This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem numsucc

Description: The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014)

Ref Expression
Hypotheses numsucc.1 
|- Y e. NN0
numsucc.2 
|- T = ( Y + 1 )
numsucc.3 
|- A e. NN0
numsucc.4 
|- ( A + 1 ) = B
numsucc.5 
|- N = ( ( T x. A ) + Y )
Assertion numsucc 
|- ( N + 1 ) = ( ( T x. B ) + 0 )

Proof

Step Hyp Ref Expression
1 numsucc.1 
 |-  Y e. NN0
2 numsucc.2 
 |-  T = ( Y + 1 )
3 numsucc.3 
 |-  A e. NN0
4 numsucc.4 
 |-  ( A + 1 ) = B
5 numsucc.5 
 |-  N = ( ( T x. A ) + Y )
6 1nn0 
 |-  1 e. NN0
7 1 6 nn0addcli 
 |-  ( Y + 1 ) e. NN0
8 2 7 eqeltri 
 |-  T e. NN0
9 8 nn0cni 
 |-  T e. CC
10 9 mulridi 
 |-  ( T x. 1 ) = T
11 10 oveq2i 
 |-  ( ( T x. A ) + ( T x. 1 ) ) = ( ( T x. A ) + T )
12 3 nn0cni 
 |-  A e. CC
13 ax-1cn 
 |-  1 e. CC
14 9 12 13 adddii 
 |-  ( T x. ( A + 1 ) ) = ( ( T x. A ) + ( T x. 1 ) )
15 2 eqcomi 
 |-  ( Y + 1 ) = T
16 8 3 1 15 5 numsuc 
 |-  ( N + 1 ) = ( ( T x. A ) + T )
17 11 14 16 3eqtr4ri 
 |-  ( N + 1 ) = ( T x. ( A + 1 ) )
18 4 oveq2i 
 |-  ( T x. ( A + 1 ) ) = ( T x. B )
19 3 6 nn0addcli 
 |-  ( A + 1 ) e. NN0
20 4 19 eqeltrri 
 |-  B e. NN0
21 8 20 num0u 
 |-  ( T x. B ) = ( ( T x. B ) + 0 )
22 17 18 21 3eqtri 
 |-  ( N + 1 ) = ( ( T x. B ) + 0 )