Metamath Proof Explorer

Theorem hashnnm

Description: The # function on _om preserves multiplication. (Contributed by Eric Schmidt, 7-Jul-2026)

Ref Expression
Assertion hashnnm ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ♯ ‘ ( 𝐴 ·o 𝐵 ) ) = ( ( ♯ ‘ 𝐴 ) · ( ♯ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 nnon ⊢ ( 𝐴 ∈ ω → 𝐴 ∈ On )
2 nnon ⊢ ( 𝐵 ∈ ω → 𝐵 ∈ On )
3 omxpen ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ·o 𝐵 ) ≈ ( 𝐴 × 𝐵 ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ·o 𝐵 ) ≈ ( 𝐴 × 𝐵 ) )
5 hasheni ⊢ ( ( 𝐴 ·o 𝐵 ) ≈ ( 𝐴 × 𝐵 ) → ( ♯ ‘ ( 𝐴 ·o 𝐵 ) ) = ( ♯ ‘ ( 𝐴 × 𝐵 ) ) )
6 4 5 syl ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ♯ ‘ ( 𝐴 ·o 𝐵 ) ) = ( ♯ ‘ ( 𝐴 × 𝐵 ) ) )
7 nnfi ⊢ ( 𝐴 ∈ ω → 𝐴 ∈ Fin )
8 nnfi ⊢ ( 𝐵 ∈ ω → 𝐵 ∈ Fin )
9 hashxp ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ♯ ‘ ( 𝐴 × 𝐵 ) ) = ( ( ♯ ‘ 𝐴 ) · ( ♯ ‘ 𝐵 ) ) )
10 7 8 9 syl2an ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ♯ ‘ ( 𝐴 × 𝐵 ) ) = ( ( ♯ ‘ 𝐴 ) · ( ♯ ‘ 𝐵 ) ) )
11 6 10 eqtrd ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ♯ ‘ ( 𝐴 ·o 𝐵 ) ) = ( ( ♯ ‘ 𝐴 ) · ( ♯ ‘ 𝐵 ) ) )