Description: A real greater than or equal to 1 raised to a nonnegative integer is greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | reexpcld.1 | |- ( ph -> A e. RR ) |
|
| reexpcld.2 | |- ( ph -> N e. NN0 ) |
||
| expge1d.3 | |- ( ph -> 1 <_ A ) |
||
| Assertion | expge1d | |- ( ph -> 1 <_ ( A ^ N ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reexpcld.1 | |- ( ph -> A e. RR ) |
|
| 2 | reexpcld.2 | |- ( ph -> N e. NN0 ) |
|
| 3 | expge1d.3 | |- ( ph -> 1 <_ A ) |
|
| 4 | expge1 | |- ( ( A e. RR /\ N e. NN0 /\ 1 <_ A ) -> 1 <_ ( A ^ N ) ) |
|
| 5 | 1 2 3 4 | syl3anc | |- ( ph -> 1 <_ ( A ^ N ) ) |