![]() |
Metamath
Proof Explorer Theorem List (p. 278 of 480) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30438) |
![]() (30439-31961) |
![]() (31962-47939) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | addscut2 27701* | Show that the cut involved in surreal addition is legitimate. (Contributed by Scott Fenton, 8-Mar-2025.) |
âĒ (ð â ð â No ) & âĒ (ð â ð â No ) â âĒ (ð â ({ð âĢ âð â ( L âð)ð = (ð +s ð)} ⊠{ð âĢ âð â ( L âð)ð = (ð +s ð)}) <<s ({ðĪ âĢ âð â ( R âð)ðĪ = (ð +s ð)} ⊠{ðĄ âĢ âð â ( R âð)ðĄ = (ð +s ð )})) | ||
Theorem | addscld 27702 | Surreal numbers are closed under addition. Theorem 6(iii) of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ (ð â ð â No ) & âĒ (ð â ð â No ) â âĒ (ð â (ð +s ð) â No ) | ||
Theorem | addscl 27703 | Surreal numbers are closed under addition. Theorem 6(iii) of [Conway[ p. 18. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ ((ðī â No â§ ðĩ â No ) â (ðī +s ðĩ) â No ) | ||
Theorem | addsf 27704 | Function statement for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ +s :( No à No )âķ No | ||
Theorem | addsfo 27705 | Surreal addition is onto. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ +s :( No à No )âontoâ No | ||
Theorem | peano2no 27706 | A theorem for surreals that is analagous to the second Peano postulate peano2 7883. (Contributed by Scott Fenton, 17-Mar-2025.) |
âĒ (ðī â No â (ðī +s 1s ) â No ) | ||
Theorem | sltadd1im 27707 | Surreal less-than is preserved under addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â (ðī <s ðĩ â (ðī +s ðķ) <s (ðĩ +s ðķ))) | ||
Theorem | sltadd2im 27708 | Surreal less-than is preserved under addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â (ðī <s ðĩ â (ðķ +s ðī) <s (ðķ +s ðĩ))) | ||
Theorem | sleadd1im 27709 | Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â ((ðī +s ðķ) âĪs (ðĩ +s ðķ) â ðī âĪs ðĩ)) | ||
Theorem | sleadd2im 27710 | Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â ((ðķ +s ðī) âĪs (ðķ +s ðĩ) â ðī âĪs ðĩ)) | ||
Theorem | sleadd1 27711 | Addition to both sides of surreal less-than or equal. Theorem 5 of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â (ðī âĪs ðĩ â (ðī +s ðķ) âĪs (ðĩ +s ðķ))) | ||
Theorem | sleadd2 27712 | Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â (ðī âĪs ðĩ â (ðķ +s ðī) âĪs (ðķ +s ðĩ))) | ||
Theorem | sltadd2 27713 | Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â (ðī <s ðĩ â (ðķ +s ðī) <s (ðķ +s ðĩ))) | ||
Theorem | sltadd1 27714 | Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â (ðī <s ðĩ â (ðī +s ðķ) <s (ðĩ +s ðķ))) | ||
Theorem | addscan2 27715 | Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â ((ðī +s ðķ) = (ðĩ +s ðķ) â ðī = ðĩ)) | ||
Theorem | addscan1 27716 | Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â ((ðķ +s ðī) = (ðķ +s ðĩ) â ðī = ðĩ)) | ||
Theorem | sleadd1d 27717 | Addition to both sides of surreal less-than or equal. Theorem 5 of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â (ðī âĪs ðĩ â (ðī +s ðķ) âĪs (ðĩ +s ðķ))) | ||
Theorem | sleadd2d 27718 | Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â (ðī âĪs ðĩ â (ðķ +s ðī) âĪs (ðķ +s ðĩ))) | ||
Theorem | sltadd2d 27719 | Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â (ðī <s ðĩ â (ðķ +s ðī) <s (ðķ +s ðĩ))) | ||
Theorem | sltadd1d 27720 | Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â (ðī <s ðĩ â (ðī +s ðķ) <s (ðĩ +s ðķ))) | ||
Theorem | addscan2d 27721 | Cancellation law for surreal addition. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðī +s ðķ) = (ðĩ +s ðķ) â ðī = ðĩ)) | ||
Theorem | addscan1d 27722 | Cancellation law for surreal addition. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðķ +s ðī) = (ðķ +s ðĩ) â ðī = ðĩ)) | ||
