Metamath Proof Explorer


Theorem rpexpcl

Description: Closure law for integer exponentiation of positive reals. (Contributed by NM, 24-Feb-2008) (Revised by Mario Carneiro, 9-Sep-2014)

Ref Expression
Assertion rpexpcl A+NAN+

Proof

Step Hyp Ref Expression
1 simpl A+NA+
2 rpne0 A+A0
3 2 adantr A+NA0
4 simpr A+NN
5 rpssre +
6 ax-resscn
7 5 6 sstri +
8 rpmulcl x+y+xy+
9 1rp 1+
10 rpreccl x+1x+
11 10 adantr x+x01x+
12 7 8 9 11 expcl2lem A+A0NAN+
13 1 3 4 12 syl3anc A+NAN+