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Theorem List for Metamath Proof Explorer - 33401-33500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsscoid 33401 A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
(𝐴 ⊆ ( I ∘ 𝐵) ↔ (Rel 𝐴𝐴𝐵))

Theoremdffun10 33402 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 33403 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐹 ∈ V       (𝐹 Funs ↔ Fun 𝐹)

Theoremelfunsg 33404 Closed form of elfuns 33403. (Contributed by Scott Fenton, 2-May-2014.)
(𝐹𝑉 → (𝐹 Funs ↔ Fun 𝐹))

Theorembrsingle 33405 The binary relation 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 33406* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥})

Theoremfnsingle 33407 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 33408 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 33409* 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 33410 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
𝐴 ∈ V       {𝐴} ∈ Singletons

Theoremdfiota3 33411 A definition of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
(℩𝑥𝜑) = ({{𝑥𝜑}} ∩ Singletons )

Theoremdffv5 33412 Another quantifier-free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐹𝐴) = ({(𝐹 “ {𝐴})} ∩ Singletons )

Theoremunisnif 33413 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 33414 Binary relation 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 33415 Closed form of brimage 33414. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Image𝑅𝐵𝐵 = (𝑅𝐴)))

Theoremfunimage 33416 Image𝐴 is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Image𝐴

Theoremfnimage 33417* 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 33418* 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 33419 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 33420 Binary relation 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 33421 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       (𝐴Domain𝐵𝐵 = dom 𝐴)

Theorembrrange 33422 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       (𝐴Range𝐵𝐵 = ran 𝐴)

Theorembrdomaing 33423 Closed form of brdomain 33421. (Contributed by Scott Fenton, 2-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Domain𝐵𝐵 = dom 𝐴))

Theorembrrangeg 33424 Closed form of brrange 33422. (Contributed by Scott Fenton, 3-May-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴Range𝐵𝐵 = ran 𝐴))

Theorembrimg 33425 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⟩Img𝐶𝐶 = (𝐴𝐵))

Theorembrapply 33426 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⟩Apply𝐶𝐶 = (𝐴𝐵))

Theorembrcup 33427 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⟩Cup𝐶𝐶 = (𝐴𝐵))

Theorembrcap 33428 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⟩Cap𝐶𝐶 = (𝐴𝐵))

Theorembrsuccf 33429 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       (𝐴Succ𝐵𝐵 = suc 𝐴)

Theoremfunpartlem 33430* Lemma for funpartfun 33431. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})

Theoremfunpartfun 33431 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

Theoremfunpartss 33432 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 33433 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 33434 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 33435 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 33436 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       (𝐴FullFun𝐹𝐵𝐵 = (𝐹𝐴))

Theorembrrestrict 33437 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⟩Restrict𝐶𝐶 = (𝐴𝐵))

Theoremdfrecs2 33438 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 33439 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)))))))

Theoremdfint3 33440 Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
𝐴 = (V ∖ ((V ∖ E ) “ 𝐴))

Theoremimagesset 33441 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 33442* Binary relation form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)
𝑆 ∈ V    &   𝐴 ∈ V       (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)

Theorembrlb 33443* Binary relation form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)
𝑆 ∈ V    &   𝐴 ∈ V       (𝑆LB𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)

20.9.31  Alternate ordered pairs

Syntaxcaltop 33444 Declare the syntax for an alternate ordered pair.
class 𝐴, 𝐵

Syntaxcaltxp 33445 Declare the syntax for an alternate Cartesian product.
class (𝐴 ×× 𝐵)

Definitiondf-altop 33446 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 33457), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)
𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}

Definitiondf-altxp 33447* Define Cartesian products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.)
(𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫}

Theoremaltopex 33448 Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)
𝐴, 𝐵⟫ ∈ V

Theoremaltopthsn 33449 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 33450 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
((𝐴 = 𝐵𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)

Theoremaltopeq1 33451 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐴 = 𝐵 → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐶⟫)

Theoremaltopeq2 33452 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
(𝐴 = 𝐵 → ⟪𝐶, 𝐴⟫ = ⟪𝐶, 𝐵⟫)

Theoremaltopth1 33453 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 33454 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 33455 Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)
((𝐴𝑉𝐵𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremaltopthbg 33456 Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)
((𝐴𝑉𝐷𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremaltopth 33457 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 5356), 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 33458 Alternate ordered pair theorem with different sethood requirements. See altopth 33457 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
𝐴 ∈ V    &   𝐷 ∈ V       (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremaltopthc 33459 Alternate ordered pair theorem with different sethood requirements. See altopth 33457 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
𝐵 ∈ V    &   𝐶 ∈ V       (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremaltopthd 33460 Alternate ordered pair theorem with different sethood requirements. See altopth 33457 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
𝐶 ∈ V    &   𝐷 ∈ V       (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremaltxpeq1 33461 Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
(𝐴 = 𝐵 → (𝐴 ×× 𝐶) = (𝐵 ×× 𝐶))

Theoremaltxpeq2 33462 Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
(𝐴 = 𝐵 → (𝐶 ×× 𝐴) = (𝐶 ×× 𝐵))

Theoremelaltxp 33463* Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.)
(𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫)

Theoremaltopelaltxp 33464 Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5579, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)
(⟪𝑋, 𝑌⟫ ∈ (𝐴 ×× 𝐵) ↔ (𝑋𝐴𝑌𝐵))

Theoremaltxpsspw 33465 An inclusion rule for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
(𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)

Theoremaltxpexg 33466 The alternate Cartesian product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)
((𝐴𝑉𝐵𝑊) → (𝐴 ×× 𝐵) ∈ V)

Theoremrankaltopb 33467 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 33468 Bound-variable hypothesis builder for alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴, 𝐵

Theoremsbcaltop 33469* Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
(𝐴 ∈ V → 𝐴 / 𝑥𝐶, 𝐷⟫ = ⟪𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟫)

20.9.32  Geometry in the Euclidean space

20.9.32.1  Congruence properties

Syntaxcofs 33470 Declare the syntax for the outer five segment configuration.
class OuterFiveSeg

Definitiondf-ofs 33471* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 26730). See brofs 33493 and 5segofs 33494 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 33472 Deduction form of axcgrrflx 26706. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))       (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐵, 𝐴⟩)

Theoremcgrtr4d 33473 Deduction form of axcgrtr 26707. (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 33474 Deduction form of axcgrtr 26707. (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 33475 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 33476 Deduction form of cgrrflx 33475. (Contributed by Scott Fenton, 13-Oct-2013.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ (𝔼‘𝑁))    &   (𝜑𝐵 ∈ (𝔼‘𝑁))       (𝜑 → ⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝐵⟩)

Theoremcgrcomim 33477 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 33478 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 33479 Deduction form of cgrcom 33478. (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 33480 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 33481 Deduction form of cgrtr 33480. (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 33482 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 33483 Deduction form of cgrtr3 33482. (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 33484 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 33485 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 33486 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 33487 Deduction form of cgrcoml 33484. (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 33488 Deduction form of cgrcoml 33484. (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 33489 Deduction form of cgrcomlr 33486. (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 33490 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 33491 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 33492 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 = 𝐵𝐶 = 𝐷)))

Theorembrofs 33493 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⟩))))

Theorem5segofs 33494 Rephrase ax5seg 26730 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 33495 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 33496 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 33497 Deduction form of cgrextend 33496. (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⟩)

Theoremsegconeq 33498 Two points that satisfy the conclusion of axsegcon 26719 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))) → ((𝑄𝐴 ∧ (𝐴 Btwn ⟨𝑄, 𝑋⟩ ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ (𝐴 Btwn ⟨𝑄, 𝑌⟩ ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)) → 𝑋 = 𝑌))

Theoremsegconeu 33499* Existential uniqueness version of segconeq 33498. (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)) ∧ 𝐶𝐷)) → ∃!𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩))

20.9.32.2  Betweenness properties

Theorembtwntriv2 33500 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⟩)

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 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