Theorem | addsuniflem 27723* | Lemma for addsunif 27724. State the whole theorem with extra distinct variable conditions. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ (ð â ðŋ <<s ð ) & âĒ (ð â ð <<s ð) & âĒ (ð â ðī = (ðŋ |s ð )) & âĒ (ð â ðĩ = (ð |s ð)) â âĒ (ð â (ðī +s ðĩ) = (({ðĶ âĢ âð â ðŋ ðĶ = (ð +s ðĩ)} ⊠{ð§ âĢ âð â ð ð§ = (ðī +s ð)}) |s ({ðĪ âĢ âð â ð ðĪ = (ð +s ðĩ)} ⊠{ðĄ âĢ âð â ð ðĄ = (ðī +s ð )}))) | ||
Theorem | addsunif 27724* | Uniformity theorem for surreal addition. This theorem states that we can use any cuts that define ðī and ðĩ in the definition of surreal addition. Theorem 3.2 of [Gonshor] p. 15. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ (ð â ðŋ <<s ð ) & âĒ (ð â ð <<s ð) & âĒ (ð â ðī = (ðŋ |s ð )) & âĒ (ð â ðĩ = (ð |s ð)) â âĒ (ð â (ðī +s ðĩ) = (({ðĶ âĢ âð â ðŋ ðĶ = (ð +s ðĩ)} ⊠{ð§ âĢ âð â ð ð§ = (ðī +s ð)}) |s ({ðĪ âĢ âð â ð ðĪ = (ð +s ðĩ)} ⊠{ðĄ âĢ âð â ð ðĄ = (ðī +s ð )}))) | ||
Theorem | addsasslem1 27725* | Lemma for addition associativity. Expand one form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðī +s ðĩ) +s ðķ) = ((({ðĶ âĢ âð â ( L âðī)ðĶ = ((ð +s ðĩ) +s ðķ)} ⊠{ð§ âĢ âð â ( L âðĩ)ð§ = ((ðī +s ð) +s ðķ)}) ⊠{ðĪ âĢ âð â ( L âðķ)ðĪ = ((ðī +s ðĩ) +s ð)}) |s (({ð âĢ âð â ( R âðī)ð = ((ð +s ðĩ) +s ðķ)} ⊠{ð âĢ âð â ( R âðĩ)ð = ((ðī +s ð) +s ðķ)}) ⊠{ð âĢ âð â ( R âðķ)ð = ((ðī +s ðĩ) +s ð)}))) | ||
Theorem | addsasslem2 27726* | Lemma for addition associativity. Expand the other form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â (ðī +s (ðĩ +s ðķ)) = ((({ðĶ âĢ âð â ( L âðī)ðĶ = (ð +s (ðĩ +s ðķ))} ⊠{ð§ âĢ âð â ( L âðĩ)ð§ = (ðī +s (ð +s ðķ))}) ⊠{ðĪ âĢ âð â ( L âðķ)ðĪ = (ðī +s (ðĩ +s ð))}) |s (({ð âĢ âð â ( R âðī)ð = (ð +s (ðĩ +s ðķ))} ⊠{ð âĢ âð â ( R âðĩ)ð = (ðī +s (ð +s ðķ))}) ⊠{ð âĢ âð â ( R âðķ)ð = (ðī +s (ðĩ +s ð))}))) | ||
Theorem | addsass 27727 | Surreal addition is associative. Part of theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 22-Jan-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â ((ðī +s ðĩ) +s ðķ) = (ðī +s (ðĩ +s ðķ))) | ||
Theorem | addsassd 27728 | Surreal addition is associative. Part of theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 22-Jan-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðī +s ðĩ) +s ðķ) = (ðī +s (ðĩ +s ðķ))) | ||
Theorem | adds32d 27729 | Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 22-Jan-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðī +s ðĩ) +s ðķ) = ((ðī +s ðķ) +s ðĩ)) | ||
Theorem | adds12d 27730 | Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Scott Fenton, 9-Mar-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â (ðī +s (ðĩ +s ðķ)) = (ðĩ +s (ðī +s ðķ))) | ||
Theorem | adds4d 27731 | Rearrangement of four terms in a surreal sum. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) & âĒ (ð â ð· â No ) â âĒ (ð â ((ðī +s ðĩ) +s (ðķ +s ð·)) = ((ðī +s ðķ) +s (ðĩ +s ð·))) | ||
Theorem | adds42d 27732 | Rearrangement of four terms in a surreal sum. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) & âĒ (ð â ð· â No ) â âĒ (ð â ((ðī +s ðĩ) +s (ðķ +s ð·)) = ((ðī +s ðķ) +s (ð· +s ðĩ))) | ||
Syntax | cnegs 27733 | Declare the syntax for surreal negation. |
class -us | ||
Syntax | csubs 27734 | Declare the syntax for surreal subtraction. |
class -s | ||
Definition | df-negs 27735* | Define surreal negation. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.) |
âĒ -us = norec ((ðĨ â V, ð â V âĶ ((ð â ( R âðĨ)) |s (ð â ( L âðĨ))))) | ||
Definition | df-subs 27736* | Define surreal subtraction. (Contributed by Scott Fenton, 20-Aug-2024.) |
âĒ -s = (ðĨ â No , ðĶ â No âĶ (ðĨ +s ( -us âðĶ))) | ||
