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Theorem List for Metamath Proof Explorer - 11001-11100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgenpcd 11001* Downward closure of an operation on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    &   ((((๐ด โˆˆ P โˆง ๐‘” โˆˆ ๐ด) โˆง (๐ต โˆˆ P โˆง โ„Ž โˆˆ ๐ต)) โˆง ๐‘ฅ โˆˆ Q) โ†’ (๐‘ฅ <Q (๐‘”๐บโ„Ž) โ†’ ๐‘ฅ โˆˆ (๐ด๐น๐ต)))    โ‡’   ((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐‘“ โˆˆ (๐ด๐น๐ต) โ†’ (๐‘ฅ <Q ๐‘“ โ†’ ๐‘ฅ โˆˆ (๐ด๐น๐ต))))
 
Theoremgenpnmax 11002* An operation on positive reals has no largest member. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    &   (๐‘ฃ โˆˆ Q โ†’ (๐‘ง <Q ๐‘ค โ†” (๐‘ฃ๐บ๐‘ง) <Q (๐‘ฃ๐บ๐‘ค)))    &   (๐‘ง๐บ๐‘ค) = (๐‘ค๐บ๐‘ง)    โ‡’   ((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐‘“ โˆˆ (๐ด๐น๐ต) โ†’ โˆƒ๐‘ฅ โˆˆ (๐ด๐น๐ต)๐‘“ <Q ๐‘ฅ))
 
Theoremgenpcl 11003* Closure of an operation on reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    &   (โ„Ž โˆˆ Q โ†’ (๐‘“ <Q ๐‘” โ†” (โ„Ž๐บ๐‘“) <Q (โ„Ž๐บ๐‘”)))    &   (๐‘ฅ๐บ๐‘ฆ) = (๐‘ฆ๐บ๐‘ฅ)    &   ((((๐ด โˆˆ P โˆง ๐‘” โˆˆ ๐ด) โˆง (๐ต โˆˆ P โˆง โ„Ž โˆˆ ๐ต)) โˆง ๐‘ฅ โˆˆ Q) โ†’ (๐‘ฅ <Q (๐‘”๐บโ„Ž) โ†’ ๐‘ฅ โˆˆ (๐ด๐น๐ต)))    โ‡’   ((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด๐น๐ต) โˆˆ P)
 
Theoremgenpass 11004* Associativity of an operation on reals. (Contributed by NM, 18-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐น = (๐‘ค โˆˆ P, ๐‘ฃ โˆˆ P โ†ฆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐‘ค โˆƒ๐‘ง โˆˆ ๐‘ฃ ๐‘ฅ = (๐‘ฆ๐บ๐‘ง)})    &   ((๐‘ฆ โˆˆ Q โˆง ๐‘ง โˆˆ Q) โ†’ (๐‘ฆ๐บ๐‘ง) โˆˆ Q)    &   dom ๐น = (P ร— P)    &   ((๐‘“ โˆˆ P โˆง ๐‘” โˆˆ P) โ†’ (๐‘“๐น๐‘”) โˆˆ P)    &   ((๐‘“๐บ๐‘”)๐บโ„Ž) = (๐‘“๐บ(๐‘”๐บโ„Ž))    โ‡’   ((๐ด๐น๐ต)๐น๐ถ) = (๐ด๐น(๐ต๐น๐ถ))
 
Theoremplpv 11005* Value of addition on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด +P ๐ต) = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐ด โˆƒ๐‘ง โˆˆ ๐ต ๐‘ฅ = (๐‘ฆ +Q ๐‘ง)})
 
Theoremmpv 11006* Value of multiplication on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด ยทP ๐ต) = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ ๐ด โˆƒ๐‘ง โˆˆ ๐ต ๐‘ฅ = (๐‘ฆ ยทQ ๐‘ง)})
 
Theoremdmplp 11007 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
dom +P = (P ร— P)
 
