Theorem ragncol

Description: Right angle implies non-colinearity. A consequence of theorem 8.9 of Schwabhauser p. 58. (Contributed by Thierry Arnoux, 1-Dec-2019)

	Ref	Expression
Hypotheses	israg.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	israg.d	⊢ − = ( dist ‘ 𝐺 )
	israg.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	israg.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	israg.s	⊢ 𝑆 = ( pInvG ‘ 𝐺 )
	israg.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	israg.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	israg.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	israg.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	ragncol.1	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )
	ragncol.2	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
	ragncol.3	⊢ ( 𝜑 → 𝐶 ≠ 𝐵 )
Assertion	ragncol	⊢ ( 𝜑 → ¬ ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		israg.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		israg.d	⊢ − = ( dist ‘ 𝐺 )
3		israg.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		israg.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
5		israg.s	⊢ 𝑆 = ( pInvG ‘ 𝐺 )
6		israg.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
7		israg.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
8		israg.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
9		israg.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
10		ragncol.1	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )
11		ragncol.2	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
12		ragncol.3	⊢ ( 𝜑 → 𝐶 ≠ 𝐵 )
13	11	neneqd	⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 )
14	12	neneqd	⊢ ( 𝜑 → ¬ 𝐶 = 𝐵 )
15		ioran	⊢ ( ¬ ( 𝐴 = 𝐵 ∨ 𝐶 = 𝐵 ) ↔ ( ¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐵 ) )
16	13 14 15	sylanbrc	⊢ ( 𝜑 → ¬ ( 𝐴 = 𝐵 ∨ 𝐶 = 𝐵 ) )
17	6	adantr	⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) ) → 𝐺 ∈ TarskiG )
18	7	adantr	⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) ) → 𝐴 ∈ 𝑃 )
19	8	adantr	⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) ) → 𝐵 ∈ 𝑃 )
20	9	adantr	⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) ) → 𝐶 ∈ 𝑃 )
21	10	adantr	⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) ) → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )
22		simpr	⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) ) → ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) )
23	1 2 3 4 5 17 18 19 20 21 22	ragflat3	⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) ) → ( 𝐴 = 𝐵 ∨ 𝐶 = 𝐵 ) )
24	16 23	mtand	⊢ ( 𝜑 → ¬ ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) )