Metamath Proof Explorer


Theorem nn0mulcom

Description: Multiplication is commutative for nonnegative integers. Proven without ax-mulcom . (Contributed by SN, 25-Jan-2025)

Ref Expression
Assertion nn0mulcom
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( A x. B ) = ( B x. 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 nnmulcom
 |-  ( ( A e. NN /\ B e. NN ) -> ( A x. B ) = ( B x. A ) )
4 nnre
 |-  ( B e. NN -> B e. RR )
5 remul02
 |-  ( B e. RR -> ( 0 x. B ) = 0 )
6 remul01
 |-  ( B e. RR -> ( B x. 0 ) = 0 )
7 5 6 eqtr4d
 |-  ( B e. RR -> ( 0 x. B ) = ( B x. 0 ) )
8 4 7 syl
 |-  ( B e. NN -> ( 0 x. B ) = ( B x. 0 ) )
9 oveq1
 |-  ( A = 0 -> ( A x. B ) = ( 0 x. B ) )
10 oveq2
 |-  ( A = 0 -> ( B x. A ) = ( B x. 0 ) )
11 9 10 eqeq12d
 |-  ( A = 0 -> ( ( A x. B ) = ( B x. A ) <-> ( 0 x. B ) = ( B x. 0 ) ) )
12 8 11 syl5ibrcom
 |-  ( B e. NN -> ( A = 0 -> ( A x. B ) = ( B x. A ) ) )
13 12 impcom
 |-  ( ( A = 0 /\ B e. NN ) -> ( A x. B ) = ( B x. A ) )
14 3 13 jaoian
 |-  ( ( ( A e. NN \/ A = 0 ) /\ B e. NN ) -> ( A x. B ) = ( B x. A ) )
15 2 14 sylanb
 |-  ( ( A e. NN0 /\ B e. NN ) -> ( A x. B ) = ( B x. A ) )
16 nn0re
 |-  ( A e. NN0 -> A e. RR )
17 remul01
 |-  ( A e. RR -> ( A x. 0 ) = 0 )
18 remul02
 |-  ( A e. RR -> ( 0 x. A ) = 0 )
19 17 18 eqtr4d
 |-  ( A e. RR -> ( A x. 0 ) = ( 0 x. A ) )
20 16 19 syl
 |-  ( A e. NN0 -> ( A x. 0 ) = ( 0 x. A ) )
21 oveq2
 |-  ( B = 0 -> ( A x. B ) = ( A x. 0 ) )
22 oveq1
 |-  ( B = 0 -> ( B x. A ) = ( 0 x. A ) )
23 21 22 eqeq12d
 |-  ( B = 0 -> ( ( A x. B ) = ( B x. A ) <-> ( A x. 0 ) = ( 0 x. A ) ) )
24 20 23 syl5ibrcom
 |-  ( A e. NN0 -> ( B = 0 -> ( A x. B ) = ( B x. A ) ) )
25 24 imp
 |-  ( ( A e. NN0 /\ B = 0 ) -> ( A x. B ) = ( B x. A ) )
26 15 25 jaodan
 |-  ( ( A e. NN0 /\ ( B e. NN \/ B = 0 ) ) -> ( A x. B ) = ( B x. A ) )
27 1 26 sylan2b
 |-  ( ( A e. NN0 /\ B e. NN0 ) -> ( A x. B ) = ( B x. A ) )