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Metamath Proof Explorer

Theorem decaddc2

Description: Add two numerals M and N (with carry). (Contributed by Mario Carneiro, 18-Feb-2014) (Revised by AV, 6-Sep-2021)

Ref Expression
Hypotheses decma.a 
|- A e. NN0
decma.b 
|- B e. NN0
decma.c 
|- C e. NN0
decma.d 
|- D e. NN0
decma.m 
|- M = ; A B
decma.n 
|- N = ; C D
decaddc.e 
|- ( ( A + C ) + 1 ) = E
decaddc2.t 
|- ( B + D ) = ; 1 0
Assertion decaddc2 
|- ( M + N ) = ; E 0

Proof

Step Hyp Ref Expression
1 decma.a 
 |-  A e. NN0
2 decma.b 
 |-  B e. NN0
3 decma.c 
 |-  C e. NN0
4 decma.d 
 |-  D e. NN0
5 decma.m 
 |-  M = ; A B
6 decma.n 
 |-  N = ; C D
7 decaddc.e 
 |-  ( ( A + C ) + 1 ) = E
8 decaddc2.t 
 |-  ( B + D ) = ; 1 0
9 0nn0 
 |-  0 e. NN0
10 1 2 3 4 5 6 7 9 8 decaddc 
 |-  ( M + N ) = ; E 0