Theorem tgellng

Description: Property of lying on the line going through points X and Y . Definition 4.10 of Schwabhauser p. 36. We choose the notation Z e. ( X ( LineGG ) Y ) instead of "colinear" because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 28-Mar-2019)

	Ref	Expression
Hypotheses	tglngval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	tglngval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	tglngval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tglngval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tglngval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
	tglngval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
	tglngval.z	⊢ ( 𝜑 → 𝑋 ≠ 𝑌 )
	tgellng.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
Assertion	tgellng	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ↔ ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		tglngval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tglngval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
3		tglngval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		tglngval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tglngval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
6		tglngval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
7		tglngval.z	⊢ ( 𝜑 → 𝑋 ≠ 𝑌 )
8		tgellng.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
9	1 2 3 4 5 6 7	tglngval	⊢ ( 𝜑 → ( 𝑋 𝐿 𝑌 ) = { 𝑧 ∈ 𝑃 ∣ ( 𝑧 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑧 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ) } )
10	9	eleq2d	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ↔ 𝑍 ∈ { 𝑧 ∈ 𝑃 ∣ ( 𝑧 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑧 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ) } ) )
11		eleq1	⊢ ( 𝑧 = 𝑍 → ( 𝑧 ∈ ( 𝑋 𝐼 𝑌 ) ↔ 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ) )
12		oveq1	⊢ ( 𝑧 = 𝑍 → ( 𝑧 𝐼 𝑌 ) = ( 𝑍 𝐼 𝑌 ) )
13	12	eleq2d	⊢ ( 𝑧 = 𝑍 → ( 𝑋 ∈ ( 𝑧 𝐼 𝑌 ) ↔ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) )
14		oveq2	⊢ ( 𝑧 = 𝑍 → ( 𝑋 𝐼 𝑧 ) = ( 𝑋 𝐼 𝑍 ) )
15	14	eleq2d	⊢ ( 𝑧 = 𝑍 → ( 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ↔ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) )
16	11 13 15	3orbi123d	⊢ ( 𝑧 = 𝑍 → ( ( 𝑧 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑧 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ) ↔ ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ) )
17	16	elrab	⊢ ( 𝑍 ∈ { 𝑧 ∈ 𝑃 ∣ ( 𝑧 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑧 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑧 ) ) } ↔ ( 𝑍 ∈ 𝑃 ∧ ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ) )
18	10 17	bitrdi	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ↔ ( 𝑍 ∈ 𝑃 ∧ ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ) ) )
19	8 18	mpbirand	⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ↔ ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ) )