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Theorem List for Metamath Proof Explorer - 32001-32100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremellimits 32001 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐴 ∈ V       (𝐴 Limits ↔ Lim 𝐴)

Theoremlimitssson 32002 The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
Limits ⊆ On

Theoremdfom5b 32003 A quantifier-free definition of ω that does not depend on ax-inf 8532. (Note: label was changed from dfom5 8544 to dfom5b 32003 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.)
ω = (On ∩ Limits )

Theoremsscoid 32004 A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
(𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))

Theoremdffun10 32005 Another potential definition of functionhood. Based on statements in http://people.math.gatech.edu/~belinfan/research/autoreas/otter/sum/fs/. (Contributed by Scott Fenton, 30-Aug-2017.)
(Fun 𝐹𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))))

Theoremelfuns 32006 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐹 ∈ V       (𝐹 Funs ↔ Fun 𝐹)

Theoremelfunsg 32007 Closed form of elfuns 32006. (Contributed by Scott Fenton, 2-May-2014.)
(𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))

Theorembrsingle 32008 The binary relationship form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Singleton𝐵𝐵 = {𝐴})

Theoremelsingles 32009* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥})

Theoremfnsingle 32010 The singleton relationship is a function over the universe. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singleton Fn V

Theoremfvsingle 32011 The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.)
(Singleton‘𝐴) = {𝐴}

Theoremdfsingles2 32012* Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singletons = {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}

Theoremsnelsingles 32013 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
𝐴 ∈ V       {𝐴} ∈ Singletons

Theoremdfiota3 32014 A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
(℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )

Theoremdffv5 32015 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐹𝐴) = ({(𝐹 “ {𝐴})} ∩ Singletons )

Theoremunisnif 32016 Express union of singleton in terms of if. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
{𝐴} = if(𝐴 ∈ V, 𝐴, ∅)

Theorembrimage 32017 Binary relationship form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴))

Theorembrimageg 32018 Closed form of brimage 32017. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴)))

Theoremfunimage 32019 Image𝐴 is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Image𝐴

Theoremfnimage 32020* Image𝑅 is a function over the set-like portion of 𝑅. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}

Theoremimageval 32021* The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image𝑅 = (𝑥 ∈ V ↦ (𝑅𝑥))

Theoremfvimage 32022 The value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴𝑉 ∧ (𝑅𝐴) ∈ 𝑊) → (Image𝑅𝐴) = (𝑅𝐴))

Theorembrcart 32023 Binary relationship form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Cart𝐶𝐶 = (𝐴 × 𝐵))

Theorembrdomain 32024 The binary relationship form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Domain𝐵𝐵 = dom 𝐴)

Theorembrrange 32025 The binary relationship form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Range𝐵𝐵 = ran 𝐴)

Theorembrdomaing 32026 Closed form of brdomain 32024. (Contributed by Scott Fenton, 2-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Domain𝐵𝐵 = dom 𝐴))

Theorembrrangeg 32027 Closed form of brrange 32025. (Contributed by Scott Fenton, 3-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))

Theorembrimg 32028 The binary relationship form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Img𝐶𝐶 = (𝐴𝐵))

Theorembrapply 32029 The binary relationship form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩Apply𝐶𝐶 = (𝐴𝐵))

Theorembrcup 32030 Binary relationship 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⟩Cup𝐶𝐶 = (𝐴𝐵))

Theorembrcap 32031 Binary relationship 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⟩Cap𝐶𝐶 = (𝐴𝐵))

Theorembrsuccf 32032 Binary relationship form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴Succ𝐵𝐵 = suc 𝐴)

Theoremfunpartlem 32033* Lemma for funpartfun 32034. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})

Theoremfunpartfun 32034 The functional part of 𝐹 is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Funpart𝐹

Theoremfunpartss 32035 The functional part of 𝐹 is a subset of 𝐹. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Funpart𝐹𝐹

Theoremfunpartfv 32036 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.)
(Funpart𝐹𝐴) = (𝐹𝐴)

Theoremfullfunfnv 32037 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

Theoremfullfunfv 32038 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.)
(FullFun𝐹𝐴) = (𝐹𝐴)

Theorembrfullfun 32039 A binary relationship form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴FullFun𝐹𝐵𝐵 = (𝐹𝐴))

Theorembrrestrict 32040 The binary relationship 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⟩Restrict𝐶𝐶 = (𝐴𝐵))

Theoremdfrecs2 32041 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))))

Theoremdfrdg4 32042 A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
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)))))))

Theoremdfint3 32043 Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
𝐴 = (V ∖ ((V ∖ E ) “ 𝐴))

Theoremimagesset 32044 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

Theorembrub 32045* Binary relationship form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)
𝑆 ∈ V    &   𝐴 ∈ V       (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)

Theorembrlb 32046* Binary relationship form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)
𝑆 ∈ V    &   𝐴 ∈ V       (𝑆LB𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)

20.8.31  Alternate ordered pairs

Syntaxcaltop 32047 Declare the syntax for an alternate ordered pair.
class 𝐴, 𝐵

Syntaxcaltxp 32048 Declare the syntax for an alternate Cartesian product.
class (𝐴 ×× 𝐵)

Definitiondf-altop 32049 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 32060), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)
𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}

Definitiondf-altxp 32050* Define Cartesian products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.)
(𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫}

Theoremaltopex 32051 Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)
𝐴, 𝐵⟫ ∈ V

Theoremaltopthsn 32052 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}))

Theoremaltopeq12 32053 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
((𝐴 = 𝐵𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)

Theoremaltopeq1 32054 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)

Theoremaltopeq2 32055 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫)

Theoremaltopth1 32056 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.)
(𝐴𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐴 = 𝐶))

Theoremaltopth2 32057 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.)
(𝐵𝑉 → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ → 𝐵 = 𝐷))

Theoremaltopthg 32058 Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)
((𝐴𝑉𝐵𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremaltopthbg 32059 Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)
((𝐴𝑉𝐷𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremaltopth 32060 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 4943), requires 𝐷 to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremaltopthb 32061 Alternate ordered pair theorem with different sethood requirements. See altopth 32060 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
𝐴 ∈ V    &   𝐷 ∈ V       (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremaltopthc 32062 Alternate ordered pair theorem with different sethood requirements. See altopth 32060 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
𝐵 ∈ V    &   𝐶 ∈ V       (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremaltopthd 32063 Alternate ordered pair theorem with different sethood requirements. See altopth 32060 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
𝐶 ∈ V    &   𝐷 ∈ V       (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremaltxpeq1 32064 Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
(𝐴 = 𝐵 → (𝐴 ×× 𝐶) = (𝐵 ×× 𝐶))

Theoremaltxpeq2 32065 Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
(𝐴 = 𝐵 → (𝐶 ×× 𝐴) = (𝐶 ×× 𝐵))

Theoremelaltxp 32066* Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.)
(𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫)

Theoremaltopelaltxp 32067 Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5144, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)
(⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋𝐴𝑌𝐵))

Theoremaltxpsspw 32068 An inclusion rule for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
(𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)

Theoremaltxpexg 32069 The alternate Cartesian product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)
((𝐴𝑉𝐵𝑊) → (𝐴 ×× 𝐵) ∈ V)

Theoremrankaltopb 32070 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)))

Theoremnfaltop 32071 Bound-variable hypothesis builder for alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴, 𝐵

Theoremsbcaltop 32072* Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
(𝐴 ∈ V → 𝐴 / 𝑥𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)

20.8.32  Geometry in the Euclidean space

20.8.32.1  Congruence properties

Syntaxcofs 32073 Declare the syntax for the outer five segment configuration.
class OuterFiveSeg

Definitiondf-ofs 32074* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 25812). See brofs 32096 and 5segofs 32097 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⟩)))}

Theoremcgrrflx2d 32075 Deduction form of axcgrrflx 25788. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))       (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐵, 𝐴⟩)

Theoremcgrtr4d 32076 Deduction form of axcgrtr 25789. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))    &   (𝜑𝐶 ∈ (𝔼‘𝑁))    &   (𝜑𝐷 ∈ (𝔼‘𝑁))    &   (𝜑𝐸 ∈ (𝔼‘𝑁))    &   (𝜑𝐹 ∈ (𝔼‘𝑁))    &   (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)    &   (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩)       (𝜑 → ⟨𝐶, 𝐷⟩Cgr⟨𝐸, 𝐹⟩)

Theoremcgrtr4and 32077 Deduction form of axcgrtr 25789. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))    &   (𝜑𝐶 ∈ (𝔼‘𝑁))    &   (𝜑𝐷 ∈ (𝔼‘𝑁))    &   (𝜑𝐸 ∈ (𝔼‘𝑁))    &   (𝜑𝐹 ∈ (𝔼‘𝑁))    &   ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)    &   ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩)       ((𝜑𝜓) → ⟨𝐶, 𝐷⟩Cgr⟨𝐸, 𝐹⟩)

