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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | cgrcomrand 36001 | Deduction form of cgrcoml 35997. (Contributed by Scott Fenton, 14-Oct-2013.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) & ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝐵〉Cgr〈𝐷, 𝐶〉) | ||
| Theorem | cgrcomlrand 36002 | Deduction form of cgrcomlr 35999. (Contributed by Scott Fenton, 14-Oct-2013.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) & ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 〈𝐵, 𝐴〉Cgr〈𝐷, 𝐶〉) | ||
| Theorem | cgrtriv 36003 | Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 〈𝐴, 𝐴〉Cgr〈𝐵, 𝐵〉) | ||
| Theorem | cgrid2 36004 | Identity law for congruence. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (〈𝐴, 𝐴〉Cgr〈𝐵, 𝐶〉 → 𝐵 = 𝐶)) | ||
| Theorem | cgrdegen 36005 | Two congruent segments are either both degenerate or both nondegenerate. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))) | ||
| Theorem | brofs 36006 | Binary relation form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-2013.) |
| ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉 OuterFiveSeg 〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 ↔ ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐹 Btwn 〈𝐸, 𝐺〉) ∧ (〈𝐴, 𝐵〉Cgr〈𝐸, 𝐹〉 ∧ 〈𝐵, 𝐶〉Cgr〈𝐹, 𝐺〉) ∧ (〈𝐴, 𝐷〉Cgr〈𝐸, 𝐻〉 ∧ 〈𝐵, 𝐷〉Cgr〈𝐹, 𝐻〉)))) | ||
| Theorem | 5segofs 36007 | Rephrase ax5seg 28953 using the outer five segment predicate. Theorem 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.) |
| ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → ((〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉 OuterFiveSeg 〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 ∧ 𝐴 ≠ 𝐵) → 〈𝐶, 𝐷〉Cgr〈𝐺, 𝐻〉)) | ||
| Theorem | ofscom 36008 | The outer five segment predicate commutes. (Contributed by Scott Fenton, 26-Sep-2013.) |
| ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉 OuterFiveSeg 〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 ↔ 〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 OuterFiveSeg 〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉)) | ||
| Theorem | cgrextend 36009 | Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐸 Btwn 〈𝐷, 𝐹〉) ∧ (〈𝐴, 𝐵〉Cgr〈𝐷, 𝐸〉 ∧ 〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉)) → 〈𝐴, 𝐶〉Cgr〈𝐷, 𝐹〉)) | ||
| Theorem | cgrextendand 36010 | Deduction form of cgrextend 36009. (Contributed by Scott Fenton, 14-Oct-2013.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐸 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐹 ∈ (𝔼‘𝑁)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐵 Btwn 〈𝐴, 𝐶〉) & ⊢ ((𝜑 ∧ 𝜓) → 𝐸 Btwn 〈𝐷, 𝐹〉) & ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝐵〉Cgr〈𝐷, 𝐸〉) & ⊢ ((𝜑 ∧ 𝜓) → 〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝐶〉Cgr〈𝐷, 𝐹〉) | ||
| Theorem | segconeq 36011 | Two points that satisfy the conclusion of axsegcon 28942 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → ((𝑄 ≠ 𝐴 ∧ (𝐴 Btwn 〈𝑄, 𝑋〉 ∧ 〈𝐴, 𝑋〉Cgr〈𝐵, 𝐶〉) ∧ (𝐴 Btwn 〈𝑄, 𝑌〉 ∧ 〈𝐴, 𝑌〉Cgr〈𝐵, 𝐶〉)) → 𝑋 = 𝑌)) | ||
| Theorem | segconeu 36012* | Existential uniqueness version of segconeq 36011. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → ∃!𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn 〈𝐶, 𝑟〉 ∧ 〈𝐷, 𝑟〉Cgr〈𝐴, 𝐵〉)) | ||
| Theorem | btwntriv2 36013 | Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐵 Btwn 〈𝐴, 𝐵〉) | ||
| Theorem | btwncomim 36014 | Betweenness commutes. Implication version. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Btwn 〈𝐵, 𝐶〉 → 𝐴 Btwn 〈𝐶, 𝐵〉)) | ||
| Theorem | btwncom 36015 | Betweenness commutes. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Btwn 〈𝐵, 𝐶〉 ↔ 𝐴 Btwn 〈𝐶, 𝐵〉)) | ||
| Theorem | btwncomand 36016 | Deduction form of btwncom 36015. (Contributed by Scott Fenton, 14-Oct-2013.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐴 Btwn 〈𝐵, 𝐶〉) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝐴 Btwn 〈𝐶, 𝐵〉) | ||
| Theorem | btwntriv1 36017 | Betweenness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐴 Btwn 〈𝐴, 𝐵〉) | ||
| Theorem | btwnswapid 36018 | If you can swap the first two arguments of a betweenness statement, then those arguments are identical. Theorem 3.4 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ((𝐴 Btwn 〈𝐵, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉) → 𝐴 = 𝐵)) | ||
| Theorem | btwnswapid2 36019 | If you can swap arguments one and three of a betweenness statement, then those arguments are identical. (Contributed by Scott Fenton, 7-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ((𝐴 Btwn 〈𝐵, 𝐶〉 ∧ 𝐶 Btwn 〈𝐵, 𝐴〉) → 𝐴 = 𝐶)) | ||
| Theorem | btwnintr 36020 | Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐷〉 ∧ 𝐶 Btwn 〈𝐵, 𝐷〉) → 𝐵 Btwn 〈𝐴, 𝐶〉)) | ||
| Theorem | btwnexch3 36021 | Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐶 Btwn 〈𝐵, 𝐷〉)) | ||
| Theorem | btwnexch3and 36022 | Deduction form of btwnexch3 36021. (Contributed by Scott Fenton, 13-Oct-2013.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐵 Btwn 〈𝐴, 𝐶〉) & ⊢ ((𝜑 ∧ 𝜓) → 𝐶 Btwn 〈𝐴, 𝐷〉) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝐶 Btwn 〈𝐵, 𝐷〉) | ||
| Theorem | btwnouttr2 36023 | Outer transitivity law for betweenness. Left-hand side of Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 ≠ 𝐶 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐵, 𝐷〉) → 𝐶 Btwn 〈𝐴, 𝐷〉)) | ||
| Theorem | btwnexch2 36024 | Exchange the outer point of two betweenness statements. Right-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 14-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐷〉 ∧ 𝐶 Btwn 〈𝐵, 𝐷〉) → 𝐶 Btwn 〈𝐴, 𝐷〉)) | ||
| Theorem | btwnouttr 36025 | Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 14-Jun-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 ≠ 𝐶 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐵, 𝐷〉) → 𝐵 Btwn 〈𝐴, 𝐷〉)) | ||
| Theorem | btwnexch 36026 | Outer transitivity law for betweenness. Right-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 24-Sep-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐵 Btwn 〈𝐴, 𝐷〉)) | ||
