Metamath Proof Explorer


Theorem tgelrnln

Description: The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019)

Ref Expression
Hypotheses tglineelsb2.p 𝐵 = ( Base ‘ 𝐺 )
tglineelsb2.i 𝐼 = ( Itv ‘ 𝐺 )
tglineelsb2.l 𝐿 = ( LineG ‘ 𝐺 )
tglineelsb2.g ( 𝜑𝐺 ∈ TarskiG )
tgelrnln.x ( 𝜑𝑋𝐵 )
tgelrnln.y ( 𝜑𝑌𝐵 )
tgelrnln.d ( 𝜑𝑋𝑌 )
Assertion tgelrnln ( 𝜑 → ( 𝑋 𝐿 𝑌 ) ∈ ran 𝐿 )

Proof

Step Hyp Ref Expression
1 tglineelsb2.p 𝐵 = ( Base ‘ 𝐺 )
2 tglineelsb2.i 𝐼 = ( Itv ‘ 𝐺 )
3 tglineelsb2.l 𝐿 = ( LineG ‘ 𝐺 )
4 tglineelsb2.g ( 𝜑𝐺 ∈ TarskiG )
5 tgelrnln.x ( 𝜑𝑋𝐵 )
6 tgelrnln.y ( 𝜑𝑌𝐵 )
7 tgelrnln.d ( 𝜑𝑋𝑌 )
8 df-ov ( 𝑋 𝐿 𝑌 ) = ( 𝐿 ‘ ⟨ 𝑋 , 𝑌 ⟩ )
9 1 3 2 tglnfn ( 𝐺 ∈ TarskiG → 𝐿 Fn ( ( 𝐵 × 𝐵 ) ∖ I ) )
10 4 9 syl ( 𝜑𝐿 Fn ( ( 𝐵 × 𝐵 ) ∖ I ) )
11 5 6 opelxpd ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐵 × 𝐵 ) )
12 df-br ( 𝑋 I 𝑌 ↔ ⟨ 𝑋 , 𝑌 ⟩ ∈ I )
13 ideqg ( 𝑌𝐵 → ( 𝑋 I 𝑌𝑋 = 𝑌 ) )
14 12 13 bitr3id ( 𝑌𝐵 → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ I ↔ 𝑋 = 𝑌 ) )
15 14 necon3bbid ( 𝑌𝐵 → ( ¬ ⟨ 𝑋 , 𝑌 ⟩ ∈ I ↔ 𝑋𝑌 ) )
16 15 biimpar ( ( 𝑌𝐵𝑋𝑌 ) → ¬ ⟨ 𝑋 , 𝑌 ⟩ ∈ I )
17 6 7 16 syl2anc ( 𝜑 → ¬ ⟨ 𝑋 , 𝑌 ⟩ ∈ I )
18 11 17 eldifd ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ ( ( 𝐵 × 𝐵 ) ∖ I ) )
19 fnfvelrn ( ( 𝐿 Fn ( ( 𝐵 × 𝐵 ) ∖ I ) ∧ ⟨ 𝑋 , 𝑌 ⟩ ∈ ( ( 𝐵 × 𝐵 ) ∖ I ) ) → ( 𝐿 ‘ ⟨ 𝑋 , 𝑌 ⟩ ) ∈ ran 𝐿 )
20 10 18 19 syl2anc ( 𝜑 → ( 𝐿 ‘ ⟨ 𝑋 , 𝑌 ⟩ ) ∈ ran 𝐿 )
21 8 20 eqeltrid ( 𝜑 → ( 𝑋 𝐿 𝑌 ) ∈ ran 𝐿 )