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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cgrtr3 32101 | Transitivity law for congruence. (Contributed by Scott Fenton, 7-Oct-2013.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((〈𝐴, 𝐵〉Cgr〈𝐸, 𝐹〉 ∧ 〈𝐶, 𝐷〉Cgr〈𝐸, 𝐹〉) → 〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉)) | ||
Theorem | cgrtr3and 32102 | Deduction form of cgrtr3 32101. (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 → 𝐸 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐹 ∈ (𝔼‘𝑁)) & ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝐵〉Cgr〈𝐸, 𝐹〉) & ⊢ ((𝜑 ∧ 𝜓) → 〈𝐶, 𝐷〉Cgr〈𝐸, 𝐹〉) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉) | ||
Theorem | cgrcoml 32103 | Congruence commutes on the left. Biconditional version of Theorem 2.4 of [Schwabhauser] p. 27. (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〉 ↔ 〈𝐵, 𝐴〉Cgr〈𝐶, 𝐷〉)) | ||
Theorem | cgrcomr 32104 | Congruence commutes on the right. Biconditional version of Theorem 2.5 of [Schwabhauser] p. 27. (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〉Cgr〈𝐷, 𝐶〉)) | ||
Theorem | cgrcomlr 32105 | Congruence commutes on both sides. (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〉 ↔ 〈𝐵, 𝐴〉Cgr〈𝐷, 𝐶〉)) | ||
Theorem | cgrcomland 32106 | Deduction form of cgrcoml 32103. (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〈𝐶, 𝐷〉) | ||
Theorem | cgrcomrand 32107 | Deduction form of cgrcoml 32103. (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 32108 | Deduction form of cgrcomlr 32105. (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 32109 | 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 32110 | 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 32111 | Two congruent segments are either both degenerate or both non-degenerate. (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 32112 | 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 32113 | Rephrase ax5seg 25818 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 32114 | 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 32115 | 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 32116 | Deduction form of cgrextend 32115. (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 32117 | Two points that satsify the conclusion of axsegcon 25807 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 32118* | Existential uniqueness version of segconeq 32117. (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 32119 | 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 32120 | 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 32121 | 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 32122 | Deduction form of btwncom 32121. (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 32123 | 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 32124 | 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 32125 | 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 32126 | 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 32127 | 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 32128 | Deduction form of btwnexch3 32127. (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 32129 | 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 32130 | 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 32131 | 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 32132 | 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 32133 | Deduction form of btwnexch 32132. (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 32134* | 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 32135* | 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 32136 | Declare the syntax for the segment transport function. |
class TransportTo | ||
Definition | df-transport 32137* | Define the segment transport function. See fvtransport 32139 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 32138 | The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ Fun TransportTo | ||
Theorem | fvtransport 32139* | 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 32140 | 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 32141 | 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 32142 | Declare the syntax for the inner five segment predicate. |
class InnerFiveSeg | ||
Syntax | ccgr3 32143 | Declare the syntax for the three place congruence predicate. |
class Cgr3 | ||
Syntax | ccolin 32144 | Declare the syntax for the colinearity predicate. |
class Colinear | ||
Syntax | cfs 32145 | Declare the syntax for the five segment predicate. |
class FiveSeg | ||
Definition | df-colinear 32146* | 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 32147* | The inner five segment configuration is an abbreviation for another congruence condition. See brifs 32150 and ifscgr 32151 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 32148* | 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 32149* | The general five segment configuration is a generalization of the outer and inner five segment configurations. See brfs 32186 and fscgr 32187 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 32150 | 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 32151 | 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 32152 | 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 32153 | 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 32154 | 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 32155 | 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 32156 | 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 32157 | 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 32158 | 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 32159 | 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 32160 | 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 32161 | 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 32162* | 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 32163 | 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 32164 | Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ Rel Colinear | ||
Theorem | brcolinear2 32165* | 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 32166 | 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 32167 | The colinear predicate exists. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ Colinear ∈ V | ||
Theorem | colineardim1 32168 | 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 32169 | 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 32170 | 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 32171 | 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 32172 | 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 32173 | 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 32174 | 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 32175 | 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 32176 | 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 32177 | 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 32178 | 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 32179 | 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 32180 | 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 32181 | 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 32182 | 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 32183* | 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 32184 | 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 32185 | 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 32186 | 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 32187 | 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 32188 | 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 32189 | Deduction form of linecgr 32188. (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 32190 | 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 32191 | 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 32192 | 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 32193 | Deduction form of endofsegid 32192. (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 32194 | Lemma for btwnconn1 32208. 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 32195 | Lemma for btwnconn1 32208. 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 32196 | Lemma for btwnconn1 32208. 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 32197 | Lemma for btwnconn1 32208. 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 32198 | Lemma for btwnconn1 32208. 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 32199 | Lemma for btwnconn1 32208. 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 32200 | Lemma for btwnconn1 32208. 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) |
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