Theorem srads

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

	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		df-ds	⊢ dist = Slot ; 1 2
4		1nn0	⊢ 1 ∈ ℕ0
5		2nn	⊢ 2 ∈ ℕ
6	4 5	decnncl	⊢ ; 1 2 ∈ ℕ
7		1nn	⊢ 1 ∈ ℕ
8		2nn0	⊢ 2 ∈ ℕ0
9		8nn0	⊢ 8 ∈ ℕ0
10		8lt10	⊢ 8 < ; 1 0
11	7 8 9 10	declti	⊢ 8 < ; 1 2
12	11	olci	⊢ ( ; 1 2 < 5 ∨ 8 < ; 1 2 )
13	1 2 3 6 12	sralem	⊢ ( 𝜑 → ( dist ‘ 𝑊 ) = ( dist ‘ 𝐴 ) )