Theorem srads

Description: Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015) (Revised by Thierry Arnoux, 16-Jun-2019) (Revised by AV, 29-Oct-2024)

	Ref	Expression
Hypotheses	srapart.a	⊢ ( 𝜑 → 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
	srapart.s	⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝑊 ) )
Assertion	srads	⊢ ( 𝜑 → ( dist ‘ 𝑊 ) = ( dist ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		srapart.a	⊢ ( 𝜑 → 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
2		srapart.s	⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝑊 ) )
3		dsid	⊢ dist = Slot ( dist ‘ ndx )
4		slotsdnscsi	⊢ ( ( dist ‘ ndx ) ≠ ( Scalar ‘ ndx ) ∧ ( dist ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( dist ‘ ndx ) ≠ ( ·𝑖 ‘ ndx ) )
5	4	simp1i	⊢ ( dist ‘ ndx ) ≠ ( Scalar ‘ ndx )
6	5	necomi	⊢ ( Scalar ‘ ndx ) ≠ ( dist ‘ ndx )
7	4	simp2i	⊢ ( dist ‘ ndx ) ≠ ( ·𝑠 ‘ ndx )
8	7	necomi	⊢ ( ·𝑠 ‘ ndx ) ≠ ( dist ‘ ndx )
9	4	simp3i	⊢ ( dist ‘ ndx ) ≠ ( ·𝑖 ‘ ndx )
10	9	necomi	⊢ ( ·𝑖 ‘ ndx ) ≠ ( dist ‘ ndx )
11	1 2 3 6 8 10	sralem	⊢ ( 𝜑 → ( dist ‘ 𝑊 ) = ( dist ‘ 𝐴 ) )