Metamath Proof Explorer


Theorem om0x

Description: Ordinal multiplication with zero. Definition 8.15 of TakeutiZaring p. 62. Unlike om0 , this version works whether or not A is an ordinal. However, since it is an artifact of our particular function value definition outside the domain, we will not use it in order to be conventional and present it only as a curiosity. (Contributed by NM, 1-Feb-1996) (New usage is discouraged.)

Ref Expression
Assertion om0x ( 𝐴 ·o ∅ ) = ∅

Proof

Step Hyp Ref Expression
1 om0 ( 𝐴 ∈ On → ( 𝐴 ·o ∅ ) = ∅ )
2 1 adantr ( ( 𝐴 ∈ On ∧ ∅ ∈ On ) → ( 𝐴 ·o ∅ ) = ∅ )
3 fnom ·o Fn ( On × On )
4 3 fndmi dom ·o = ( On × On )
5 4 ndmov ( ¬ ( 𝐴 ∈ On ∧ ∅ ∈ On ) → ( 𝐴 ·o ∅ ) = ∅ )
6 2 5 pm2.61i ( 𝐴 ·o ∅ ) = ∅