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 ) )