Theorem infxpabs

Description: Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004) (Revised by Mario Carneiro, 29-Apr-2015)

	Ref	Expression
Assertion	infxpabs	⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴 ) ) → ( 𝐴 × 𝐵 ) ≈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		infxpdom	⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ) → ( 𝐴 × 𝐵 ) ≼ 𝐴 )
2	1	3expa	⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝐵 ≼ 𝐴 ) → ( 𝐴 × 𝐵 ) ≼ 𝐴 )
3	2	adantrl	⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴 ) ) → ( 𝐴 × 𝐵 ) ≼ 𝐴 )
4		simpll	⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴 ) ) → 𝐴 ∈ dom card )
5		numdom	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴 ) → 𝐵 ∈ dom card )
6	5	ad2ant2rl	⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴 ) ) → 𝐵 ∈ dom card )
7		simprl	⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴 ) ) → 𝐵 ≠ ∅ )
8		xpdom3	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ 𝐵 ≠ ∅ ) → 𝐴 ≼ ( 𝐴 × 𝐵 ) )
9	4 6 7 8	syl3anc	⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴 ) ) → 𝐴 ≼ ( 𝐴 × 𝐵 ) )
10		sbth	⊢ ( ( ( 𝐴 × 𝐵 ) ≼ 𝐴 ∧ 𝐴 ≼ ( 𝐴 × 𝐵 ) ) → ( 𝐴 × 𝐵 ) ≈ 𝐴 )
11	3 9 10	syl2anc	⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴 ) ) → ( 𝐴 × 𝐵 ) ≈ 𝐴 )