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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | funpartlem 33301* | Lemma for funpartfun 33302. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.) |
⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}) | ||
Theorem | funpartfun 33302 | The functional part of 𝐹 is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ Fun Funpart𝐹 | ||
Theorem | funpartss 33303 | The functional part of 𝐹 is a subset of 𝐹. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ Funpart𝐹 ⊆ 𝐹 | ||
Theorem | funpartfv 33304 | The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (Funpart𝐹‘𝐴) = (𝐹‘𝐴) | ||
Theorem | fullfunfnv 33305 | The full functional part of 𝐹 is a function over V. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ FullFun𝐹 Fn V | ||
Theorem | fullfunfv 33306 | The function value of the full function of 𝐹 agrees with 𝐹. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (FullFun𝐹‘𝐴) = (𝐹‘𝐴) | ||
Theorem | brfullfun 33307 | A binary relation form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴FullFun𝐹𝐵 ↔ 𝐵 = (𝐹‘𝐴)) | ||
Theorem | brrestrict 33308 | Binary relation form of the Restrict function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉Restrict𝐶 ↔ 𝐶 = (𝐴 ↾ 𝐵)) | ||
Theorem | dfrecs2 33309 | A quantifier-free definition of recs. (Contributed by Scott Fenton, 17-Jul-2020.) |
⊢ recs(𝐹) = ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (FullFun𝐹 ∘ Restrict)))) | ||
Theorem | dfrdg4 33310 | A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ rec(𝐹, 𝐴) = ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) | ||
Theorem | dfint3 33311 | Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.) |
⊢ ∩ 𝐴 = (V ∖ (◡(V ∖ E ) “ 𝐴)) | ||
Theorem | imagesset 33312 | The Image functor applied to the converse of the subset relationship yields a subset of the subset relationship. (Contributed by Scott Fenton, 14-Apr-2018.) |
⊢ Image◡ SSet ⊆ SSet | ||
Theorem | brub 33313* | Binary relation form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.) |
⊢ 𝑆 ∈ V & ⊢ 𝐴 ∈ V ⇒ ⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴) | ||
Theorem | brlb 33314* | Binary relation form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.) |
⊢ 𝑆 ∈ V & ⊢ 𝐴 ∈ V ⇒ ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥) | ||
Syntax | caltop 33315 | Declare the syntax for an alternate ordered pair. |
class ⟪𝐴, 𝐵⟫ | ||
Syntax | caltxp 33316 | Declare the syntax for an alternate Cartesian product. |
class (𝐴 ×× 𝐵) | ||
Definition | df-altop 33317 | An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 33328), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}} | ||
Definition | df-altxp 33318* | Define Cartesian products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.) |
⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫} | ||
Theorem | altopex 33319 | Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ ⟪𝐴, 𝐵⟫ ∈ V | ||
Theorem | altopthsn 33320 | Two alternate ordered pairs are equal iff the singletons of their respective elements are equal. Note that this holds regardless of sethood of any of the elements. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷})) | ||
Theorem | altopeq12 33321 | Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫) | ||
Theorem | altopeq1 33322 | Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ (𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫) | ||
Theorem | altopeq2 33323 | Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ (𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫) | ||
Theorem | altopth1 33324 | Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ (𝐴 ∈ 𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐴 = 𝐶)) | ||
Theorem | altopth2 33325 | Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ (𝐵 ∈ 𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐵 = 𝐷)) | ||
Theorem | altopthg 33326 | Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | altopthbg 33327 | Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | altopth 33328 | The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that 𝐶 and 𝐷 are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 5360), requires 𝐷 to be a set. (Contributed by Scott Fenton, 23-Mar-2012.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | altopthb 33329 | Alternate ordered pair theorem with different sethood requirements. See altopth 33328 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | altopthc 33330 | Alternate ordered pair theorem with different sethood requirements. See altopth 33328 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | altopthd 33331 | Alternate ordered pair theorem with different sethood requirements. See altopth 33328 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.) |
⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | altxpeq1 33332 | Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.) |
⊢ (𝐴 = 𝐵 → (𝐴 ×× 𝐶) = (𝐵 ×× 𝐶)) | ||
Theorem | altxpeq2 33333 | Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.) |
⊢ (𝐴 = 𝐵 → (𝐶 ×× 𝐴) = (𝐶 ×× 𝐵)) | ||
Theorem | elaltxp 33334* | Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.) |
⊢ (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫) | ||
Theorem | altopelaltxp 33335 | Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5585, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) | ||
Theorem | altxpsspw 33336 | An inclusion rule for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.) |
⊢ (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) | ||
Theorem | altxpexg 33337 | The alternate Cartesian product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ×× 𝐵) ∈ V) | ||
Theorem | rankaltopb 33338 | Compute the rank of an alternate ordered pair. (Contributed by Scott Fenton, 18-Dec-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵))) | ||
Theorem | nfaltop 33339 | Bound-variable hypothesis builder for alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥⟪𝐴, 𝐵⟫ | ||
Theorem | sbcaltop 33340* | Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.) |
⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫) | ||
Syntax | cofs 33341 | Declare the syntax for the outer five segment configuration. |
class OuterFiveSeg | ||