Theoremdmmp 11008 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
dom ยทP = (P ร— P)
 
Theoremnqpr 11009* The canonical embedding of the rationals into the reals. (Contributed by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
(๐ด โˆˆ Q โ†’ {๐‘ฅ โˆฃ ๐‘ฅ <Q ๐ด} โˆˆ P)
 
Theorem1pr 11010 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
1P โˆˆ P
 
Theoremaddclprlem1 11011 Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
(((๐ด โˆˆ P โˆง ๐‘” โˆˆ ๐ด) โˆง ๐‘ฅ โˆˆ Q) โ†’ (๐‘ฅ <Q (๐‘” +Q โ„Ž) โ†’ ((๐‘ฅ ยทQ (*Qโ€˜(๐‘” +Q โ„Ž))) ยทQ ๐‘”) โˆˆ ๐ด))
 
Theoremaddclprlem2 11012* Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
((((๐ด โˆˆ P โˆง ๐‘” โˆˆ ๐ด) โˆง (๐ต โˆˆ P โˆง โ„Ž โˆˆ ๐ต)) โˆง ๐‘ฅ โˆˆ Q) โ†’ (๐‘ฅ <Q (๐‘” +Q โ„Ž) โ†’ ๐‘ฅ โˆˆ (๐ด +P ๐ต)))
 
Theoremaddclpr 11013 Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด +P ๐ต) โˆˆ P)
 
Theoremmulclprlem 11014* Lemma to prove downward closure in positive real multiplication. Part of proof of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 14-Mar-1996.) (New usage is discouraged.)
((((๐ด โˆˆ P โˆง ๐‘” โˆˆ ๐ด) โˆง (๐ต โˆˆ P โˆง โ„Ž โˆˆ ๐ต)) โˆง ๐‘ฅ โˆˆ Q) โ†’ (๐‘ฅ <Q (๐‘” ยทQ โ„Ž) โ†’ ๐‘ฅ โˆˆ (๐ด ยทP ๐ต)))
 
Theoremmulclpr 11015 Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด ยทP ๐ต) โˆˆ P)
 
Theoremaddcompr 11016 Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
(๐ด +P ๐ต) = (๐ต +P ๐ด)
 
Theoremaddasspr 11017 Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
((๐ด +P ๐ต) +P ๐ถ) = (๐ด +P (๐ต +P ๐ถ))
 
Theoremmulcompr 11018 Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
(๐ด ยทP ๐ต) = (๐ต ยทP ๐ด)
 
Theoremmulasspr 11019 Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
((๐ด ยทP ๐ต) ยทP ๐ถ) = (๐ด ยทP (๐ต ยทP ๐ถ))
 
Theoremdistrlem1pr 11020 Lemma for distributive law for positive reals. (Contributed by NM, 1-May-1996.) (Revised by Mario Carneiro, 13-Jun-2013.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P โˆง ๐ถ โˆˆ P) โ†’ (๐ด ยทP (๐ต +P ๐ถ)) โŠ† ((๐ด ยทP ๐ต) +P (๐ด ยทP ๐ถ)))
 
Theoremdistrlem4pr 11021* Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
(((๐ด โˆˆ P โˆง ๐ต โˆˆ P โˆง ๐ถ โˆˆ P) โˆง ((๐‘ฅ โˆˆ ๐ด โˆง ๐‘ฆ โˆˆ ๐ต) โˆง (๐‘“ โˆˆ ๐ด โˆง ๐‘ง โˆˆ ๐ถ))) โ†’ ((๐‘ฅ ยทQ ๐‘ฆ) +Q (๐‘“ ยทQ ๐‘ง)) โˆˆ (๐ด ยทP (๐ต +P ๐ถ)))
 
Theoremdistrlem5pr 11022 Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P โˆง ๐ถ โˆˆ P) โ†’ ((๐ด ยทP ๐ต) +P (๐ด ยทP ๐ถ)) โŠ† (๐ด ยทP (๐ต +P ๐ถ)))
 
Theoremdistrpr 11023 Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
(๐ด ยทP (๐ต +P ๐ถ)) = ((๐ด ยทP ๐ต) +P (๐ด ยทP ๐ถ))
 
Theorem1idpr 11024 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.) (New usage is discouraged.)
(๐ด โˆˆ P โ†’ (๐ด ยทP 1P) = ๐ด)
 
Theoremltprord 11025 Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด<P ๐ต โ†” ๐ด โŠŠ ๐ต))
 
Theorempsslinpr 11026 Proper subset is a linear ordering on positive reals. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ (๐ด โŠŠ ๐ต โˆจ ๐ด = ๐ต โˆจ ๐ต โŠŠ ๐ด))
 
Theoremltsopr 11027 Positive real 'less than' is a strict ordering. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
<P Or P
 
Theoremprlem934 11028* Lemma 9-3.4 of [Gleason] p. 122. (Contributed by NM, 25-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
๐ต โˆˆ V    โ‡’   (๐ด โˆˆ P โ†’ โˆƒ๐‘ฅ โˆˆ ๐ด ยฌ (๐‘ฅ +Q ๐ต) โˆˆ ๐ด)
 
Theoremltaddpr 11029 The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ ๐ด<P (๐ด +P ๐ต))
 
Theoremltaddpr2 11030 The sum of two positive reals is greater than one of them. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
(๐ถ โˆˆ P โ†’ ((๐ด +P ๐ต) = ๐ถ โ†’ ๐ด<P ๐ถ))
 
Theoremltexprlem1 11031* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)
๐ถ = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ(ยฌ ๐‘ฆ โˆˆ ๐ด โˆง (๐‘ฆ +Q ๐‘ฅ) โˆˆ ๐ต)}    โ‡’   (๐ต โˆˆ P โ†’ (๐ด โŠŠ ๐ต โ†’ ๐ถ โ‰  โˆ…))
 
Theoremltexprlem2 11032* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)
๐ถ = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ(ยฌ ๐‘ฆ โˆˆ ๐ด โˆง (๐‘ฆ +Q ๐‘ฅ) โˆˆ ๐ต)}    โ‡’   (๐ต โˆˆ P โ†’ ๐ถ โŠŠ Q)
 
Theoremltexprlem3 11033* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
๐ถ = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ(ยฌ ๐‘ฆ โˆˆ ๐ด โˆง (๐‘ฆ +Q ๐‘ฅ) โˆˆ ๐ต)}    โ‡’   (๐ต โˆˆ P โ†’ (๐‘ฅ โˆˆ ๐ถ โ†’ โˆ€๐‘ง(๐‘ง <Q ๐‘ฅ โ†’ ๐‘ง โˆˆ ๐ถ)))
 
Theoremltexprlem4 11034* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
๐ถ = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ(ยฌ ๐‘ฆ โˆˆ ๐ด โˆง (๐‘ฆ +Q ๐‘ฅ) โˆˆ ๐ต)}    โ‡’   (๐ต โˆˆ P โ†’ (๐‘ฅ โˆˆ ๐ถ โ†’ โˆƒ๐‘ง(๐‘ง โˆˆ ๐ถ โˆง ๐‘ฅ <Q ๐‘ง)))
 
Theoremltexprlem5 11035* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
๐ถ = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ(ยฌ ๐‘ฆ โˆˆ ๐ด โˆง (๐‘ฆ +Q ๐‘ฅ) โˆˆ ๐ต)}    โ‡’   ((๐ต โˆˆ P โˆง ๐ด โŠŠ ๐ต) โ†’ ๐ถ โˆˆ P)
 
Theoremltexprlem6 11036* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐ถ = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ(ยฌ ๐‘ฆ โˆˆ ๐ด โˆง (๐‘ฆ +Q ๐‘ฅ) โˆˆ ๐ต)}    โ‡’   (((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โˆง ๐ด โŠŠ ๐ต) โ†’ (๐ด +P ๐ถ) โŠ† ๐ต)
 