| Theorem | btwnexchand 36027 | Deduction form of btwnexch 36026. (Contributed by Scott Fenton, 13-Oct-2013.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐵 Btwn 〈𝐴, 𝐶〉) & ⊢ ((𝜑 ∧ 𝜓) → 𝐶 Btwn 〈𝐴, 𝐷〉) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝐵 Btwn 〈𝐴, 𝐷〉) | ||
| Theorem | btwndiff 36028* | There is always a 𝑐 distinct from 𝐵 such that 𝐵 lies between 𝐴 and 𝑐. Theorem 3.14 of [Schwabhauser] p. 32. (Contributed by Scott Fenton, 24-Sep-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ∃𝑐 ∈ (𝔼‘𝑁)(𝐵 Btwn 〈𝐴, 𝑐〉 ∧ 𝐵 ≠ 𝑐)) | ||
| Theorem | trisegint 36029* | A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Scott Fenton, 24-Sep-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐸 Btwn 〈𝐷, 𝐶〉 ∧ 𝑃 Btwn 〈𝐴, 𝐷〉) → ∃𝑞 ∈ (𝔼‘𝑁)(𝑞 Btwn 〈𝑃, 𝐶〉 ∧ 𝑞 Btwn 〈𝐵, 𝐸〉))) | ||
| Syntax | ctransport 36030 | Declare the syntax for the segment transport function. |
| class TransportTo | ||
| Definition | df-transport 36031* | Define the segment transport function. See fvtransport 36033 for an explanation of the function. (Contributed by Scott Fenton, 18-Oct-2013.) |
| ⊢ TransportTo = {〈〈𝑝, 𝑞〉, 𝑥〉 ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn 〈(1st ‘𝑞), 𝑟〉 ∧ 〈(2nd ‘𝑞), 𝑟〉Cgr𝑝)))} | ||
| Theorem | funtransport 36032 | The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ Fun TransportTo | ||
| Theorem | fvtransport 36033* | Calculate the value of the TransportTo function. This function takes four points, 𝐴 through 𝐷, where 𝐶 and 𝐷 are distinct. It then returns the point that extends 𝐶𝐷 by the length of 𝐴𝐵. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉) = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn 〈𝐶, 𝑟〉 ∧ 〈𝐷, 𝑟〉Cgr〈𝐴, 𝐵〉))) | ||
| Theorem | transportcl 36034 | Closure law for segment transport. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉) ∈ (𝔼‘𝑁)) | ||
| Theorem | transportprops 36035 | Calculate the defining properties of the transport function. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (𝐷 Btwn 〈𝐶, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉 ∧ 〈𝐷, (〈𝐴, 𝐵〉TransportTo〈𝐶, 𝐷〉)〉Cgr〈𝐴, 𝐵〉)) | ||
| Syntax | cifs 36036 | Declare the syntax for the inner five segment predicate. |
| class InnerFiveSeg | ||
| Syntax | ccgr3 36037 | Declare the syntax for the three place congruence predicate. |
| class Cgr3 | ||
| Syntax | ccolin 36038 | Declare the syntax for the colinearity predicate. |
| class Colinear | ||
| Syntax | cfs 36039 | Declare the syntax for the five segment predicate. |
| class FiveSeg | ||
| Definition | df-colinear 36040* | The colinearity predicate states that the three points in its arguments sit on one line. Definition 4.10 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 25-Oct-2013.) |
| ⊢ Colinear = ◡{〈〈𝑏, 𝑐〉, 𝑎〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn 〈𝑏, 𝑐〉 ∨ 𝑏 Btwn 〈𝑐, 𝑎〉 ∨ 𝑐 Btwn 〈𝑎, 𝑏〉))} | ||
| Definition | df-ifs 36041* | The inner five segment configuration is an abbreviation for another congruence condition. See brifs 36044 and ifscgr 36045 for how it is used. Definition 4.1 of [Schwabhauser] p. 34. (Contributed by Scott Fenton, 26-Sep-2013.) |
