Metamath Proof Explorer


Theorem tgsas

Description: First congruence theorem: SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent. Theorem 11.49 of Schwabhauser p. 107. (Contributed by Thierry Arnoux, 1-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 )
tgsas.1 ( 𝜑 → ( 𝐴 𝐵 ) = ( 𝐷 𝐸 ) )
tgsas.2 ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
tgsas.3 ( 𝜑 → ( 𝐵 𝐶 ) = ( 𝐸 𝐹 ) )
Assertion tgsas ( 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 tgsas.1 ( 𝜑 → ( 𝐴 𝐵 ) = ( 𝐷 𝐸 ) )
12 tgsas.2 ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
13 tgsas.3 ( 𝜑 → ( 𝐵 𝐶 ) = ( 𝐸 𝐹 ) )
14 eqid ( cgrG ‘ 𝐺 ) = ( cgrG ‘ 𝐺 )
15 1 2 3 4 5 6 7 8 9 10 11 12 13 tgsas1 ( 𝜑 → ( 𝐶 𝐴 ) = ( 𝐹 𝐷 ) )
16 1 2 14 4 5 6 7 8 9 10 11 13 15 trgcgr ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ )