Description: Third congruence theorem: SSS (Side-Side-Side): If the three pairs of sides of two triangles are equal in length, then the triangles are congruent. Theorem 11.51 of Schwabhauser p. 109. (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 | ⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) | ||
| tgsas.a | ⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) | ||
| tgsas.b | ⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) | ||
| tgsas.c | ⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) | ||
| tgsas.d | ⊢ ( 𝜑 → 𝐷 ∈ 𝑃 ) | ||
| tgsas.e | ⊢ ( 𝜑 → 𝐸 ∈ 𝑃 ) | ||
| tgsas.f | ⊢ ( 𝜑 → 𝐹 ∈ 𝑃 ) | ||
| tgsss.1 | ⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐷 − 𝐸 ) ) | ||
| tgsss.2 | ⊢ ( 𝜑 → ( 𝐵 − 𝐶 ) = ( 𝐸 − 𝐹 ) ) | ||
| tgsss.3 | ⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( 𝐹 − 𝐷 ) ) | ||
| tgsss.4 | ⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) | ||
| tgsss.5 | ⊢ ( 𝜑 → 𝐵 ≠ 𝐶 ) | ||
| tgsss.6 | ⊢ ( 𝜑 → 𝐶 ≠ 𝐴 ) | ||
| Assertion | tgsss1 | ⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tgsas.p | ⊢ 𝑃 = ( Base ‘ 𝐺 ) | |
| 2 | tgsas.m | ⊢ − = ( dist ‘ 𝐺 ) | |
| 3 | tgsas.i | ⊢ 𝐼 = ( Itv ‘ 𝐺 ) | |
| 4 | tgsas.g | ⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) | |
| 5 | tgsas.a | ⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) | |
| 6 | tgsas.b | ⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) | |
| 7 | tgsas.c | ⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) | |
| 8 | tgsas.d | ⊢ ( 𝜑 → 𝐷 ∈ 𝑃 ) | |
| 9 | tgsas.e | ⊢ ( 𝜑 → 𝐸 ∈ 𝑃 ) | |
| 10 | tgsas.f | ⊢ ( 𝜑 → 𝐹 ∈ 𝑃 ) | |
| 11 | tgsss.1 | ⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐷 − 𝐸 ) ) | |
| 12 | tgsss.2 | ⊢ ( 𝜑 → ( 𝐵 − 𝐶 ) = ( 𝐸 − 𝐹 ) ) | |
| 13 | tgsss.3 | ⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( 𝐹 − 𝐷 ) ) | |
| 14 | tgsss.4 | ⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) | |
| 15 | tgsss.5 | ⊢ ( 𝜑 → 𝐵 ≠ 𝐶 ) | |
| 16 | tgsss.6 | ⊢ ( 𝜑 → 𝐶 ≠ 𝐴 ) | |
| 17 | eqid | ⊢ ( hlG ‘ 𝐺 ) = ( hlG ‘ 𝐺 ) | |
| 18 | eqid | ⊢ ( cgrG ‘ 𝐺 ) = ( cgrG ‘ 𝐺 ) | |
| 19 | 1 2 18 4 5 6 7 8 9 10 11 12 13 | trgcgr | ⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ ) |
| 20 | 1 3 4 17 5 6 7 8 9 10 14 15 19 | cgrcgra | ⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐷 𝐸 𝐹 ”⟩ ) |