Theoremltexprlem7 11037* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐ถ = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ(ยฌ ๐‘ฆ โˆˆ ๐ด โˆง (๐‘ฆ +Q ๐‘ฅ) โˆˆ ๐ต)}    โ‡’   (((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โˆง ๐ด โŠŠ ๐ต) โ†’ ๐ต โŠ† (๐ด +P ๐ถ))
 
Theoremltexpri 11038* Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
(๐ด<P ๐ต โ†’ โˆƒ๐‘ฅ โˆˆ P (๐ด +P ๐‘ฅ) = ๐ต)
 
Theoremltaprlem 11039 Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
(๐ถ โˆˆ P โ†’ (๐ด<P ๐ต โ†’ (๐ถ +P ๐ด)<P (๐ถ +P ๐ต)))
 
Theoremltapr 11040 Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
(๐ถ โˆˆ P โ†’ (๐ด<P ๐ต โ†” (๐ถ +P ๐ด)<P (๐ถ +P ๐ต)))
 
Theoremaddcanpr 11041 Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by NM, 9-Apr-1996.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โ†’ ((๐ด +P ๐ต) = (๐ด +P ๐ถ) โ†’ ๐ต = ๐ถ))
 
Theoremprlem936 11042* Lemma 9-3.6 of [Gleason] p. 124. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
((๐ด โˆˆ P โˆง 1Q <Q ๐ต) โ†’ โˆƒ๐‘ฅ โˆˆ ๐ด ยฌ (๐‘ฅ ยทQ ๐ต) โˆˆ ๐ด)
 
Theoremreclem2pr 11043* Lemma for Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐ต = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ(๐‘ฅ <Q ๐‘ฆ โˆง ยฌ (*Qโ€˜๐‘ฆ) โˆˆ ๐ด)}    โ‡’   (๐ด โˆˆ P โ†’ ๐ต โˆˆ P)
 
Theoremreclem3pr 11044* Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
๐ต = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ(๐‘ฅ <Q ๐‘ฆ โˆง ยฌ (*Qโ€˜๐‘ฆ) โˆˆ ๐ด)}    โ‡’   (๐ด โˆˆ P โ†’ 1P โŠ† (๐ด ยทP ๐ต))
 
Theoremreclem4pr 11045* Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (New usage is discouraged.)
๐ต = {๐‘ฅ โˆฃ โˆƒ๐‘ฆ(๐‘ฅ <Q ๐‘ฆ โˆง ยฌ (*Qโ€˜๐‘ฆ) โˆˆ ๐ด)}    โ‡’   (๐ด โˆˆ P โ†’ (๐ด ยทP ๐ต) = 1P)
 
Theoremrecexpr 11046* The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
(๐ด โˆˆ P โ†’ โˆƒ๐‘ฅ โˆˆ P (๐ด ยทP ๐‘ฅ) = 1P)
 
Theoremsuplem1pr 11047* The union of a nonempty, bounded set of positive reals is a positive real. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
((๐ด โ‰  โˆ… โˆง โˆƒ๐‘ฅ โˆˆ P โˆ€๐‘ฆ โˆˆ ๐ด ๐‘ฆ<P ๐‘ฅ) โ†’ โˆช ๐ด โˆˆ P)
 
Theoremsuplem2pr 11048* The union of a set of positive reals (if a positive real) is its supremum (the least upper bound). Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
(๐ด โŠ† P โ†’ ((๐‘ฆ โˆˆ ๐ด โ†’ ยฌ โˆช ๐ด<P ๐‘ฆ) โˆง (๐‘ฆ<P โˆช ๐ด โ†’ โˆƒ๐‘ง โˆˆ ๐ด ๐‘ฆ<P ๐‘ง)))
 
