Metamath Proof Explorer


Theorem tgasa

Description: Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of Schwabhauser p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020)

Ref Expression
Hypotheses tgsas.p 𝑃 = ( Base ‘ 𝐺 )
tgsas.m = ( dist ‘ 𝐺 )
tgsas.i 𝐼 = ( Itv ‘ 𝐺 )
tgsas.g ( 𝜑𝐺 ∈ TarskiG )
tgsas.a ( 𝜑𝐴𝑃 )
tgsas.b ( 𝜑𝐵𝑃 )
tgsas.c ( 𝜑𝐶𝑃 )
tgsas.d ( 𝜑𝐷𝑃 )
tgsas.e ( 𝜑𝐸𝑃 )
tgsas.f ( 𝜑𝐹𝑃 )
tgasa.l 𝐿 = ( LineG ‘ 𝐺 )
tgasa.1 ( 𝜑 → ¬ ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) )
tgasa.2 ( 𝜑 → ( 𝐴 𝐵 ) = ( 𝐷 𝐸 ) )
tgasa.3 ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
tgasa.4 ( 𝜑 → ⟨“ 𝐶 𝐴 𝐵 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐹 𝐷 𝐸 ”⟩ )
Assertion tgasa ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ )

Proof

Step Hyp Ref Expression
1 tgsas.p 𝑃 = ( Base ‘ 𝐺 )
2 tgsas.m = ( dist ‘ 𝐺 )
3 tgsas.i 𝐼 = ( Itv ‘ 𝐺 )
4 tgsas.g ( 𝜑𝐺 ∈ TarskiG )
5 tgsas.a ( 𝜑𝐴𝑃 )
6 tgsas.b ( 𝜑𝐵𝑃 )
7 tgsas.c ( 𝜑𝐶𝑃 )
8 tgsas.d ( 𝜑𝐷𝑃 )
9 tgsas.e ( 𝜑𝐸𝑃 )
10 tgsas.f ( 𝜑𝐹𝑃 )
11 tgasa.l 𝐿 = ( LineG ‘ 𝐺 )
12 tgasa.1 ( 𝜑 → ¬ ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) )
13 tgasa.2 ( 𝜑 → ( 𝐴 𝐵 ) = ( 𝐷 𝐸 ) )
14 tgasa.3 ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
15 tgasa.4 ( 𝜑 → ⟨“ 𝐶 𝐴 𝐵 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐹 𝐷 𝐸 ”⟩ )
16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 tgasa1 ( 𝜑 → ( 𝐵 𝐶 ) = ( 𝐸 𝐹 ) )
17 1 2 3 4 5 6 7 8 9 10 13 14 16 tgsas ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ )