Theorem nnmulcom

Description: Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024)

	Ref	Expression
Assertion	nnmulcom	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝑥 = 1 → ( 𝑥 · 𝐵 ) = ( 1 · 𝐵 ) )
2		oveq2	⊢ ( 𝑥 = 1 → ( 𝐵 · 𝑥 ) = ( 𝐵 · 1 ) )
3	1 2	eqeq12d	⊢ ( 𝑥 = 1 → ( ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ↔ ( 1 · 𝐵 ) = ( 𝐵 · 1 ) ) )
4	3	imbi2d	⊢ ( 𝑥 = 1 → ( ( 𝐵 ∈ ℕ → ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ) ↔ ( 𝐵 ∈ ℕ → ( 1 · 𝐵 ) = ( 𝐵 · 1 ) ) ) )
5		oveq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 · 𝐵 ) = ( 𝑦 · 𝐵 ) )
6		oveq2	⊢ ( 𝑥 = 𝑦 → ( 𝐵 · 𝑥 ) = ( 𝐵 · 𝑦 ) )
7	5 6	eqeq12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ↔ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) )
8	7	imbi2d	⊢ ( 𝑥 = 𝑦 → ( ( 𝐵 ∈ ℕ → ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ) ↔ ( 𝐵 ∈ ℕ → ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) ) )
9		oveq1	⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑥 · 𝐵 ) = ( ( 𝑦 + 1 ) · 𝐵 ) )
10		oveq2	⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝐵 · 𝑥 ) = ( 𝐵 · ( 𝑦 + 1 ) ) )
11	9 10	eqeq12d	⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ↔ ( ( 𝑦 + 1 ) · 𝐵 ) = ( 𝐵 · ( 𝑦 + 1 ) ) ) )
12	11	imbi2d	⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝐵 ∈ ℕ → ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ) ↔ ( 𝐵 ∈ ℕ → ( ( 𝑦 + 1 ) · 𝐵 ) = ( 𝐵 · ( 𝑦 + 1 ) ) ) ) )
13		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 · 𝐵 ) = ( 𝐴 · 𝐵 ) )
14		oveq2	⊢ ( 𝑥 = 𝐴 → ( 𝐵 · 𝑥 ) = ( 𝐵 · 𝐴 ) )
15	13 14	eqeq12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ↔ ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) )
16	15	imbi2d	⊢ ( 𝑥 = 𝐴 → ( ( 𝐵 ∈ ℕ → ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ) ↔ ( 𝐵 ∈ ℕ → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) ) )
17		nnmul1com	⊢ ( 𝐵 ∈ ℕ → ( 1 · 𝐵 ) = ( 𝐵 · 1 ) )
18		simp3	⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) )
19	17	3ad2ant2	⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( 1 · 𝐵 ) = ( 𝐵 · 1 ) )
20	18 19	oveq12d	⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( ( 𝑦 · 𝐵 ) + ( 1 · 𝐵 ) ) = ( ( 𝐵 · 𝑦 ) + ( 𝐵 · 1 ) ) )
21		simp1	⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → 𝑦 ∈ ℕ )
22		1nn	⊢ 1 ∈ ℕ
23	22	a1i	⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → 1 ∈ ℕ )
24		simp2	⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → 𝐵 ∈ ℕ )
25		nnadddir	⊢ ( ( 𝑦 ∈ ℕ ∧ 1 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝑦 + 1 ) · 𝐵 ) = ( ( 𝑦 · 𝐵 ) + ( 1 · 𝐵 ) ) )
26	21 23 24 25	syl3anc	⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( ( 𝑦 + 1 ) · 𝐵 ) = ( ( 𝑦 · 𝐵 ) + ( 1 · 𝐵 ) ) )
27	24	nncnd	⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → 𝐵 ∈ ℂ )
28	21	nncnd	⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → 𝑦 ∈ ℂ )
29		1cnd	⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → 1 ∈ ℂ )
30	27 28 29	adddid	⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( 𝐵 · ( 𝑦 + 1 ) ) = ( ( 𝐵 · 𝑦 ) + ( 𝐵 · 1 ) ) )
31	20 26 30	3eqtr4d	⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( ( 𝑦 + 1 ) · 𝐵 ) = ( 𝐵 · ( 𝑦 + 1 ) ) )
32	31	3exp	⊢ ( 𝑦 ∈ ℕ → ( 𝐵 ∈ ℕ → ( ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) → ( ( 𝑦 + 1 ) · 𝐵 ) = ( 𝐵 · ( 𝑦 + 1 ) ) ) ) )
33	32	a2d	⊢ ( 𝑦 ∈ ℕ → ( ( 𝐵 ∈ ℕ → ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( 𝐵 ∈ ℕ → ( ( 𝑦 + 1 ) · 𝐵 ) = ( 𝐵 · ( 𝑦 + 1 ) ) ) ) )
34	4 8 12 16 17 33	nnind	⊢ ( 𝐴 ∈ ℕ → ( 𝐵 ∈ ℕ → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) )
35	34	imp	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )