Metamath Proof Explorer


Theorem cxple3d

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses rpcxpcld.1 φA+
rpcxpcld.2 φB
cxplt3d.3 φA<1
cxplt3d.4 φC
Assertion cxple3d φBCACAB

Proof

Step Hyp Ref Expression
1 rpcxpcld.1 φA+
2 rpcxpcld.2 φB
3 cxplt3d.3 φA<1
4 cxplt3d.4 φC
5 cxple3 A+A<1BCBCACAB
6 1 3 2 4 5 syl22anc φBCACAB