| ⊢ InnerFiveSeg = {〈𝑝, 𝑞〉 ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑥 ∈ (𝔼‘𝑛)∃𝑦 ∈ (𝔼‘𝑛)∃𝑧 ∈ (𝔼‘𝑛)∃𝑤 ∈ (𝔼‘𝑛)(𝑝 = 〈〈𝑎, 𝑏〉, 〈𝑐, 𝑑〉〉 ∧ 𝑞 = 〈〈𝑥, 𝑦〉, 〈𝑧, 𝑤〉〉 ∧ ((𝑏 Btwn 〈𝑎, 𝑐〉 ∧ 𝑦 Btwn 〈𝑥, 𝑧〉) ∧ (〈𝑎, 𝑐〉Cgr〈𝑥, 𝑧〉 ∧ 〈𝑏, 𝑐〉Cgr〈𝑦, 𝑧〉) ∧ (〈𝑎, 𝑑〉Cgr〈𝑥, 𝑤〉 ∧ 〈𝑐, 𝑑〉Cgr〈𝑧, 𝑤〉)))} | ||
| Definition | df-cgr3 36042* | The three place congruence predicate. This is an abbreviation for saying that all three pair in a triple are congruent with each other. Three place form of Definition 4.4 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.) |
| ⊢ Cgr3 = {〈𝑝, 𝑞〉 ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑒 ∈ (𝔼‘𝑛)∃𝑓 ∈ (𝔼‘𝑛)(𝑝 = 〈𝑎, 〈𝑏, 𝑐〉〉 ∧ 𝑞 = 〈𝑑, 〈𝑒, 𝑓〉〉 ∧ (〈𝑎, 𝑏〉Cgr〈𝑑, 𝑒〉 ∧ 〈𝑎, 𝑐〉Cgr〈𝑑, 𝑓〉 ∧ 〈𝑏, 𝑐〉Cgr〈𝑒, 𝑓〉))} | ||
| Definition | df-fs 36043* | The general five segment configuration is a generalization of the outer and inner five segment configurations. See brfs 36080 and fscgr 36081 for its use. Definition 4.15 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ FiveSeg = {〈𝑝, 𝑞〉 ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑥 ∈ (𝔼‘𝑛)∃𝑦 ∈ (𝔼‘𝑛)∃𝑧 ∈ (𝔼‘𝑛)∃𝑤 ∈ (𝔼‘𝑛)(𝑝 = 〈〈𝑎, 𝑏〉, 〈𝑐, 𝑑〉〉 ∧ 𝑞 = 〈〈𝑥, 𝑦〉, 〈𝑧, 𝑤〉〉 ∧ (𝑎 Colinear 〈𝑏, 𝑐〉 ∧ 〈𝑎, 〈𝑏, 𝑐〉〉Cgr3〈𝑥, 〈𝑦, 𝑧〉〉 ∧ (〈𝑎, 𝑑〉Cgr〈𝑥, 𝑤〉 ∧ 〈𝑏, 𝑑〉Cgr〈𝑦, 𝑤〉)))} | ||
| Theorem | brifs 36044 | Binary relation form of the inner five segment predicate. (Contributed by Scott Fenton, 26-Sep-2013.) |
| ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉 InnerFiveSeg 〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 ↔ ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐹 Btwn 〈𝐸, 𝐺〉) ∧ (〈𝐴, 𝐶〉Cgr〈𝐸, 𝐺〉 ∧ 〈𝐵, 𝐶〉Cgr〈𝐹, 𝐺〉) ∧ (〈𝐴, 𝐷〉Cgr〈𝐸, 𝐻〉 ∧ 〈𝐶, 𝐷〉Cgr〈𝐺, 𝐻〉)))) | ||
| Theorem | ifscgr 36045 | Inner five segment congruence. Take two triangles, 𝐴𝐷𝐶 and 𝐸𝐻𝐺, with 𝐵 between 𝐴 and 𝐶 and 𝐹 between 𝐸 and 𝐺. If the other components of the triangles are congruent, then so are 𝐵𝐷 and 𝐹𝐻. Theorem 4.2 of [Schwabhauser] p. 34. (Contributed by Scott Fenton, 27-Sep-2013.) |
| ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉 InnerFiveSeg 〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 → 〈𝐵, 𝐷〉Cgr〈𝐹, 𝐻〉)) | ||
| Theorem | cgrsub 36046 | Removing identical parts from the end of a line segment preserves congruence. Theorem 4.3 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐸 Btwn 〈𝐷, 𝐹〉) ∧ (〈𝐴, 𝐶〉Cgr〈𝐷, 𝐹〉 ∧ 〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉)) → 〈𝐴, 𝐵〉Cgr〈𝐷, 𝐸〉)) | ||
| Theorem | brcgr3 36047 | Binary relation form of the three-place congruence predicate. (Contributed by Scott Fenton, 4-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉 ↔ (〈𝐴, 𝐵〉Cgr〈𝐷, 𝐸〉 ∧ 〈𝐴, 𝐶〉Cgr〈𝐷, 𝐹〉 ∧ 〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉))) | ||
| Theorem | cgr3permute3 36048 | Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉 ↔ 〈𝐵, 〈𝐶, 𝐴〉〉Cgr3〈𝐸, 〈𝐹, 𝐷〉〉)) | ||
| Theorem | cgr3permute1 36049 | Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉 ↔ 〈𝐴, 〈𝐶, 𝐵〉〉Cgr3〈𝐷, 〈𝐹, 𝐸〉〉)) | ||
| Theorem | cgr3permute2 36050 | Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉 ↔ 〈𝐵, 〈𝐴, 𝐶〉〉Cgr3〈𝐸, 〈𝐷, 𝐹〉〉)) | ||
