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Metamath Proof Explorer

Theorem nn0addcom

Description: Addition is commutative for nonnegative integers. Proven without ax-mulcom . (Contributed by SN, 1-Feb-2025)

Ref Expression
Assertion nn0addcom 
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( A + B ) = ( B + A ) )

Proof

Step Hyp Ref Expression
1 elnn0 
 |-  ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
2 elnn0 
 |-  ( A e. NN0 <-> ( A e. NN \/ A = 0 ) )
3 nnaddcom 
 |-  ( ( A e. NN /\ B e. NN ) -> ( A + B ) = ( B + A ) )
4 nnre 
 |-  ( B e. NN -> B e. RR )
5 readdlid 
 |-  ( B e. RR -> ( 0 + B ) = B )
6 readdrid 
 |-  ( B e. RR -> ( B + 0 ) = B )
7 5 6 eqtr4d 
 |-  ( B e. RR -> ( 0 + B ) = ( B + 0 ) )
8 4 7 syl 
 |-  ( B e. NN -> ( 0 + B ) = ( B + 0 ) )
9 oveq1 
 |-  ( A = 0 -> ( A + B ) = ( 0 + B ) )
10 oveq2 
 |-  ( A = 0 -> ( B + A ) = ( B + 0 ) )
11 9 10 eqeq12d 
 |-  ( A = 0 -> ( ( A + B ) = ( B + A ) <-> ( 0 + B ) = ( B + 0 ) ) )
12 8 11 syl5ibrcom 
 |-  ( B e. NN -> ( A = 0 -> ( A + B ) = ( B + A ) ) )
13 12 impcom 
 |-  ( ( A = 0 /\ B e. NN ) -> ( A + B ) = ( B + A ) )
14 3 13 jaoian 
 |-  ( ( ( A e. NN \/ A = 0 ) /\ B e. NN ) -> ( A + B ) = ( B + A ) )
15 2 14 sylanb 
 |-  ( ( A e. NN0 /\ B e. NN ) -> ( A + B ) = ( B + A ) )
16 nn0re 
 |-  ( A e. NN0 -> A e. RR )
17 readdrid 
 |-  ( A e. RR -> ( A + 0 ) = A )
18 readdlid 
 |-  ( A e. RR -> ( 0 + A ) = A )
19 17 18 eqtr4d 
 |-  ( A e. RR -> ( A + 0 ) = ( 0 + A ) )
20 16 19 syl 
 |-  ( A e. NN0 -> ( A + 0 ) = ( 0 + A ) )
21 oveq2 
 |-  ( B = 0 -> ( A + B ) = ( A + 0 ) )
22 oveq1 
 |-  ( B = 0 -> ( B + A ) = ( 0 + A ) )
23 21 22 eqeq12d 
 |-  ( B = 0 -> ( ( A + B ) = ( B + A ) <-> ( A + 0 ) = ( 0 + A ) ) )
24 20 23 syl5ibrcom 
 |-  ( A e. NN0 -> ( B = 0 -> ( A + B ) = ( B + A ) ) )
25 24 imp 
 |-  ( ( A e. NN0 /\ B = 0 ) -> ( A + B ) = ( B + A ) )
26 15 25 jaodan 
 |-  ( ( A e. NN0 /\ ( B e. NN \/ B = 0 ) ) -> ( A + B ) = ( B + A ) )
27 1 26 sylan2b 
 |-  ( ( A e. NN0 /\ B e. NN0 ) -> ( A + B ) = ( B + A ) )