Theoremsupexpr 11049* The union of a nonempty, bounded set of positive reals has a supremum. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (New usage is discouraged.)
((๐ด โ‰  โˆ… โˆง โˆƒ๐‘ฅ โˆˆ P โˆ€๐‘ฆ โˆˆ ๐ด ๐‘ฆ<P ๐‘ฅ) โ†’ โˆƒ๐‘ฅ โˆˆ P (โˆ€๐‘ฆ โˆˆ ๐ด ยฌ ๐‘ฅ<P ๐‘ฆ โˆง โˆ€๐‘ฆ โˆˆ P (๐‘ฆ<P ๐‘ฅ โ†’ โˆƒ๐‘ง โˆˆ ๐ด ๐‘ฆ<P ๐‘ง)))
 
Definitiondf-enr 11050* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
~R = {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ((๐‘ฅ โˆˆ (P ร— P) โˆง ๐‘ฆ โˆˆ (P ร— P)) โˆง โˆƒ๐‘งโˆƒ๐‘คโˆƒ๐‘ฃโˆƒ๐‘ข((๐‘ฅ = โŸจ๐‘ง, ๐‘คโŸฉ โˆง ๐‘ฆ = โŸจ๐‘ฃ, ๐‘ขโŸฉ) โˆง (๐‘ง +P ๐‘ข) = (๐‘ค +P ๐‘ฃ)))}
 
Definitiondf-nr 11051 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
R = ((P ร— P) / ~R )
 
Definitiondf-plr 11052* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
+R = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ((๐‘ฅ โˆˆ R โˆง ๐‘ฆ โˆˆ R) โˆง โˆƒ๐‘คโˆƒ๐‘ฃโˆƒ๐‘ขโˆƒ๐‘“((๐‘ฅ = [โŸจ๐‘ค, ๐‘ฃโŸฉ] ~R โˆง ๐‘ฆ = [โŸจ๐‘ข, ๐‘“โŸฉ] ~R ) โˆง ๐‘ง = [โŸจ(๐‘ค +P ๐‘ข), (๐‘ฃ +P ๐‘“)โŸฉ] ~R ))}
 
Definitiondf-mr 11053* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
ยทR = {โŸจโŸจ๐‘ฅ, ๐‘ฆโŸฉ, ๐‘งโŸฉ โˆฃ ((๐‘ฅ โˆˆ R โˆง ๐‘ฆ โˆˆ R) โˆง โˆƒ๐‘คโˆƒ๐‘ฃโˆƒ๐‘ขโˆƒ๐‘“((๐‘ฅ = [โŸจ๐‘ค, ๐‘ฃโŸฉ] ~R โˆง ๐‘ฆ = [โŸจ๐‘ข, ๐‘“โŸฉ] ~R ) โˆง ๐‘ง = [โŸจ((๐‘ค ยทP ๐‘ข) +P (๐‘ฃ ยทP ๐‘“)), ((๐‘ค ยทP ๐‘“) +P (๐‘ฃ ยทP ๐‘ข))โŸฉ] ~R ))}
 
Definitiondf-ltr 11054* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
<R = {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ((๐‘ฅ โˆˆ R โˆง ๐‘ฆ โˆˆ R) โˆง โˆƒ๐‘งโˆƒ๐‘คโˆƒ๐‘ฃโˆƒ๐‘ข((๐‘ฅ = [โŸจ๐‘ง, ๐‘คโŸฉ] ~R โˆง ๐‘ฆ = [โŸจ๐‘ฃ, ๐‘ขโŸฉ] ~R ) โˆง (๐‘ง +P ๐‘ข)<P (๐‘ค +P ๐‘ฃ)))}
 
Definitiondf-0r 11055 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
0R = [โŸจ1P, 1PโŸฉ] ~R
 
Definitiondf-1r 11056 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
1R = [โŸจ(1P +P 1P), 1PโŸฉ] ~R
 