| Theorem | cgr3permute4 36051 | Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉 ↔ 〈𝐶, 〈𝐴, 𝐵〉〉Cgr3〈𝐹, 〈𝐷, 𝐸〉〉)) | ||
| Theorem | cgr3permute5 36052 | Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉 ↔ 〈𝐶, 〈𝐵, 𝐴〉〉Cgr3〈𝐹, 〈𝐸, 𝐷〉〉)) | ||
| Theorem | cgr3tr4 36053 | Transitivity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)) ∧ (𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁) ∧ 𝐼 ∈ (𝔼‘𝑁)))) → ((〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉 ∧ 〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐺, 〈𝐻, 𝐼〉〉) → 〈𝐷, 〈𝐸, 𝐹〉〉Cgr3〈𝐺, 〈𝐻, 𝐼〉〉)) | ||
| Theorem | cgr3com 36054 | Commutativity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉 ↔ 〈𝐷, 〈𝐸, 𝐹〉〉Cgr3〈𝐴, 〈𝐵, 𝐶〉〉)) | ||
| Theorem | cgr3rflx 36055 | Identity law for three-place congruence. (Contributed by Scott Fenton, 6-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐴, 〈𝐵, 𝐶〉〉) | ||
| Theorem | cgrxfr 36056* | A line segment can be divided at the same place as a congruent line segment is divided. Theorem 4.5 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐶〉Cgr〈𝐷, 𝐹〉) → ∃𝑒 ∈ (𝔼‘𝑁)(𝑒 Btwn 〈𝐷, 𝐹〉 ∧ 〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝑒, 𝐹〉〉))) | ||
| Theorem | btwnxfr 36057 | A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 4-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉) → 𝐸 Btwn 〈𝐷, 𝐹〉)) | ||
| Theorem | colinrel 36058 | Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ Rel Colinear | ||
| Theorem | brcolinear2 36059* | Alternate colinearity binary relation. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑃 Colinear 〈𝑄, 𝑅〉 ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑃 Btwn 〈𝑄, 𝑅〉 ∨ 𝑄 Btwn 〈𝑅, 𝑃〉 ∨ 𝑅 Btwn 〈𝑃, 𝑄〉)))) | ||
| Theorem | brcolinear 36060 | The binary relation form of the colinearity predicate. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear 〈𝐵, 𝐶〉 ↔ (𝐴 Btwn 〈𝐵, 𝐶〉 ∨ 𝐵 Btwn 〈𝐶, 𝐴〉 ∨ 𝐶 Btwn 〈𝐴, 𝐵〉))) | ||
| Theorem | colinearex 36061 | The colinear predicate exists. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ Colinear ∈ V | ||
| Theorem | colineardim1 36062 | If 𝐴 is colinear with 𝐵 and 𝐶, then 𝐴 is in the same space as 𝐵. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (𝐴 Colinear 〈𝐵, 𝐶〉 → 𝐴 ∈ (𝔼‘𝑁))) | ||
| Theorem | colinearperm1 36063 | Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear 〈𝐵, 𝐶〉 ↔ 𝐴 Colinear 〈𝐶, 𝐵〉)) | ||
| Theorem | colinearperm3 36064 | Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear 〈𝐵, 𝐶〉 ↔ 𝐵 Colinear 〈𝐶, 𝐴〉)) | ||
| Theorem | colinearperm2 36065 | Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear 〈𝐵, 𝐶〉 ↔ 𝐵 Colinear 〈𝐴, 𝐶〉)) | ||
| Theorem | colinearperm4 36066 | Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear 〈𝐵, 𝐶〉 ↔ 𝐶 Colinear 〈𝐴, 𝐵〉)) | ||
| Theorem | colinearperm5 36067 | Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear 〈𝐵, 𝐶〉 ↔ 𝐶 Colinear 〈𝐵, 𝐴〉)) | ||
| Theorem | colineartriv1 36068 | Trivial case of colinearity. Theorem 4.12 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐴 Colinear 〈𝐴, 𝐵〉) | ||
| Theorem | colineartriv2 36069 | Trivial case of colinearity. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐴 Colinear 〈𝐵, 𝐵〉) | ||
