P. 365 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	brdomain 36401	Binary relation form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)
Theorem	brrange 36402	Binary relation form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)
Theorem	brdomaing 36403	Closed form of brdomain 36401. (Contributed by Scott Fenton, 2-May-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴))
Theorem	brrangeg 36404	Closed form of brrange 36402. (Contributed by Scott Fenton, 3-May-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴))
Theorem	brimg 36405	Binary relation form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩Img𝐶 ↔ 𝐶 = (𝐴 “ 𝐵))
Theorem	brapply 36406	Binary relation form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩Apply𝐶 ↔ 𝐶 = (𝐴‘𝐵))
Theorem	brcup 36407	Binary relation form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩Cup𝐶 ↔ 𝐶 = (𝐴 ∪ 𝐵))
Theorem	brcap 36408	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	lemsuccf 36409*	Lemma for unfolding different forms of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (∃𝑥(𝐴( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝐵) ↔ 𝐵 = suc 𝐴)
Theorem	brsuccf 36410	Binary relation form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴Succ𝐵 ↔ 𝐵 = suc 𝐴)
Theorem	dfsuccf2 36411*	Alternate definition of Scott Fenton's version of Succ, cf. df-sucmap 39089. (Contributed by Peter Mazsa, 6-Jan-2026.)
⊢ Succ = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
Theorem	funpartlem 36412*	Lemma for funpartfun 36413. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
Theorem	funpartfun 36413	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 36414	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 36415	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 36416	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 36417	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 36418	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 36419	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 36420	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 36421	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 36422	Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
⊢ ∩ 𝐴 = (V ∖ (◡(V ∖ E ) “ 𝐴))
Theorem	imagesset 36423	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 36424*	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 36425*	Binary relation form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)
⊢ 𝑆 ∈ V    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)
21.11.16  Alternate ordered pairs
Syntax	caltop 36426	Declare the syntax for an alternate ordered pair.
class ⟪𝐴, 𝐵⟫
Syntax	caltxp 36427	Declare the syntax for an alternate Cartesian product.
class (𝐴 ×× 𝐵)
Definition	df-altop 36428	An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 36439), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
Definition	df-altxp 36429*	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 36430	Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ ⟪𝐴, 𝐵⟫ ∈ V
Theorem	altopthsn 36431	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 36432	Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)
Theorem	altopeq1 36433	Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ (𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)
Theorem	altopeq2 36434	Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ (𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫)
Theorem	altopth1 36435	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 36436	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 36437	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 36438	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 36439	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 5460), 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 36440	Alternate ordered pair theorem with different sethood requirements. See altopth 36439 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	altopthc 36441	Alternate ordered pair theorem with different sethood requirements. See altopth 36439 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	altopthd 36442	Alternate ordered pair theorem with different sethood requirements. See altopth 36439 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	altxpeq1 36443	Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
⊢ (𝐴 = 𝐵 → (𝐴 ×× 𝐶) = (𝐵 ×× 𝐶))
Theorem	altxpeq2 36444	Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
⊢ (𝐴 = 𝐵 → (𝐶 ×× 𝐴) = (𝐶 ×× 𝐵))
Theorem	elaltxp 36445*	Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.)
⊢ (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫)
Theorem	altopelaltxp 36446	Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5699, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)
⊢ (⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
Theorem	altxpsspw 36447	An inclusion rule for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
⊢ (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
Theorem	altxpexg 36448	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 36449	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 36450	Bound-variable hypothesis builder for alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥⟪𝐴, 𝐵⟫
Theorem	sbcaltop 36451*	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⟫)
21.11.17  Geometry in the Euclidean space
21.11.17.1  Congruence properties
Syntax	cofs 36452	Declare the syntax for the outer five segment configuration.
class OuterFiveSeg
Definition	df-ofs 36453*	The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 29266). See brofs 36475 and 5segofs 36476 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 36454	Deduction form of axcgrrflx 29242. (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 36455	Deduction form of axcgrtr 29243. (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 36456	Deduction form of axcgrtr 29243. (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 36457	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 36458	Deduction form of cgrrflx 36457. (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 36459	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 36460	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 36461	Deduction form of cgrcom 36460. (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 36462	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 36463	Deduction form of cgrtr 36462. (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 36464	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 36465	Deduction form of cgrtr3 36464. (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 36466	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 36467	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 36468	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 36469	Deduction form of cgrcoml 36466. (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 36470	Deduction form of cgrcoml 36466. (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 36471	Deduction form of cgrcomlr 36468. (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 36472	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 36473	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 36474	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 36475	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 36476	Rephrase ax5seg 29266 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 36477	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 36478	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 36479	Deduction form of cgrextend 36478. (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 36480	Two points that satisfy the conclusion of axsegcon 29255 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 36481*	Existential uniqueness version of segconeq 36480. (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⟩))
21.11.17.2  Betweenness properties
Theorem	btwntriv2 36482	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 36483	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 36484	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 36485	Deduction form of btwncom 36484. (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 36486	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 36487	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 36488	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 36489	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 36490	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 36491	Deduction form of btwnexch3 36490. (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 36492	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 36493	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 36494	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 36495	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 36496	Deduction form of btwnexch 36495. (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 36497*	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 36498*	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⟩)))
21.11.17.3  Segment Transportation
Syntax	ctransport 36499	Declare the syntax for the segment transport function.
class TransportTo
Definition	df-transport 36500*	Define the segment transport function. See fvtransport 36502 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)))}