Definitiondf-m1r 11057 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
-1R = [โŸจ1P, (1P +P 1P)โŸฉ] ~R
 
Theoremenrer 11058 The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.)
~R Er (P ร— P)
 
Theoremnrex1 11059 The class of signed reals is a set. Note that a shorter proof is possible using qsex 8770 (and not requiring enrer 11058), but it would add a dependency on ax-rep 5286. (Contributed by Mario Carneiro, 17-Nov-2014.) Extract proof from that of axcnex 11142. (Revised by BJ, 4-Feb-2023.) (New usage is discouraged.)
R โˆˆ V
 
Theoremenrbreq 11060 Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
(((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โˆง (๐ถ โˆˆ P โˆง ๐ท โˆˆ P)) โ†’ (โŸจ๐ด, ๐ตโŸฉ ~R โŸจ๐ถ, ๐ทโŸฉ โ†” (๐ด +P ๐ท) = (๐ต +P ๐ถ)))
 
Theoremenreceq 11061 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) (New usage is discouraged.)
(((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โˆง (๐ถ โˆˆ P โˆง ๐ท โˆˆ P)) โ†’ ([โŸจ๐ด, ๐ตโŸฉ] ~R = [โŸจ๐ถ, ๐ทโŸฉ] ~R โ†” (๐ด +P ๐ท) = (๐ต +P ๐ถ)))
 
Theoremenrex 11062 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
~R โˆˆ V
 
Theoremltrelsr 11063 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
<R โŠ† (R ร— R)
 
Theoremaddcmpblnr 11064 Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
((((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โˆง (๐ถ โˆˆ P โˆง ๐ท โˆˆ P)) โˆง ((๐น โˆˆ P โˆง ๐บ โˆˆ P) โˆง (๐‘… โˆˆ P โˆง ๐‘† โˆˆ P))) โ†’ (((๐ด +P ๐ท) = (๐ต +P ๐ถ) โˆง (๐น +P ๐‘†) = (๐บ +P ๐‘…)) โ†’ โŸจ(๐ด +P ๐น), (๐ต +P ๐บ)โŸฉ ~R โŸจ(๐ถ +P ๐‘…), (๐ท +P ๐‘†)โŸฉ))
 
Theoremmulcmpblnrlem 11065 Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.)
(((๐ด +P ๐ท) = (๐ต +P ๐ถ) โˆง (๐น +P ๐‘†) = (๐บ +P ๐‘…)) โ†’ ((๐ท ยทP ๐น) +P (((๐ด ยทP ๐น) +P (๐ต ยทP ๐บ)) +P ((๐ถ ยทP ๐‘†) +P (๐ท ยทP ๐‘…)))) = ((๐ท ยทP ๐น) +P (((๐ด ยทP ๐บ) +P (๐ต ยทP ๐น)) +P ((๐ถ ยทP ๐‘…) +P (๐ท ยทP ๐‘†)))))
 
Theoremmulcmpblnr 11066 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.)
((((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โˆง (๐ถ โˆˆ P โˆง ๐ท โˆˆ P)) โˆง ((๐น โˆˆ P โˆง ๐บ โˆˆ P) โˆง (๐‘… โˆˆ P โˆง ๐‘† โˆˆ P))) โ†’ (((๐ด +P ๐ท) = (๐ต +P ๐ถ) โˆง (๐น +P ๐‘†) = (๐บ +P ๐‘…)) โ†’ โŸจ((๐ด ยทP ๐น) +P (๐ต ยทP ๐บ)), ((๐ด ยทP ๐บ) +P (๐ต ยทP ๐น))โŸฉ ~R โŸจ((๐ถ ยทP ๐‘…) +P (๐ท ยทP ๐‘†)), ((๐ถ ยทP ๐‘†) +P (๐ท ยทP ๐‘…))โŸฉ))
 
