Metamath Proof Explorer

Theorem cgrane4

Description: Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020)

Ref Expression
Hypotheses iscgra.p ⊢ 𝑃 = ( Base ‘ 𝐺 )
iscgra.i ⊢ 𝐼 = ( Itv ‘ 𝐺 )
iscgra.k ⊢ 𝐾 = ( hlG ‘ 𝐺 )
iscgra.g ⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
iscgra.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
iscgra.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
iscgra.c ⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
iscgra.d ⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
iscgra.e ⊢ ( 𝜑 → 𝐸 ∈ 𝑃 )
iscgra.f ⊢ ( 𝜑 → 𝐹 ∈ 𝑃 )
cgrahl1.2 ⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
Assertion cgrane4 ( 𝜑 → 𝐸 ≠ 𝐹 )

Proof

Step Hyp Ref Expression
1 iscgra.p ⊢ 𝑃 = ( Base ‘ 𝐺 )
2 iscgra.i ⊢ 𝐼 = ( Itv ‘ 𝐺 )
3 iscgra.k ⊢ 𝐾 = ( hlG ‘ 𝐺 )
4 iscgra.g ⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5 iscgra.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6 iscgra.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7 iscgra.c ⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8 iscgra.d ⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
9 iscgra.e ⊢ ( 𝜑 → 𝐸 ∈ 𝑃 )
10 iscgra.f ⊢ ( 𝜑 → 𝐹 ∈ 𝑃 )
11 cgrahl1.2 ⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
12 simplr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ 𝑦 ∈ 𝑃 ) ∧ ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝑥 𝐸 𝑦 ”⟩ ∧ 𝑥 ( 𝐾 ‘ 𝐸 ) 𝐷 ∧ 𝑦 ( 𝐾 ‘ 𝐸 ) 𝐹 ) ) → 𝑦 ∈ 𝑃 )
13 10 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ 𝑦 ∈ 𝑃 ) ∧ ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝑥 𝐸 𝑦 ”⟩ ∧ 𝑥 ( 𝐾 ‘ 𝐸 ) 𝐷 ∧ 𝑦 ( 𝐾 ‘ 𝐸 ) 𝐹 ) ) → 𝐹 ∈ 𝑃 )
14 9 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ 𝑦 ∈ 𝑃 ) ∧ ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝑥 𝐸 𝑦 ”⟩ ∧ 𝑥 ( 𝐾 ‘ 𝐸 ) 𝐷 ∧ 𝑦 ( 𝐾 ‘ 𝐸 ) 𝐹 ) ) → 𝐸 ∈ 𝑃 )
15 4 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ 𝑦 ∈ 𝑃 ) ∧ ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝑥 𝐸 𝑦 ”⟩ ∧ 𝑥 ( 𝐾 ‘ 𝐸 ) 𝐷 ∧ 𝑦 ( 𝐾 ‘ 𝐸 ) 𝐹 ) ) → 𝐺 ∈ TarskiG )
16 simpr3 ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ 𝑦 ∈ 𝑃 ) ∧ ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝑥 𝐸 𝑦 ”⟩ ∧ 𝑥 ( 𝐾 ‘ 𝐸 ) 𝐷 ∧ 𝑦 ( 𝐾 ‘ 𝐸 ) 𝐹 ) ) → 𝑦 ( 𝐾 ‘ 𝐸 ) 𝐹 )
17 1 2 3 12 13 14 15 16 hlne2 ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ 𝑦 ∈ 𝑃 ) ∧ ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝑥 𝐸 𝑦 ”⟩ ∧ 𝑥 ( 𝐾 ‘ 𝐸 ) 𝐷 ∧ 𝑦 ( 𝐾 ‘ 𝐸 ) 𝐹 ) ) → 𝐹 ≠ 𝐸 )
18 17 necomd ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ 𝑦 ∈ 𝑃 ) ∧ ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝑥 𝐸 𝑦 ”⟩ ∧ 𝑥 ( 𝐾 ‘ 𝐸 ) 𝐷 ∧ 𝑦 ( 𝐾 ‘ 𝐸 ) 𝐹 ) ) → 𝐸 ≠ 𝐹 )
19 1 2 3 4 5 6 7 8 9 10 iscgra ⊢ ( 𝜑 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ ↔ ∃ 𝑥 ∈ 𝑃 ∃ 𝑦 ∈ 𝑃 ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝑥 𝐸 𝑦 ”⟩ ∧ 𝑥 ( 𝐾 ‘ 𝐸 ) 𝐷 ∧ 𝑦 ( 𝐾 ‘ 𝐸 ) 𝐹 ) ) )
20 11 19 mpbid ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑃 ∃ 𝑦 ∈ 𝑃 ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝑥 𝐸 𝑦 ”⟩ ∧ 𝑥 ( 𝐾 ‘ 𝐸 ) 𝐷 ∧ 𝑦 ( 𝐾 ‘ 𝐸 ) 𝐹 ) )
21 18 20 r19.29vva ⊢ ( 𝜑 → 𝐸 ≠ 𝐹 )