Metamath Proof Explorer

Theorem 9t5e45

Description: 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015)

Ref Expression
Assertion 9t5e45 
|- ( 9 x. 5 ) = ; 4 5

Proof

Step Hyp Ref Expression
1 9nn0 
 |-  9 e. NN0
2 4nn0 
 |-  4 e. NN0
3 df-5 
 |-  5 = ( 4 + 1 )
4 9t4e36 
 |-  ( 9 x. 4 ) = ; 3 6
5 3nn0 
 |-  3 e. NN0
6 6nn0 
 |-  6 e. NN0
7 eqid 
 |-  ; 3 6 = ; 3 6
8 3p1e4 
 |-  ( 3 + 1 ) = 4
9 5nn0 
 |-  5 e. NN0
10 1 nn0cni 
 |-  9 e. CC
11 6 nn0cni 
 |-  6 e. CC
12 9p6e15 
 |-  ( 9 + 6 ) = ; 1 5
13 10 11 12 addcomli 
 |-  ( 6 + 9 ) = ; 1 5
14 5 6 1 7 8 9 13 decaddci 
 |-  ( ; 3 6 + 9 ) = ; 4 5
15 1 2 3 4 14 4t3lem 
 |-  ( 9 x. 5 ) = ; 4 5