Metamath Proof Explorer


Theorem hpgne1

Description: Points on the open half plane cannot lie on its border. (Contributed by Thierry Arnoux, 4-Mar-2020)

Ref Expression
Hypotheses ishpg.p 𝑃 = ( Base ‘ 𝐺 )
ishpg.i 𝐼 = ( Itv ‘ 𝐺 )
ishpg.l 𝐿 = ( LineG ‘ 𝐺 )
ishpg.o 𝑂 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃𝐷 ) ∧ 𝑏 ∈ ( 𝑃𝐷 ) ) ∧ ∃ 𝑡𝐷 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) }
ishpg.g ( 𝜑𝐺 ∈ TarskiG )
ishpg.d ( 𝜑𝐷 ∈ ran 𝐿 )
hpgbr.a ( 𝜑𝐴𝑃 )
hpgbr.b ( 𝜑𝐵𝑃 )
hpgne1.1 ( 𝜑𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 )
Assertion hpgne1 ( 𝜑 → ¬ 𝐴𝐷 )

Proof

Step Hyp Ref Expression
1 ishpg.p 𝑃 = ( Base ‘ 𝐺 )
2 ishpg.i 𝐼 = ( Itv ‘ 𝐺 )
3 ishpg.l 𝐿 = ( LineG ‘ 𝐺 )
4 ishpg.o 𝑂 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃𝐷 ) ∧ 𝑏 ∈ ( 𝑃𝐷 ) ) ∧ ∃ 𝑡𝐷 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) }
5 ishpg.g ( 𝜑𝐺 ∈ TarskiG )
6 ishpg.d ( 𝜑𝐷 ∈ ran 𝐿 )
7 hpgbr.a ( 𝜑𝐴𝑃 )
8 hpgbr.b ( 𝜑𝐵𝑃 )
9 hpgne1.1 ( 𝜑𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 )
10 eqid ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
11 6 ad2antrr ( ( ( 𝜑𝑐𝑃 ) ∧ ( 𝐴 𝑂 𝑐𝐵 𝑂 𝑐 ) ) → 𝐷 ∈ ran 𝐿 )
12 5 ad2antrr ( ( ( 𝜑𝑐𝑃 ) ∧ ( 𝐴 𝑂 𝑐𝐵 𝑂 𝑐 ) ) → 𝐺 ∈ TarskiG )
13 7 ad2antrr ( ( ( 𝜑𝑐𝑃 ) ∧ ( 𝐴 𝑂 𝑐𝐵 𝑂 𝑐 ) ) → 𝐴𝑃 )
14 simplr ( ( ( 𝜑𝑐𝑃 ) ∧ ( 𝐴 𝑂 𝑐𝐵 𝑂 𝑐 ) ) → 𝑐𝑃 )
15 simprl ( ( ( 𝜑𝑐𝑃 ) ∧ ( 𝐴 𝑂 𝑐𝐵 𝑂 𝑐 ) ) → 𝐴 𝑂 𝑐 )
16 1 10 2 4 3 11 12 13 14 15 oppne1 ( ( ( 𝜑𝑐𝑃 ) ∧ ( 𝐴 𝑂 𝑐𝐵 𝑂 𝑐 ) ) → ¬ 𝐴𝐷 )
17 1 2 3 4 5 6 7 8 hpgbr ( 𝜑 → ( 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 ↔ ∃ 𝑐𝑃 ( 𝐴 𝑂 𝑐𝐵 𝑂 𝑐 ) ) )
18 9 17 mpbid ( 𝜑 → ∃ 𝑐𝑃 ( 𝐴 𝑂 𝑐𝐵 𝑂 𝑐 ) )
19 16 18 r19.29a ( 𝜑 → ¬ 𝐴𝐷 )