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Metamath Proof Explorer

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 )