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Description: The degree of the zero polynomial is zero. Note: this differs from some other definitions of the degree of the zero polynomial, such as -u 1 , -oo or undefined. But it is convenient for us to define it this way, so that we have dgrcl , dgreq0 and coeid without having to special-case zero, although plydivalg is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | dgr0 | |- ( deg ` 0p ) = 0 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-0p | |- 0p = ( CC X. { 0 } ) |
|
| 2 | 1 | fveq2i | |- ( deg ` 0p ) = ( deg ` ( CC X. { 0 } ) ) |
| 3 | 0cn | |- 0 e. CC |
|
| 4 | 0dgr | |- ( 0 e. CC -> ( deg ` ( CC X. { 0 } ) ) = 0 ) |
|
| 5 | 3 4 | ax-mp | |- ( deg ` ( CC X. { 0 } ) ) = 0 |
| 6 | 2 5 | eqtri | |- ( deg ` 0p ) = 0 |