| Theorem | btwncolinear1 36070 | Betweenness implies colinearity. (Contributed by Scott Fenton, 7-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn 〈𝐴, 𝐵〉 → 𝐴 Colinear 〈𝐵, 𝐶〉)) | ||
| Theorem | btwncolinear2 36071 | Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn 〈𝐴, 𝐵〉 → 𝐴 Colinear 〈𝐶, 𝐵〉)) | ||
| Theorem | btwncolinear3 36072 | Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn 〈𝐴, 𝐵〉 → 𝐵 Colinear 〈𝐴, 𝐶〉)) | ||
| Theorem | btwncolinear4 36073 | Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn 〈𝐴, 𝐵〉 → 𝐵 Colinear 〈𝐶, 𝐴〉)) | ||
| Theorem | btwncolinear5 36074 | Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn 〈𝐴, 𝐵〉 → 𝐶 Colinear 〈𝐴, 𝐵〉)) | ||
| Theorem | btwncolinear6 36075 | Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn 〈𝐴, 𝐵〉 → 𝐶 Colinear 〈𝐵, 𝐴〉)) | ||
| Theorem | colinearxfr 36076 | Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((𝐵 Colinear 〈𝐴, 𝐶〉 ∧ 〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉) → 𝐸 Colinear 〈𝐷, 𝐹〉)) | ||
| Theorem | lineext 36077* | Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → ((𝐴 Colinear 〈𝐵, 𝐶〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐷, 𝐸〉) → ∃𝑓 ∈ (𝔼‘𝑁)〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝑓〉〉)) | ||
| Theorem | brofs2 36078 | Change some conditions for outer five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013.) |
| ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉 OuterFiveSeg 〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 ↔ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐸, 〈𝐹, 𝐺〉〉 ∧ (〈𝐴, 𝐷〉Cgr〈𝐸, 𝐻〉 ∧ 〈𝐵, 𝐷〉Cgr〈𝐹, 𝐻〉)))) | ||
| Theorem | brifs2 36079 | Change some conditions for inner five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013.) |
| ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉 InnerFiveSeg 〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 ↔ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐸, 〈𝐹, 𝐺〉〉 ∧ (〈𝐴, 𝐷〉Cgr〈𝐸, 𝐻〉 ∧ 〈𝐶, 𝐷〉Cgr〈𝐺, 𝐻〉)))) | ||
| Theorem | brfs 36080 | Binary relation form of the general five segment predicate. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉 FiveSeg 〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 ↔ (𝐴 Colinear 〈𝐵, 𝐶〉 ∧ 〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐸, 〈𝐹, 𝐺〉〉 ∧ (〈𝐴, 𝐷〉Cgr〈𝐸, 𝐻〉 ∧ 〈𝐵, 𝐷〉Cgr〈𝐹, 𝐻〉)))) | ||
| Theorem | fscgr 36081 | Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.) |
| ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → ((〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉 FiveSeg 〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 ∧ 𝐴 ≠ 𝐵) → 〈𝐶, 𝐷〉Cgr〈𝐺, 𝐻〉)) | ||
| Theorem | linecgr 36082 | Congruence rule for lines. Theorem 4.17 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁))) → (((𝐴 ≠ 𝐵 ∧ 𝐴 Colinear 〈𝐵, 𝐶〉) ∧ (〈𝐴, 𝑃〉Cgr〈𝐴, 𝑄〉 ∧ 〈𝐵, 𝑃〉Cgr〈𝐵, 𝑄〉)) → 〈𝐶, 𝑃〉Cgr〈𝐶, 𝑄〉)) | ||
