This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem metreslem

Description: Lemma for metres . (Contributed by Mario Carneiro, 24-Aug-2015)

Ref Expression
Assertion metreslem ( dom 𝐷 = ( 𝑋 × 𝑋 ) → ( 𝐷 ↾ ( 𝑅 × 𝑅 ) ) = ( 𝐷 ↾ ( ( 𝑋 ∩ 𝑅 ) × ( 𝑋 ∩ 𝑅 ) ) ) )

Proof

Step Hyp Ref Expression
1 resdmres ⊢ ( 𝐷 ↾ dom ( 𝐷 ↾ ( 𝑅 × 𝑅 ) ) ) = ( 𝐷 ↾ ( 𝑅 × 𝑅 ) )
2 ineq2 ⊢ ( dom 𝐷 = ( 𝑋 × 𝑋 ) → ( ( 𝑅 × 𝑅 ) ∩ dom 𝐷 ) = ( ( 𝑅 × 𝑅 ) ∩ ( 𝑋 × 𝑋 ) ) )
3 dmres ⊢ dom ( 𝐷 ↾ ( 𝑅 × 𝑅 ) ) = ( ( 𝑅 × 𝑅 ) ∩ dom 𝐷 )
4 inxp ⊢ ( ( 𝑋 × 𝑋 ) ∩ ( 𝑅 × 𝑅 ) ) = ( ( 𝑋 ∩ 𝑅 ) × ( 𝑋 ∩ 𝑅 ) )
5 incom ⊢ ( ( 𝑋 × 𝑋 ) ∩ ( 𝑅 × 𝑅 ) ) = ( ( 𝑅 × 𝑅 ) ∩ ( 𝑋 × 𝑋 ) )
6 4 5 eqtr3i ⊢ ( ( 𝑋 ∩ 𝑅 ) × ( 𝑋 ∩ 𝑅 ) ) = ( ( 𝑅 × 𝑅 ) ∩ ( 𝑋 × 𝑋 ) )
7 2 3 6 3eqtr4g ⊢ ( dom 𝐷 = ( 𝑋 × 𝑋 ) → dom ( 𝐷 ↾ ( 𝑅 × 𝑅 ) ) = ( ( 𝑋 ∩ 𝑅 ) × ( 𝑋 ∩ 𝑅 ) ) )
8 7 reseq2d ⊢ ( dom 𝐷 = ( 𝑋 × 𝑋 ) → ( 𝐷 ↾ dom ( 𝐷 ↾ ( 𝑅 × 𝑅 ) ) ) = ( 𝐷 ↾ ( ( 𝑋 ∩ 𝑅 ) × ( 𝑋 ∩ 𝑅 ) ) ) )
9 1 8 eqtr3id ⊢ ( dom 𝐷 = ( 𝑋 × 𝑋 ) → ( 𝐷 ↾ ( 𝑅 × 𝑅 ) ) = ( 𝐷 ↾ ( ( 𝑋 ∩ 𝑅 ) × ( 𝑋 ∩ 𝑅 ) ) ) )