Theoremprsrlem1 11067* Decomposing signed reals into positive reals. Lemma for addsrpr 11070 and mulsrpr 11071. (Contributed by Jim Kingdon, 30-Dec-2019.)
(((๐ด โˆˆ ((P ร— P) / ~R ) โˆง ๐ต โˆˆ ((P ร— P) / ~R )) โˆง ((๐ด = [โŸจ๐‘ค, ๐‘ฃโŸฉ] ~R โˆง ๐ต = [โŸจ๐‘ข, ๐‘กโŸฉ] ~R ) โˆง (๐ด = [โŸจ๐‘ , ๐‘“โŸฉ] ~R โˆง ๐ต = [โŸจ๐‘”, โ„ŽโŸฉ] ~R ))) โ†’ ((((๐‘ค โˆˆ P โˆง ๐‘ฃ โˆˆ P) โˆง (๐‘  โˆˆ P โˆง ๐‘“ โˆˆ P)) โˆง ((๐‘ข โˆˆ P โˆง ๐‘ก โˆˆ P) โˆง (๐‘” โˆˆ P โˆง โ„Ž โˆˆ P))) โˆง ((๐‘ค +P ๐‘“) = (๐‘ฃ +P ๐‘ ) โˆง (๐‘ข +P โ„Ž) = (๐‘ก +P ๐‘”))))
 
Theoremaddsrmo 11068* There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
((๐ด โˆˆ ((P ร— P) / ~R ) โˆง ๐ต โˆˆ ((P ร— P) / ~R )) โ†’ โˆƒ*๐‘งโˆƒ๐‘คโˆƒ๐‘ฃโˆƒ๐‘ขโˆƒ๐‘ก((๐ด = [โŸจ๐‘ค, ๐‘ฃโŸฉ] ~R โˆง ๐ต = [โŸจ๐‘ข, ๐‘กโŸฉ] ~R ) โˆง ๐‘ง = [โŸจ(๐‘ค +P ๐‘ข), (๐‘ฃ +P ๐‘ก)โŸฉ] ~R ))
 
Theoremmulsrmo 11069* There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
((๐ด โˆˆ ((P ร— P) / ~R ) โˆง ๐ต โˆˆ ((P ร— P) / ~R )) โ†’ โˆƒ*๐‘งโˆƒ๐‘คโˆƒ๐‘ฃโˆƒ๐‘ขโˆƒ๐‘ก((๐ด = [โŸจ๐‘ค, ๐‘ฃโŸฉ] ~R โˆง ๐ต = [โŸจ๐‘ข, ๐‘กโŸฉ] ~R ) โˆง ๐‘ง = [โŸจ((๐‘ค ยทP ๐‘ข) +P (๐‘ฃ ยทP ๐‘ก)), ((๐‘ค ยทP ๐‘ก) +P (๐‘ฃ ยทP ๐‘ข))โŸฉ] ~R ))
 
Theoremaddsrpr 11070 Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
(((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โˆง (๐ถ โˆˆ P โˆง ๐ท โˆˆ P)) โ†’ ([โŸจ๐ด, ๐ตโŸฉ] ~R +R [โŸจ๐ถ, ๐ทโŸฉ] ~R ) = [โŸจ(๐ด +P ๐ถ), (๐ต +P ๐ท)โŸฉ] ~R )
 
Theoremmulsrpr 11071 Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
(((๐ด โˆˆ P โˆง ๐ต โˆˆ P) โˆง (๐ถ โˆˆ P โˆง ๐ท โˆˆ P)) โ†’ ([โŸจ๐ด, ๐ตโŸฉ] ~R ยทR [โŸจ๐ถ, ๐ทโŸฉ] ~R ) = [โŸจ((๐ด ยทP ๐ถ) +P (๐ต ยทP ๐ท)), ((๐ด ยทP ๐ท) +P (๐ต ยทP ๐ถ))โŸฉ] ~R )
 