| Theorem | linecgrand 36083 | Deduction form of linecgr 36082. (Contributed by Scott Fenton, 14-Oct-2013.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝑃 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝑄 ∈ (𝔼‘𝑁)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ≠ 𝐵) & ⊢ ((𝜑 ∧ 𝜓) → 𝐴 Colinear 〈𝐵, 𝐶〉) & ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝑃〉Cgr〈𝐴, 𝑄〉) & ⊢ ((𝜑 ∧ 𝜓) → 〈𝐵, 𝑃〉Cgr〈𝐵, 𝑄〉) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 〈𝐶, 𝑃〉Cgr〈𝐶, 𝑄〉) | ||
| Theorem | lineid 36084 | Identity law for points on lines. Theorem 4.18 of [Schwabhauser] p. 38. (Contributed by Scott Fenton, 7-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (((𝐴 ≠ 𝐵 ∧ 𝐴 Colinear 〈𝐵, 𝐶〉) ∧ (〈𝐴, 𝐶〉Cgr〈𝐴, 𝐷〉 ∧ 〈𝐵, 𝐶〉Cgr〈𝐵, 𝐷〉)) → 𝐶 = 𝐷)) | ||
| Theorem | idinside 36085 | Law for finding a point inside a segment. Theorem 4.19 of [Schwabhauser] p. 38. (Contributed by Scott Fenton, 7-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐶 Btwn 〈𝐴, 𝐵〉 ∧ 〈𝐴, 𝐶〉Cgr〈𝐴, 𝐷〉 ∧ 〈𝐵, 𝐶〉Cgr〈𝐵, 𝐷〉) → 𝐶 = 𝐷)) | ||
| Theorem | endofsegid 36086 | If 𝐴, 𝐵, and 𝐶 fall in order on a line, and 𝐴𝐵 and 𝐴𝐶 are congruent, then 𝐶 = 𝐵. (Contributed by Scott Fenton, 7-Oct-2013.) |
| ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 〈𝐴, 𝐶〉Cgr〈𝐴, 𝐵〉) → 𝐶 = 𝐵)) | ||
| Theorem | endofsegidand 36087 | Deduction form of endofsegid 36086. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐶 Btwn 〈𝐴, 𝐵〉) & ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐶〉) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = 𝐶) | ||
| Theorem | btwnconn1lem1 36088 | Lemma for btwnconn1 36102. The next several lemmas introduce various properties of hypothetical points that end up eliminating alternatives to connectivity. We begin by showing a congruence property of those hypothetical points. (Contributed by Scott Fenton, 8-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑋〉 ∧ 〈𝑑, 𝑋〉Cgr〈𝐷, 𝐵〉)))) → 〈𝐵, 𝑐〉Cgr〈𝑋, 𝐶〉) | ||
| Theorem | btwnconn1lem2 36089 | Lemma for btwnconn1 36102. Now, we show that two of the hypotheticals we introduced in the first lemma are identical. (Contributed by Scott Fenton, 8-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑋〉 ∧ 〈𝑑, 𝑋〉Cgr〈𝐷, 𝐵〉)))) → 𝑋 = 𝑏) | ||
| Theorem | btwnconn1lem3 36090 | Lemma for btwnconn1 36102. Establish the next congruence in the series. (Contributed by Scott Fenton, 8-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑑, 𝑏〉Cgr〈𝐷, 𝐵〉)))) → 〈𝐵, 𝑑〉Cgr〈𝑏, 𝐷〉) | ||
| Theorem | btwnconn1lem4 36091 | Lemma for btwnconn1 36102. Assuming 𝐶 ≠ 𝑐, we now attempt to force 𝐷 = 𝑑 from here out via a series of congruences. (Contributed by Scott Fenton, 8-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑑, 𝑏〉Cgr〈𝐷, 𝐵〉)))) → 〈𝑑, 𝑐〉Cgr〈𝐷, 𝐶〉) | ||
| Theorem | btwnconn1lem5 36092 | Lemma for btwnconn1 36102. Now, we introduce 𝐸, the intersection of 𝐶𝑐 and 𝐷𝑑. We begin by showing that it is the midpoint of 𝐶 and 𝑐. (Contributed by Scott Fenton, 8-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑑, 𝑏〉Cgr〈𝐷, 𝐵〉))) ∧ (𝐸 Btwn 〈𝐶, 𝑐〉 ∧ 𝐸 Btwn 〈𝐷, 𝑑〉))) → 〈𝐸, 𝐶〉Cgr〈𝐸, 𝑐〉) | ||
| Theorem | btwnconn1lem6 36093 | Lemma for btwnconn1 36102. Next, we show that 𝐸 is the midpoint of 𝐷 and 𝑑. (Contributed by Scott Fenton, 8-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑑, 𝑏〉Cgr〈𝐷, 𝐵〉))) ∧ (𝐸 Btwn 〈𝐶, 𝑐〉 ∧ 𝐸 Btwn 〈𝐷, 𝑑〉))) → 〈𝐸, 𝐷〉Cgr〈𝐸, 𝑑〉) | ||
