr.2 | |- .x. = ( .r ` G ) |
||
| Assertion | tngmulr | |- ( N e. V -> .x. = ( .r ` T ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tngbas.t | |- T = ( G toNrmGrp N ) |
|
| 2 | tngmulr.2 | |- .x. = ( .r ` G ) |
|
| 3 | mulrid | |- .r = Slot ( .r ` ndx ) |
|
| 4 | tsetndxnmulrndx | |- ( TopSet ` ndx ) =/= ( .r ` ndx ) |
|
| 5 | 4 | necomi | |- ( .r ` ndx ) =/= ( TopSet ` ndx ) |
| 6 | dsndxnmulrndx | |- ( dist ` ndx ) =/= ( .r ` ndx ) |
|
| 7 | 6 | necomi | |- ( .r ` ndx ) =/= ( dist ` ndx ) |
| 8 | 1 3 5 7 | tnglem | |- ( N e. V -> ( .r ` G ) = ( .r ` T ) ) |
| 9 | 2 8 | eqtrid | |- ( N e. V -> .x. = ( .r ` T ) ) |