Definition | df-ofs 33342* | The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 26652). See brofs 33364 and 5segofs 33365 for how it is used. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.) |
⊢ OuterFiveSeg = {〈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, 𝑏〉Cgr〈𝑥, 𝑦〉 ∧ 〈𝑏, 𝑐〉Cgr〈𝑦, 𝑧〉) ∧ (〈𝑎, 𝑑〉Cgr〈𝑥, 𝑤〉 ∧ 〈𝑏, 𝑑〉Cgr〈𝑦, 𝑤〉)))} | ||
Theorem | cgrrflx2d 33343 | Deduction form of axcgrrflx 26628. (Contributed by Scott Fenton, 13-Oct-2013.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐵〉Cgr〈𝐵, 𝐴〉) | ||
Theorem | cgrtr4d 33344 | Deduction form of axcgrtr 26629. (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 → 〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉) & ⊢ (𝜑 → 〈𝐴, 𝐵〉Cgr〈𝐸, 𝐹〉) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐷〉Cgr〈𝐸, 𝐹〉) | ||
Theorem | cgrtr4and 33345 | Deduction form of axcgrtr 26629. (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〈𝐶, 𝐷〉) & ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝐵〉Cgr〈𝐸, 𝐹〉) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 〈𝐶, 𝐷〉Cgr〈𝐸, 𝐹〉) | ||
Theorem | cgrrflx 33346 | Reflexivity law for congruence. Theorem 2.1 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉) | ||
Theorem | cgrrflxd 33347 | Deduction form of cgrrflx 33346. (Contributed by Scott Fenton, 13-Oct-2013.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉) | ||
Theorem | cgrcomim 33348 | Congruence commutes on the two sides. Implication version. Theorem 2.2 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〉 → 〈𝐶, 𝐷〉Cgr〈𝐴, 𝐵〉)) | ||
Theorem | cgrcom 33349 | Congruence commutes between the two 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〉 ↔ 〈𝐶, 𝐷〉Cgr〈𝐴, 𝐵〉)) | ||
Theorem | cgrcomand 33350 | Deduction form of cgrcom 33349. (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) → 〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 〈𝐶, 𝐷〉Cgr〈𝐴, 𝐵〉) | ||
Theorem | cgrtr 33351 | Transitivity law for congruence. Theorem 2.3 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 24-Sep-2013.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉 ∧ 〈𝐶, 𝐷〉Cgr〈𝐸, 𝐹〉) → 〈𝐴, 𝐵〉Cgr〈𝐸, 𝐹〉)) | ||
Theorem | cgrtrand 33352 | Deduction form of cgrtr 33351. (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〈𝐶, 𝐷〉) & ⊢ ((𝜑 ∧ 𝜓) → 〈𝐶, 𝐷〉Cgr〈𝐸, 𝐹〉) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝐵〉Cgr〈𝐸, 𝐹〉) | ||
Theorem | cgrtr3 33353 | 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 33354 | Deduction form of cgrtr3 33353. (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 33355 | 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 33356 | 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 33357 | 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 33358 | Deduction form of cgrcoml 33355. (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 33359 | Deduction form of cgrcoml 33355. (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 33360 | Deduction form of cgrcomlr 33357. (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 33361 | 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 33362 | 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 33363 | 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 33364 | 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 33365 | Rephrase ax5seg 26652 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 33366 | 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 33367 | 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 33368 | Deduction form of cgrextend 33367. (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 33369 | Two points that satisfy the conclusion of axsegcon 26641 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 33370* | Existential uniqueness version of segconeq 33369. (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 33371 | 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 33372 | 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 33373 | 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 33374 | Deduction form of btwncom 33373. (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 33375 | 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 33376 | 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 33377 | 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 33378 | 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 33379 | 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 33380 | Deduction form of btwnexch3 33379. (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 33381 | 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 33382 | 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 33383 | 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 33384 | 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 33385 | Deduction form of btwnexch 33384. (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 33386* | 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 33387* | 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 33388 | Declare the syntax for the segment transport function. |
class TransportTo | ||
Definition | df-transport 33389* | Define the segment transport function. See fvtransport 33391 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 33390 | The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ Fun TransportTo | ||
Theorem | fvtransport 33391* | 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 33392 | 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 33393 | 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 33394 | Declare the syntax for the inner five segment predicate. |
class InnerFiveSeg | ||
Syntax | ccgr3 33395 | Declare the syntax for the three place congruence predicate. |
class Cgr3 | ||
Syntax | ccolin 33396 | Declare the syntax for the colinearity predicate. |
class Colinear | ||
Syntax | cfs 33397 | Declare the syntax for the five segment predicate. |
class FiveSeg | ||
Definition | df-colinear 33398* | 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 33399* | The inner five segment configuration is an abbreviation for another congruence condition. See brifs 33402 and ifscgr 33403 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 33400* | 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〉))} |
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