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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | brcap 33401 | Binary relation form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉Cap𝐶 ↔ 𝐶 = (𝐴 ∩ 𝐵)) | ||
Theorem | brsuccf 33402 | Binary relation form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴Succ𝐵 ↔ 𝐵 = suc 𝐴) | ||
Theorem | funpartlem 33403* | Lemma for funpartfun 33404. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.) |
⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}) | ||
Theorem | funpartfun 33404 | 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 33405 | 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 33406 | 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 33407 | 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 33408 | 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 33409 | 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 33410 | 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 33411 | 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 33412 | 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 33413 | Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.) |
⊢ ∩ 𝐴 = (V ∖ (◡(V ∖ E ) “ 𝐴)) | ||
Theorem | imagesset 33414 | 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 33415* | 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 33416* | 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 33417 | Declare the syntax for an alternate ordered pair. |
class ⟪𝐴, 𝐵⟫ | ||
Syntax | caltxp 33418 | Declare the syntax for an alternate Cartesian product. |
class (𝐴 ×× 𝐵) | ||
Definition | df-altop 33419 | An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 33430), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}} | ||
Definition | df-altxp 33420* | 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 33421 | Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ ⟪𝐴, 𝐵⟫ ∈ V | ||
Theorem | altopthsn 33422 | 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 33423 | Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫) | ||
Theorem | altopeq1 33424 | Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ (𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫) | ||
Theorem | altopeq2 33425 | Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ (𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫) | ||
Theorem | altopth1 33426 | 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 33427 | 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 33428 | 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 33429 | 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 33430 | 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 5368), 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 33431 | Alternate ordered pair theorem with different sethood requirements. See altopth 33430 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | altopthc 33432 | Alternate ordered pair theorem with different sethood requirements. See altopth 33430 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | altopthd 33433 | Alternate ordered pair theorem with different sethood requirements. See altopth 33430 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.) |
⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | altxpeq1 33434 | Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.) |
⊢ (𝐴 = 𝐵 → (𝐴 ×× 𝐶) = (𝐵 ×× 𝐶)) | ||
Theorem | altxpeq2 33435 | Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.) |
⊢ (𝐴 = 𝐵 → (𝐶 ×× 𝐴) = (𝐶 ×× 𝐵)) | ||
Theorem | elaltxp 33436* | Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.) |
⊢ (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫) | ||
Theorem | altopelaltxp 33437 | Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5591, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.) |
⊢ (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) | ||
Theorem | altxpsspw 33438 | An inclusion rule for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.) |
⊢ (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) | ||
Theorem | altxpexg 33439 | 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 33440 | 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 33441 | Bound-variable hypothesis builder for alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥⟪𝐴, 𝐵⟫ | ||
Theorem | sbcaltop 33442* | 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 33443 | Declare the syntax for the outer five segment configuration. |
class OuterFiveSeg | ||
Definition | df-ofs 33444* | The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 26724). See brofs 33466 and 5segofs 33467 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 33445 | Deduction form of axcgrrflx 26700. (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 33446 | Deduction form of axcgrtr 26701. (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 33447 | Deduction form of axcgrtr 26701. (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 33448 | 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 33449 | Deduction form of cgrrflx 33448. (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 33450 | 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 33451 | 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 33452 | Deduction form of cgrcom 33451. (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 33453 | 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 33454 | Deduction form of cgrtr 33453. (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 33455 | 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 33456 | Deduction form of cgrtr3 33455. (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 33457 | 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 33458 | 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 33459 | 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 33460 | Deduction form of cgrcoml 33457. (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 33461 | Deduction form of cgrcoml 33457. (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 33462 | Deduction form of cgrcomlr 33459. (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 33463 | 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 33464 | 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 33465 | 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 33466 | 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 33467 | Rephrase ax5seg 26724 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 33468 | 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 33469 | 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 33470 | Deduction form of cgrextend 33469. (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 33471 | Two points that satisfy the conclusion of axsegcon 26713 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 33472* | Existential uniqueness version of segconeq 33471. (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 33473 | 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 33474 | 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 33475 | 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 33476 | Deduction form of btwncom 33475. (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 33477 | 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 33478 | 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 33479 | 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 33480 | 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 33481 | 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 33482 | Deduction form of btwnexch3 33481. (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 33483 | 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 33484 | 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 33485 | 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 33486 | 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 33487 | Deduction form of btwnexch 33486. (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 33488* | 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 33489* | 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 33490 | Declare the syntax for the segment transport function. |
class TransportTo | ||
Definition | df-transport 33491* | Define the segment transport function. See fvtransport 33493 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 33492 | The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ Fun TransportTo | ||
Theorem | fvtransport 33493* | 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 33494 | 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 33495 | 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 33496 | Declare the syntax for the inner five segment predicate. |
class InnerFiveSeg | ||
Syntax | ccgr3 33497 | Declare the syntax for the three place congruence predicate. |
class Cgr3 | ||
Syntax | ccolin 33498 | Declare the syntax for the colinearity predicate. |
class Colinear | ||
Syntax | cfs 33499 | Declare the syntax for the five segment predicate. |
class FiveSeg | ||
Definition | df-colinear 33500* | 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〉))} |
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