Description: Double of a positive real is larger than the real itself. (Contributed by Glauco Siliprandi, 11-Dec-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | 2timesgt | |- ( A e. RR+ -> A < ( 2 x. A ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre | |- ( A e. RR+ -> A e. RR ) |
|
2 | id | |- ( A e. RR+ -> A e. RR+ ) |
|
3 | 1 2 | ltaddrp2d | |- ( A e. RR+ -> A < ( A + A ) ) |
4 | rpcn | |- ( A e. RR+ -> A e. CC ) |
|
5 | 2times | |- ( A e. CC -> ( 2 x. A ) = ( A + A ) ) |
|
6 | 5 | eqcomd | |- ( A e. CC -> ( A + A ) = ( 2 x. A ) ) |
7 | 4 6 | syl | |- ( A e. RR+ -> ( A + A ) = ( 2 x. A ) ) |
8 | 3 7 | breqtrd | |- ( A e. RR+ -> A < ( 2 x. A ) ) |