| Theorem | btwnconn1lem7 36094 | Lemma for btwnconn1 36102. Under our assumptions, 𝐶 and 𝑑 are distinct. (Contributed by Scott Fenton, 8-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑑, 𝑏〉Cgr〈𝐷, 𝐵〉))) ∧ (𝐸 Btwn 〈𝐶, 𝑐〉 ∧ 𝐸 Btwn 〈𝐷, 𝑑〉))) → 𝐶 ≠ 𝑑) | ||
| Theorem | btwnconn1lem8 36095 | Lemma for btwnconn1 36102. Now, we introduce the last three points used in the construction: 𝑃, 𝑄, and 𝑅 will turn out to be equal further down, and will provide us with the key to the final statement. We begin by establishing congruence of 𝑅𝑃 and 𝐸𝑑. (Contributed by Scott Fenton, 8-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑑, 𝑏〉Cgr〈𝐷, 𝐵〉))) ∧ ((𝐸 Btwn 〈𝐶, 𝑐〉 ∧ 𝐸 Btwn 〈𝐷, 𝑑〉) ∧ ((𝐶 Btwn 〈𝑐, 𝑃〉 ∧ 〈𝐶, 𝑃〉Cgr〈𝐶, 𝑑〉) ∧ (𝐶 Btwn 〈𝑑, 𝑅〉 ∧ 〈𝐶, 𝑅〉Cgr〈𝐶, 𝐸〉) ∧ (𝑅 Btwn 〈𝑃, 𝑄〉 ∧ 〈𝑅, 𝑄〉Cgr〈𝑅, 𝑃〉))))) → 〈𝑅, 𝑃〉Cgr〈𝐸, 𝑑〉) | ||
| Theorem | btwnconn1lem9 36096 | Lemma for btwnconn1 36102. Now, a quick use of transitivity to establish congruence on 𝑅𝑄 and 𝐸𝐷. (Contributed by Scott Fenton, 8-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑑, 𝑏〉Cgr〈𝐷, 𝐵〉))) ∧ ((𝐸 Btwn 〈𝐶, 𝑐〉 ∧ 𝐸 Btwn 〈𝐷, 𝑑〉) ∧ ((𝐶 Btwn 〈𝑐, 𝑃〉 ∧ 〈𝐶, 𝑃〉Cgr〈𝐶, 𝑑〉) ∧ (𝐶 Btwn 〈𝑑, 𝑅〉 ∧ 〈𝐶, 𝑅〉Cgr〈𝐶, 𝐸〉) ∧ (𝑅 Btwn 〈𝑃, 𝑄〉 ∧ 〈𝑅, 𝑄〉Cgr〈𝑅, 𝑃〉))))) → 〈𝑅, 𝑄〉Cgr〈𝐸, 𝐷〉) | ||
| Theorem | btwnconn1lem10 36097 | Lemma for btwnconn1 36102. Now we establish a congruence that will give us 𝐷 = 𝑑 when we compute 𝑃 = 𝑄 later on. (Contributed by Scott Fenton, 8-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑑, 𝑏〉Cgr〈𝐷, 𝐵〉))) ∧ ((𝐸 Btwn 〈𝐶, 𝑐〉 ∧ 𝐸 Btwn 〈𝐷, 𝑑〉) ∧ ((𝐶 Btwn 〈𝑐, 𝑃〉 ∧ 〈𝐶, 𝑃〉Cgr〈𝐶, 𝑑〉) ∧ (𝐶 Btwn 〈𝑑, 𝑅〉 ∧ 〈𝐶, 𝑅〉Cgr〈𝐶, 𝐸〉) ∧ (𝑅 Btwn 〈𝑃, 𝑄〉 ∧ 〈𝑅, 𝑄〉Cgr〈𝑅, 𝑃〉))))) → 〈𝑑, 𝐷〉Cgr〈𝑃, 𝑄〉) | ||
| Theorem | btwnconn1lem11 36098 | Lemma for btwnconn1 36102. Now, we establish that 𝐷 and 𝑄 are equidistant from 𝐶. (Contributed by Scott Fenton, 8-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑑, 𝑏〉Cgr〈𝐷, 𝐵〉))) ∧ ((𝐸 Btwn 〈𝐶, 𝑐〉 ∧ 𝐸 Btwn 〈𝐷, 𝑑〉) ∧ ((𝐶 Btwn 〈𝑐, 𝑃〉 ∧ 〈𝐶, 𝑃〉Cgr〈𝐶, 𝑑〉) ∧ (𝐶 Btwn 〈𝑑, 𝑅〉 ∧ 〈𝐶, 𝑅〉Cgr〈𝐶, 𝐸〉) ∧ (𝑅 Btwn 〈𝑃, 𝑄〉 ∧ 〈𝑅, 𝑄〉Cgr〈𝑅, 𝑃〉))))) → 〈𝐷, 𝐶〉Cgr〈𝑄, 𝐶〉) | ||
| Theorem | btwnconn1lem12 36099 | Lemma for btwnconn1 36102. Using a long string of invocations of linecgr 36082, we show that 𝐷 = 𝑑. (Contributed by Scott Fenton, 9-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑑, 𝑏〉Cgr〈𝐷, 𝐵〉))) ∧ ((𝐸 Btwn 〈𝐶, 𝑐〉 ∧ 𝐸 Btwn 〈𝐷, 𝑑〉) ∧ ((𝐶 Btwn 〈𝑐, 𝑃〉 ∧ 〈𝐶, 𝑃〉Cgr〈𝐶, 𝑑〉) ∧ (𝐶 Btwn 〈𝑑, 𝑅〉 ∧ 〈𝐶, 𝑅〉Cgr〈𝐶, 𝐸〉) ∧ (𝑅 Btwn 〈𝑃, 𝑄〉 ∧ 〈𝑅, 𝑄〉Cgr〈𝑅, 𝑃〉))))) → 𝐷 = 𝑑) | ||
| Theorem | btwnconn1lem13 36100 | Lemma for btwnconn1 36102. Begin back-filling and eliminating hypotheses. (Contributed by Scott Fenton, 9-Oct-2013.) |
| ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁)))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ ((𝐷 Btwn 〈𝐴, 𝑐〉 ∧ 〈𝐷, 𝑐〉Cgr〈𝐶, 𝐷〉) ∧ (𝐶 Btwn 〈𝐴, 𝑑〉 ∧ 〈𝐶, 𝑑〉Cgr〈𝐶, 𝐷〉)) ∧ ((𝑐 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑐, 𝑏〉Cgr〈𝐶, 𝐵〉) ∧ (𝑑 Btwn 〈𝐴, 𝑏〉 ∧ 〈𝑑, 𝑏〉Cgr〈𝐷, 𝐵〉)))) → (𝐶 = 𝑐 ∨ 𝐷 = 𝑑)) | ||
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