Description: Obsolete proof of srads as of 29-Oct-2024. Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015) (Revised by Thierry Arnoux, 16-Jun-2019) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | srapart.a | |- ( ph -> A = ( ( subringAlg ` W ) ` S ) ) |
|
| srapart.s | |- ( ph -> S C_ ( Base ` W ) ) |
||
| Assertion | sradsOLD | |- ( ph -> ( dist ` W ) = ( dist ` A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | srapart.a | |- ( ph -> A = ( ( subringAlg ` W ) ` S ) ) |
|
| 2 | srapart.s | |- ( ph -> S C_ ( Base ` W ) ) |
|
| 3 | df-ds | |- dist = Slot ; 1 2 |
|
| 4 | 1nn0 | |- 1 e. NN0 |
|
| 5 | 2nn | |- 2 e. NN |
|
| 6 | 4 5 | decnncl | |- ; 1 2 e. NN |
| 7 | 1nn | |- 1 e. NN |
|
| 8 | 2nn0 | |- 2 e. NN0 |
|
| 9 | 8nn0 | |- 8 e. NN0 |
|
| 10 | 8lt10 | |- 8 < ; 1 0 |
|
| 11 | 7 8 9 10 | declti | |- 8 < ; 1 2 |
| 12 | 11 | olci | |- ( ; 1 2 < 5 \/ 8 < ; 1 2 ) |
| 13 | 1 2 3 6 12 | sralemOLD | |- ( ph -> ( dist ` W ) = ( dist ` A ) ) |