Theorem infxpdom

Description: Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006) (Revised by Mario Carneiro, 29-Apr-2015)

	Ref	Expression
Assertion	infxpdom	⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ) → ( 𝐴 × 𝐵 ) ≼ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		xpdom2g	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴 ) → ( 𝐴 × 𝐵 ) ≼ ( 𝐴 × 𝐴 ) )
2	1	3adant2	⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ) → ( 𝐴 × 𝐵 ) ≼ ( 𝐴 × 𝐴 ) )
3		infxpidm2	⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝐴 × 𝐴 ) ≈ 𝐴 )
4	3	3adant3	⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ) → ( 𝐴 × 𝐴 ) ≈ 𝐴 )
5		domentr	⊢ ( ( ( 𝐴 × 𝐵 ) ≼ ( 𝐴 × 𝐴 ) ∧ ( 𝐴 × 𝐴 ) ≈ 𝐴 ) → ( 𝐴 × 𝐵 ) ≼ 𝐴 )
6	2 4 5	syl2anc	⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ) → ( 𝐴 × 𝐵 ) ≼ 𝐴 )