Theorem nn0mulcom

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

	Ref	Expression
Assertion	nn0mulcom	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		elnn0	⊢ ( 𝐵 ∈ ℕ0 ↔ ( 𝐵 ∈ ℕ ∨ 𝐵 = 0 ) )
2		elnn0	⊢ ( 𝐴 ∈ ℕ0 ↔ ( 𝐴 ∈ ℕ ∨ 𝐴 = 0 ) )
3		nnmulcom	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
4		nnre	⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℝ )
5		remul02	⊢ ( 𝐵 ∈ ℝ → ( 0 · 𝐵 ) = 0 )
6		remul01	⊢ ( 𝐵 ∈ ℝ → ( 𝐵 · 0 ) = 0 )
7	5 6	eqtr4d	⊢ ( 𝐵 ∈ ℝ → ( 0 · 𝐵 ) = ( 𝐵 · 0 ) )
8	4 7	syl	⊢ ( 𝐵 ∈ ℕ → ( 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 ∈ ℕ → ( 𝐴 = 0 → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) )
13	12	impcom	⊢ ( ( 𝐴 = 0 ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
14	3 13	jaoian	⊢ ( ( ( 𝐴 ∈ ℕ ∨ 𝐴 = 0 ) ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
15	2 14	sylanb	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
16		nn0re	⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ )
17		remul01	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 · 0 ) = 0 )
18		remul02	⊢ ( 𝐴 ∈ ℝ → ( 0 · 𝐴 ) = 0 )
19	17 18	eqtr4d	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 · 0 ) = ( 0 · 𝐴 ) )
20	16 19	syl	⊢ ( 𝐴 ∈ ℕ0 → ( 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 ∈ ℕ0 → ( 𝐵 = 0 → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) )
25	24	imp	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 = 0 ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
26	15 25	jaodan	⊢ ( ( 𝐴 ∈ ℕ0 ∧ ( 𝐵 ∈ ℕ ∨ 𝐵 = 0 ) ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
27	1 26	sylan2b	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )