P. 129 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12801-12900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	s8cli 12801	A length 8 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C D E F G H "> e. Word _V
Theorem	s2fv0 12802	Extract the first symbol from a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
|- ( A e. V -> ( <" A B "> ` 0 ) = A )
Theorem	s2fv1 12803	Extract the second symbol from a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
|- ( B e. V -> ( <" A B "> ` 1 ) = B )
Theorem	s2len 12804	The length of a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
|- ( # ` <" A B "> ) = 2
Theorem	s3fv0 12805	Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
|- ( A e. V -> ( <" A B C "> ` 0 ) = A )
Theorem	s3fv1 12806	Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
|- ( B e. V -> ( <" A B C "> ` 1 ) = B )
Theorem	s3fv2 12807	Extract the third symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
|- ( C e. V -> ( <" A B C "> ` 2 ) = C )
Theorem	s3len 12808	The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- ( # ` <" A B C "> ) = 3
Theorem	s4len 12809	The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- ( # ` <" A B C D "> ) = 4
Theorem	s5len 12810	The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- ( # ` <" A B C D E "> ) = 5
Theorem	s6len 12811	The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- ( # ` <" A B C D E F "> ) = 6
Theorem	s7len 12812	The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- ( # ` <" A B C D E F G "> ) = 7
Theorem	s8len 12813	The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- ( # ` <" A B C D E F G H "> ) = 8
Theorem	s2prop 12814	A length 2 word is an unordered pair of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
|- ( ( A e. S /\ B e. S ) -> <" A B "> = { <. 0 , A >. , <. 1 , B >. } )
Theorem	s4prop 12815	A length 4 word is a union of two unordered pairs of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
|- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> <" A B C D "> = ( { <. 0 , A >. , <. 1 , B >. } u. { <. 2 , C >. , <. 3 , D >. } ) )
Theorem	s2f1o 12816	A length 2 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
|- ( ( A e. S /\ B e. S /\ A =/= B ) -> ( E = <" A B "> -> E : { 0 , 1 } -1-1-onto-> { A , B } ) )
Theorem	f1oun2prg 12817	A union of unordered pairs of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) -> ( { <. 0 , A >. , <. 1 , B >. } u. { <. 2 , C >. , <. 3 , D >. } ) : ( { 0 , 1 } u. { 2 , 3 } ) -1-1-onto-> ( { A , B } u. { C , D } ) ) )
Theorem	s4f1o 12818	A length 4 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
|- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) -> ( E = <" A B C D "> -> E : dom E -1-1-onto-> ( { A , B } u. { C , D } ) ) ) )
Theorem	s4dom 12819	The domain of a length 4 word is the union of two (disjunct) pairs. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
|- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( E = <" A B C D "> -> dom E = ( { 0 , 1 } u. { 2 , 3 } ) ) )
Theorem	s2co 12820	Mapping a doubleton by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> F : X --> Y )   &    |- ( ph -> A e. X )   &    |- ( ph -> B e. X )   =>    |- ( ph -> ( F o. <" A B "> ) = <" ( F ` A ) ( F ` B ) "> )
Theorem	s3co 12821	Mapping a length 3 string by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> F : X --> Y )   &    |- ( ph -> A e. X )   &    |- ( ph -> B e. X )   &    |- ( ph -> C e. X )   =>    |- ( ph -> ( F o. <" A B C "> ) = <" ( F ` A ) ( F ` B ) ( F ` C ) "> )
Theorem	s0s1 12822	Concatenation of fixed length strings. (This special case of ccatlid 12557 is provided to complete the pattern s0s1 12822, df-s2 12765, df-s3 12766, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
|- <" A "> = ( (/) concat <" A "> )
Theorem	s1s2 12823	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C "> = ( <" A "> concat <" B C "> )
Theorem	s1s3 12824	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C D "> = ( <" A "> concat <" B C D "> )
Theorem	s1s4 12825	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C D E "> = ( <" A "> concat <" B C D E "> )
Theorem	s1s5 12826	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C D E F "> = ( <" A "> concat <" B C D E F "> )
Theorem	s1s6 12827	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C D E F G "> = ( <" A "> concat <" B C D E F G "> )
Theorem	s1s7 12828	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C D E F G H "> = ( <" A "> concat <" B C D E F G H "> )
Theorem	s2s2 12829	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C D "> = ( <" A B "> concat <" C D "> )
Theorem	s4s2 12830	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C D E F "> = ( <" A B C D "> concat <" E F "> )
Theorem	s4s3 12831	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C D E F G "> = ( <" A B C D "> concat <" E F G "> )
Theorem	s4s4 12832	Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C D E F G H "> = ( <" A B C D "> concat <" E F G H "> )
Theorem	s2eq2s1eq 12833	Two length 2 words are equal iff the corresponding singleton words consisting of their symbols are equal. (Contributed by Alexander van der Vekens, 24-Sep-2018.)
|- ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( <" A B "> = <" C D "> <-> ( <" A "> = <" C "> /\ <" B "> = <" D "> ) ) )
Theorem	s2eq2seq 12834	Two length 2 words are equal iff the corresponding symbols are equal. (Contributed by AV, 20-Oct-2018.)
|- ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( <" A B "> = <" C D "> <-> ( A = C /\ B = D ) ) )
Theorem	swrds2 12835	Extract two adjacent symbols from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
|- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W substr <. I , ( I + 2 ) >. ) = <" ( W ` I ) ( W ` ( I + 1 ) ) "> )
Theorem	wrdlen2i 12836	Implications of a word of length 2. (Contributed by AV, 27-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.)
|- ( ( S e. V /\ T e. V ) -> ( W = { <. 0 , S >. , <. 1 , T >. } -> ( ( W e. Word V /\ ( # ` W ) = 2 ) /\ ( ( W ` 0 ) = S /\ ( W ` 1 ) = T ) ) ) )
Theorem	wrd2pr2op 12837	A word of length 2 represented as unordered pair of ordered pairs. (Contributed by AV, 20-Oct-2018.)
|- ( ( W e. Word V /\ ( # ` W ) = 2 ) -> W = { <. 0 , ( W ` 0 ) >. , <. 1 , ( W ` 1 ) >. } )
Theorem	wrdlen2 12838	A word of length 2. (Contributed by AV, 20-Oct-2018.)
|- ( ( S e. V /\ T e. V ) -> ( W = { <. 0 , S >. , <. 1 , T >. } <-> ( ( W e. Word V /\ ( # ` W ) = 2 ) /\ ( ( W ` 0 ) = S /\ ( W ` 1 ) = T ) ) ) )
Theorem	wrdlen2s2 12839	A word of length 2 as doubleton. (Contributed by AV, 20-Oct-2018.)
|- ( ( W e. Word V /\ ( # ` W ) = 2 ) -> W = <" ( W ` 0 ) ( W ` 1 ) "> )
Theorem	repsw2 12840	The "repeated symbol word" of length 2. (Contributed by AV, 6-Nov-2018.)
|- ( S e. V -> ( S repeatS 2 ) = <" S S "> )
Theorem	repsw3 12841	The "repeated symbol word" of length 3. (Contributed by AV, 6-Nov-2018.)
|- ( S e. V -> ( S repeatS 3 ) = <" S S S "> )
Theorem	swrd2lsw 12842	Extract the last two single symbol from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
|- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> )
Theorem	2swrd2eqwrdeq 12843	Two words of length at least 2 are equal if and only if they have the same prefix and the same two single symbols suffix. (Contributed by AV, 24-Sep-2018.) (Revised by Mario Carneiro/AV, 23-Oct-2018.)
|- ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) -> ( W = U <-> ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) ) )
Theorem	ccatw2s1ccatws2 12844	The concatenation of a word with two singleton words equals the concatenation of the word with the doubleton consisting of the symbols of the two singletons. (Contributed by Mario Carneiro/AV, 21-Oct-2018.)
|- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> ( ( W concat <" X "> ) concat <" Y "> ) = ( W concat <" X Y "> ) )
Theorem	ccat2s1fvwALT 12845	Alternate proof of ccat2s1fvw 12594 using words of length 2, see df-s2 12765. A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018.) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018.) (New usage is discouraged.)
|- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W concat <" X "> ) concat <" Y "> ) ` I ) = ( W ` I ) )
Theorem	wwlktovf 12846*	Lemma 1 for wrd2f1tovbij 12850. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
|- D = { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) }   &    |- R = { n e. V | { P , n } e. X }   &    |- F = ( t e. D |-> ( t ` 1 ) )   =>    |- F : D --> R
Theorem	wwlktovf1 12847*	Lemma 2 for wrd2f1tovbij 12850. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
|- D = { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) }   &    |- R = { n e. V | { P , n } e. X }   &    |- F = ( t e. D |-> ( t ` 1 ) )   =>    |- F : D -1-1-> R
Theorem	wwlktovfo 12848*	Lemma 3 for wrd2f1tovbij 12850. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
|- D = { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) }   &    |- R = { n e. V | { P , n } e. X }   &    |- F = ( t e. D |-> ( t ` 1 ) )   =>    |- ( P e. V -> F : D -onto-> R )
Theorem	wwlktovf1o 12849*	Lemma 4 for wrd2f1tovbij 12850. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
|- D = { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) }   &    |- R = { n e. V | { P , n } e. X }   &    |- F = ( t e. D |-> ( t ` 1 ) )   =>    |- ( P e. V -> F : D -1-1-onto-> R )
Theorem	wrd2f1tovbij 12850*	There is a bijection between words of length two with a fixed first symbol contained in a pair and the symbols contained in a pair together with the fixed symbol. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
|- ( ( V e. Y /\ P e. V ) -> E. f f : { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) } -1-1-onto-> { n e. V | { P , n } e. X } )
5.8  Elementary real and complex functions
5.8.1  The "shift" operation
Syntax	cshi 12851	Extend class notation with function shifter.
class shift
Definition	df-shft 12852*	Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of CC) and produces a new function on CC. See shftval 12859 for its value. (Contributed by NM, 20-Jul-2005.)
|- shift = ( f e. _V , x e. CC |-> { <. y , z >. | ( y e. CC /\ ( y - x ) f z ) } )
Theorem	shftlem 12853*	Two ways to write a shifted set ( B + A ). (Contributed by Mario Carneiro, 3-Nov-2013.)
|- ( ( A e. CC /\ B C_ CC ) -> { x e. CC | ( x - A ) e. B } = { x | E. y e. B x = ( y + A ) } )
Theorem	shftuz 12854*	A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> { x e. CC | ( x - A ) e. ( ZZ>= ` B ) } = ( ZZ>= ` ( B + A ) ) )
Theorem	shftfval 12855*	The value of the sequence shifter operation is a function on CC. A is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- F e. _V   =>    |- ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
Theorem	shftdm 12856*	Domain of a relation shifted by A. The set on the right is more commonly notated as ( dom F + A ) (meaning add A to every element of dom F). (Contributed by Mario Carneiro, 3-Nov-2013.)
|- F e. _V   =>    |- ( A e. CC -> dom ( F shift A ) = { x e. CC | ( x - A ) e. dom F } )
Theorem	shftfib 12857	Value of a fiber of the relation F. (Contributed by Mario Carneiro, 4-Nov-2013.)
|- F e. _V   =>    |- ( ( A e. CC /\ B e. CC ) -> ( ( F shift A ) " { B } ) = ( F " { ( B - A ) } ) )
Theorem	shftfn 12858*	Functionality and domain of a sequence shifted by A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- F e. _V   =>    |- ( ( F Fn B /\ A e. CC ) -> ( F shift A ) Fn { x e. CC | ( x - A ) e. B } )
Theorem	shftval 12859	Value of a sequence shifted by A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)
|- F e. _V   =>    |- ( ( A e. CC /\ B e. CC ) -> ( ( F shift A ) ` B ) = ( F ` ( B - A ) ) )
Theorem	shftval2 12860	Value of a sequence shifted by A - B. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( F shift ( A - B ) ) ` ( A + C ) ) = ( F ` ( B + C ) ) )
Theorem	shftval3 12861	Value of a sequence shifted by A - B. (Contributed by NM, 20-Jul-2005.)
|- F e. _V   =>    |- ( ( A e. CC /\ B e. CC ) -> ( ( F shift ( A - B ) ) ` A ) = ( F ` B ) )
Theorem	shftval4 12862	Value of a sequence shifted by -u A. (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( ( A e. CC /\ B e. CC ) -> ( ( F shift -u A ) ` B ) = ( F ` ( A + B ) ) )
Theorem	shftval5 12863	Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( ( A e. CC /\ B e. CC ) -> ( ( F shift A ) ` ( B + A ) ) = ( F ` B ) )
Theorem	shftf 12864*	Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( ( F : B --> C /\ A e. CC ) -> ( F shift A ) : { x e. CC | ( x - A ) e. B } --> C )
Theorem	2shfti 12865	Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( ( A e. CC /\ B e. CC ) -> ( ( F shift A ) shift B ) = ( F shift ( A + B ) ) )
Theorem	shftidt2 12866	Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( F shift 0 ) = ( F |` CC )
Theorem	shftidt 12867	Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( A e. CC -> ( ( F shift 0 ) ` A ) = ( F ` A ) )
Theorem	shftcan1 12868	Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( ( A e. CC /\ B e. CC ) -> ( ( ( F shift A ) shift -u A ) ` B ) = ( F ` B ) )
Theorem	shftcan2 12869	Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( ( A e. CC /\ B e. CC ) -> ( ( ( F shift -u A ) shift A ) ` B ) = ( F ` B ) )
Theorem	seqshft 12870	Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-Feb-2014.)
|- F e. _V   =>    |- ( ( M e. ZZ /\ N e. ZZ ) -> seq M ( .+ , ( F shift N ) ) = ( seq ( M - N ) ( .+ , F ) shift N ) )
5.8.2  Signum (sgn or sign) function
Syntax	csgn 12871	Extend class notation to include the Signum function.
class sgn
Definition	df-sgn 12872	Signum function. Pronounced "signum" , otherwise it might be confused with "sine". Defined as "sgn" in ISO 80000-2:2009(E) operation 2-9.13. It is named "sign" (with the same definition) in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. We define this over RR* (df-xr 9623) instead of RR so that it can accept +oo and -oo. Note that df-psgn 16307 defines the sign of a permutation, which is different. Value shown in sgnval 12873. (Contributed by David A. Wheeler, 15-May-2015.)
|- sgn = ( x e. RR* |-> if ( x = 0 , 0 , if ( x < 0 , -u 1 , 1 ) ) )
Theorem	sgnval 12873	Value of Signum function. Pronounced "signum" . See df-sgn 12872. (Contributed by David A. Wheeler, 15-May-2015.)
|- ( A e. RR* -> (sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) )
Theorem	sgn0 12874	Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
|- (sgn ` 0 ) = 0
Theorem	sgnp 12875	Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
|- ( ( A e. RR* /\ 0 < A ) -> (sgn ` A ) = 1 )
Theorem	sgnrrp 12876	Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.)
|- ( A e. RR+ -> (sgn ` A ) = 1 )
Theorem	sgn1 12877	Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
|- (sgn ` 1 ) = 1
Theorem	sgnpnf 12878	Proof that the signum of +oo is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
|- (sgn ` +oo ) = 1
Theorem	sgnn 12879	Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
|- ( ( A e. RR* /\ A < 0 ) -> (sgn ` A ) = -u 1 )
Theorem	sgnmnf 12880	Proof that the signum of -oo is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
|- (sgn ` -oo ) = -u 1
5.8.3  Real and imaginary parts; conjugate
Syntax	ccj 12881	Extend class notation to include complex conjugate function.
class *
Syntax	cre 12882	Extend class notation to include real part of a complex number.
class Re
Syntax	cim 12883	Extend class notation to include imaginary part of a complex number.
class Im
Definition	df-cj 12884*	Define the complex conjugate function. See cjcli 12954 for its closure and cjval 12887 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
|- * = ( x e. CC |-> ( iota_ y e. CC ( ( x + y ) e. RR /\ ( _i x. ( x - y ) ) e. RR ) ) )
Definition	df-re 12885	Define a function whose value is the real part of a complex number. See reval 12891 for its value, recli 12952 for its closure, and replim 12901 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
|- Re = ( x e. CC |-> ( ( x + ( * ` x ) ) / 2 ) )
Definition	df-im 12886	Define a function whose value is the imaginary part of a complex number. See imval 12892 for its value, imcli 12953 for its closure, and replim 12901 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
|- Im = ( x e. CC |-> ( Re ` ( x / _i ) ) )
Theorem	cjval 12887*	The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
|- ( A e. CC -> ( * ` A ) = ( iota_ x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) ) )
Theorem	cjth 12888	The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
|- ( A e. CC -> ( ( A + ( * ` A ) ) e. RR /\ ( _i x. ( A - ( * ` A ) ) ) e. RR ) )
Theorem	cjf 12889	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
|- * : CC --> CC
Theorem	cjcl 12890	The conjugate of a complex number is a complex number (closure law). (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
|- ( A e. CC -> ( * ` A ) e. CC )
Theorem	reval 12891	The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
|- ( A e. CC -> ( Re ` A ) = ( ( A + ( * ` A ) ) / 2 ) )
Theorem	imval 12892	The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
|- ( A e. CC -> ( Im ` A ) = ( Re ` ( A / _i ) ) )
Theorem	imre 12893	The imaginary part of a complex number in terms of the real part function. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 6-Nov-2013.)
|- ( A e. CC -> ( Im ` A ) = ( Re ` ( -u _i x. A ) ) )
Theorem	reim 12894	The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)
|- ( A e. CC -> ( Re ` A ) = ( Im ` ( _i x. A ) ) )
Theorem	recl 12895	The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
|- ( A e. CC -> ( Re ` A ) e. RR )
Theorem	imcl 12896	The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
|- ( A e. CC -> ( Im ` A ) e. RR )
Theorem	ref 12897	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
|- Re : CC --> RR
Theorem	imf 12898	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
|- Im : CC --> RR
Theorem	crre 12899	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
|- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( A + ( _i x. B ) ) ) = A )
Theorem	crim 12900	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
|- ( ( A e. RR /\ B e. RR ) -> ( Im ` ( A + ( _i x. B ) ) ) = B )