Theoremcgrrflx 32078 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⟩)

Theoremcgrrflxd 32079 Deduction form of cgrrflx 32078. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))       (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝐵⟩)

Theoremcgrcomim 32080 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⟩))

Theoremcgrcom 32081 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⟩))

Theoremcgrcomand 32082 Deduction form of cgrcom 32081. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))    &   (𝜑𝐶 ∈ (𝔼‘𝑁))    &   (𝜑𝐷 ∈ (𝔼‘𝑁))    &   ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)       ((𝜑𝜓) → ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝐵⟩)

Theoremcgrtr 32083 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⟩))

Theoremcgrtrand 32084 Deduction form of cgrtr 32083. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))    &   (𝜑𝐶 ∈ (𝔼‘𝑁))    &   (𝜑𝐷 ∈ (𝔼‘𝑁))    &   (𝜑𝐸 ∈ (𝔼‘𝑁))    &   (𝜑𝐹 ∈ (𝔼‘𝑁))    &   ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)    &   ((𝜑𝜓) → ⟨𝐶, 𝐷⟩Cgr⟨𝐸, 𝐹⟩)       ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩)

Theoremcgrtr3 32085 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⟩))

Theoremcgrtr3and 32086 Deduction form of cgrtr3 32085. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))    &   (𝜑𝐶 ∈ (𝔼‘𝑁))    &   (𝜑𝐷 ∈ (𝔼‘𝑁))    &   (𝜑𝐸 ∈ (𝔼‘𝑁))    &   (𝜑𝐹 ∈ (𝔼‘𝑁))    &   ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩)    &   ((𝜑𝜓) → ⟨𝐶, 𝐷⟩Cgr⟨𝐸, 𝐹⟩)       ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)

Theoremcgrcoml 32087 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⟩))

Theoremcgrcomr 32088 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⟩))

Theoremcgrcomlr 32089 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⟩))

Theoremcgrcomland 32090 Deduction form of cgrcoml 32087. (Contributed by Scott Fenton, 14-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))    &   (𝜑𝐶 ∈ (𝔼‘𝑁))    &   (𝜑𝐷 ∈ (𝔼‘𝑁))    &   ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)       ((𝜑𝜓) → ⟨𝐵, 𝐴⟩Cgr⟨𝐶, 𝐷⟩)

Theoremcgrcomrand 32091 Deduction form of cgrcoml 32087. (Contributed by Scott Fenton, 14-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))    &   (𝜑𝐶 ∈ (𝔼‘𝑁))    &   (𝜑𝐷 ∈ (𝔼‘𝑁))    &   ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)       ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐶⟩)

Theoremcgrcomlrand 32092 Deduction form of cgrcomlr 32089. (Contributed by Scott Fenton, 14-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))    &   (𝜑𝐶 ∈ (𝔼‘𝑁))    &   (𝜑𝐷 ∈ (𝔼‘𝑁))    &   ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩)       ((𝜑𝜓) → ⟨𝐵, 𝐴⟩Cgr⟨𝐷, 𝐶⟩)

Theoremcgrtriv 32093 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⟩)

Theoremcgrid2 32094 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))

Theoremcgrdegen 32095 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 = 𝐵𝐶 = 𝐷)))

Theorembrofs 32096 Binary relationship 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⟩))))

Theorem5segofs 32097 Rephrase ax5seg 25812 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⟩⟩ ∧ 𝐴𝐵) → ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩))

Theoremofscom 32098 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⟩⟩))

Theoremcgrextend 32099 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⟩))

Theoremcgrextendand 32100 Deduction form of cgrextend 32099. (Contributed by Scott Fenton, 14-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))    &   (𝜑𝐶 ∈ (𝔼‘𝑁))    &   (𝜑𝐷 ∈ (𝔼‘𝑁))    &   (𝜑𝐸 ∈ (𝔼‘𝑁))    &   (𝜑𝐹 ∈ (𝔼‘𝑁))    &   ((𝜑𝜓) → 𝐵 Btwn ⟨𝐴, 𝐶⟩)    &   ((𝜑𝜓) → 𝐸 Btwn ⟨𝐷, 𝐹⟩)    &   ((𝜑𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩)    &   ((𝜑𝜓) → ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)       ((𝜑𝜓) → ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩)

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42322
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