Theorem trkgstr

Description: Functionality of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017)

	Ref	Expression
Hypothesis	trkgstr.w	⊢ 𝑊 = { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ , ⟨ ( Itv ‘ ndx ) , 𝐼 ⟩ }
Assertion	trkgstr	⊢ 𝑊 Struct ⟨ 1 , ; 1 6 ⟩

Proof

Step	Hyp	Ref	Expression
1		trkgstr.w	⊢ 𝑊 = { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ , ⟨ ( Itv ‘ ndx ) , 𝐼 ⟩ }
2		1nn	⊢ 1 ∈ ℕ
3		basendx	⊢ ( Base ‘ ndx ) = 1
4		2nn0	⊢ 2 ∈ ℕ0
5		1nn0	⊢ 1 ∈ ℕ0
6		1lt10	⊢ 1 < ; 1 0
7	2 4 5 6	declti	⊢ 1 < ; 1 2
8		2nn	⊢ 2 ∈ ℕ
9	5 8	decnncl	⊢ ; 1 2 ∈ ℕ
10		dsndx	⊢ ( dist ‘ ndx ) = ; 1 2
11		6nn	⊢ 6 ∈ ℕ
12		2lt6	⊢ 2 < 6
13	5 4 11 12	declt	⊢ ; 1 2 < ; 1 6
14	5 11	decnncl	⊢ ; 1 6 ∈ ℕ
15		itvndx	⊢ ( Itv ‘ ndx ) = ; 1 6
16	2 3 7 9 10 13 14 15	strle3	⊢ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ , ⟨ ( Itv ‘ ndx ) , 𝐼 ⟩ } Struct ⟨ 1 , ; 1 6 ⟩
17	1 16	eqbrtri	⊢ 𝑊 Struct ⟨ 1 , ; 1 6 ⟩