Theorem colperp

Description: Deduce a perpendicularity from perpendicularity and colinearity. (Contributed by Thierry Arnoux, 8-Dec-2019)

	Ref	Expression
Hypotheses	colperpex.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	colperpex.d	⊢ − = ( dist ‘ 𝐺 )
	colperpex.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	colperpex.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	colperpex.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	colperp.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	colperp.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	colperp.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	colperp.1	⊢ ( 𝜑 → ( 𝐴 𝐿 𝐵 ) ( ⟂G ‘ 𝐺 ) 𝐷 )
	colperp.2	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) )
	colperp.3	⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )
Assertion	colperp	⊢ ( 𝜑 → ( 𝐴 𝐿 𝐶 ) ( ⟂G ‘ 𝐺 ) 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		colperpex.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		colperpex.d	⊢ − = ( dist ‘ 𝐺 )
3		colperpex.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		colperpex.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
5		colperpex.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
6		colperp.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
7		colperp.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
8		colperp.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
9		colperp.1	⊢ ( 𝜑 → ( 𝐴 𝐿 𝐵 ) ( ⟂G ‘ 𝐺 ) 𝐷 )
10		colperp.2	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) )
11		colperp.3	⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )
12	4 5 9	perpln1	⊢ ( 𝜑 → ( 𝐴 𝐿 𝐵 ) ∈ ran 𝐿 )
13	1 3 4 5 6 7 12	tglnne	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
14	1 3 4 5 6 7 13	tglinerflx1	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐴 𝐿 𝐵 ) )
15	13	neneqd	⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 )
16	10	orcomd	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ∨ 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ) )
17	16	ord	⊢ ( 𝜑 → ( ¬ 𝐴 = 𝐵 → 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ) )
18	15 17	mpd	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) )
19	1 3 4 5 6 8 11 11 12 14 18	tglinethru	⊢ ( 𝜑 → ( 𝐴 𝐿 𝐵 ) = ( 𝐴 𝐿 𝐶 ) )
20	19 9	eqbrtrrd	⊢ ( 𝜑 → ( 𝐴 𝐿 𝐶 ) ( ⟂G ‘ 𝐺 ) 𝐷 )