Theoremltsrpr 11072 Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
([โŸจ๐ด, ๐ตโŸฉ] ~R <R [โŸจ๐ถ, ๐ทโŸฉ] ~R โ†” (๐ด +P ๐ท)<P (๐ต +P ๐ถ))
 
Theoremgt0srpr 11073 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
(0R <R [โŸจ๐ด, ๐ตโŸฉ] ~R โ†” ๐ต<P ๐ด)
 
Theorem0nsr 11074 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
ยฌ โˆ… โˆˆ R
 
Theorem0r 11075 The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
0R โˆˆ R
 
Theorem1sr 11076 The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
1R โˆˆ R
 
Theoremm1r 11077 The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
-1R โˆˆ R
 
Theoremaddclsr 11078 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
((๐ด โˆˆ R โˆง ๐ต โˆˆ R) โ†’ (๐ด +R ๐ต) โˆˆ R)
 
Theoremmulclsr 11079 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
((๐ด โˆˆ R โˆง ๐ต โˆˆ R) โ†’ (๐ด ยทR ๐ต) โˆˆ R)
 
Theoremdmaddsr 11080 Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
dom +R = (R ร— R)
 
Theoremdmmulsr 11081 Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
dom ยทR = (R ร— R)
 
Theoremaddcomsr 11082 Addition of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
(๐ด +R ๐ต) = (๐ต +R ๐ด)
 
Theoremaddasssr 11083 Addition of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
((๐ด +R ๐ต) +R ๐ถ) = (๐ด +R (๐ต +R ๐ถ))
 
Theoremmulcomsr 11084 Multiplication of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
(๐ด ยทR ๐ต) = (๐ต ยทR ๐ด)
 
Theoremmulasssr 11085 Multiplication of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
((๐ด ยทR ๐ต) ยทR ๐ถ) = (๐ด ยทR (๐ต ยทR ๐ถ))
 
Theoremdistrsr 11086 Multiplication of signed reals is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
(๐ด ยทR (๐ต +R ๐ถ)) = ((๐ด ยทR ๐ต) +R (๐ด ยทR ๐ถ))
 
Theoremm1p1sr 11087 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
(-1R +R 1R) = 0R
 
Theoremm1m1sr 11088 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
(-1R ยทR -1R) = 1R
 
Theoremltsosr 11089 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
<R Or R
 
Theorem0lt1sr 11090 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
0R <R 1R
 
Theorem1ne0sr 11091 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
ยฌ 1R = 0R
 
Theorem0idsr 11092 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
(๐ด โˆˆ R โ†’ (๐ด +R 0R) = ๐ด)
 
Theorem1idsr 11093 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
(๐ด โˆˆ R โ†’ (๐ด ยทR 1R) = ๐ด)
 
Theorem00sr 11094 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
(๐ด โˆˆ R โ†’ (๐ด ยทR 0R) = 0R)
 
Theoremltasr 11095 Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
(๐ถ โˆˆ R โ†’ (๐ด <R ๐ต โ†” (๐ถ +R ๐ด) <R (๐ถ +R ๐ต)))
 
Theorempn0sr 11096 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
(๐ด โˆˆ R โ†’ (๐ด +R (๐ด ยทR -1R)) = 0R)
 
Theoremnegexsr 11097* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
(๐ด โˆˆ R โ†’ โˆƒ๐‘ฅ โˆˆ R (๐ด +R ๐‘ฅ) = 0R)
 
Theoremrecexsrlem 11098* The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
(0R <R ๐ด โ†’ โˆƒ๐‘ฅ โˆˆ R (๐ด ยทR ๐‘ฅ) = 1R)
 
Theoremaddgt0sr 11099 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
((0R <R ๐ด โˆง 0R <R ๐ต) โ†’ 0R <R (๐ด +R ๐ต))
 
Theoremmulgt0sr 11100 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
((0R <R ๐ด โˆง 0R <R ๐ต) โ†’ 0R <R (๐ด ยทR ๐ต))
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