Theorem | negsfn 27737 | Surreal negation is a function over surreals. (Contributed by Scott Fenton, 20-Aug-2024.) |
âĒ -us Fn No | ||
Theorem | subsfn 27738 | Surreal subtraction is a function over pairs of surreals. (Contributed by Scott Fenton, 22-Jan-2025.) |
âĒ -s Fn ( No à No ) | ||
Theorem | negsval 27739 | The value of the surreal negation function. (Contributed by Scott Fenton, 20-Aug-2024.) |
âĒ (ðī â No â ( -us âðī) = (( -us â ( R âðī)) |s ( -us â ( L âðī)))) | ||
Theorem | negs0s 27740 | Negative surreal zero is surreal zero. (Contributed by Scott Fenton, 20-Aug-2024.) |
âĒ ( -us â 0s ) = 0s | ||
Theorem | negsproplem1 27741* | Lemma for surreal negation. We prove a pair of properties of surreal negation simultaneously. First, we instantiate some quantifiers. (Contributed by Scott Fenton, 2-Feb-2025.) |
âĒ (ð â âðĨ â No âðĶ â No ((( bday âðĨ) ⊠( bday âðĶ)) â (( bday âðī) ⊠( bday âðĩ)) â (( -us âðĨ) â No â§ (ðĨ <s ðĶ â ( -us âðĶ) <s ( -us âðĨ))))) & âĒ (ð â ð â No ) & âĒ (ð â ð â No ) & âĒ (ð â (( bday âð) ⊠( bday âð)) â (( bday âðī) ⊠( bday âðĩ))) â âĒ (ð â (( -us âð) â No â§ (ð <s ð â ( -us âð) <s ( -us âð)))) | ||
Theorem | negsproplem2 27742* | Lemma for surreal negation. Show that the cut that defines negation is legitimate. (Contributed by Scott Fenton, 2-Feb-2025.) |
âĒ (ð â âðĨ â No âðĶ â No ((( bday âðĨ) ⊠( bday âðĶ)) â (( bday âðī) ⊠( bday âðĩ)) â (( -us âðĨ) â No â§ (ðĨ <s ðĶ â ( -us âðĶ) <s ( -us âðĨ))))) & âĒ (ð â ðī â No ) â âĒ (ð â ( -us â ( R âðī)) <<s ( -us â ( L âðī))) | ||
Theorem | negsproplem3 27743* | Lemma for surreal negation. Give the cut properties of surreal negation. (Contributed by Scott Fenton, 2-Feb-2025.) |
âĒ (ð â âðĨ â No âðĶ â No ((( bday âðĨ) ⊠( bday âðĶ)) â (( bday âðī) ⊠( bday âðĩ)) â (( -us âðĨ) â No â§ (ðĨ <s ðĶ â ( -us âðĶ) <s ( -us âðĨ))))) & âĒ (ð â ðī â No ) â âĒ (ð â (( -us âðī) â No â§ ( -us â ( R âðī)) <<s {( -us âðī)} â§ {( -us âðī)} <<s ( -us â ( L âðī)))) | ||
Theorem | negsproplem4 27744* | Lemma for surreal negation. Show the second half of the inductive hypothesis when ðī is simpler than ðĩ. (Contributed by Scott Fenton, 2-Feb-2025.) |
âĒ (ð â âðĨ â No âðĶ â No ((( bday âðĨ) ⊠( bday âðĶ)) â (( bday âðī) ⊠( bday âðĩ)) â (( -us âðĨ) â No â§ (ðĨ <s ðĶ â ( -us âðĶ) <s ( -us âðĨ))))) & âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðī <s ðĩ) & âĒ (ð â ( bday âðī) â ( bday âðĩ)) â âĒ (ð â ( -us âðĩ) <s ( -us âðī)) | ||
Theorem | negsproplem5 27745* | Lemma for surreal negation. Show the second half of the inductive hypothesis when ðĩ is simpler than ðī. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ (ð â âðĨ â No âðĶ â No ((( bday âðĨ) ⊠( bday âðĶ)) â (( bday âðī) ⊠( bday âðĩ)) â (( -us âðĨ) â No â§ (ðĨ <s ðĶ â ( -us âðĶ) <s ( -us âðĨ))))) & âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðī <s ðĩ) & âĒ (ð â ( bday âðĩ) â ( bday âðī)) â âĒ (ð â ( -us âðĩ) <s ( -us âðī)) | ||
Theorem | negsproplem6 27746* | Lemma for surreal negation. Show the second half of the inductive hypothesis when ðī is the same age as ðĩ. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ (ð â âðĨ â No âðĶ â No ((( bday âðĨ) ⊠( bday âðĶ)) â (( bday âðī) ⊠( bday âðĩ)) â (( -us âðĨ) â No â§ (ðĨ <s ðĶ â ( -us âðĶ) <s ( -us âðĨ))))) & âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðī <s ðĩ) & âĒ (ð â ( bday âðī) = ( bday âðĩ)) â âĒ (ð â ( -us âðĩ) <s ( -us âðī)) | ||
Theorem | negsproplem7 27747* | Lemma for surreal negation. Show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ (ð â âðĨ â No âðĶ â No ((( bday âðĨ) ⊠( bday âðĶ)) â (( bday âðī) ⊠( bday âðĩ)) â (( -us âðĨ) â No â§ (ðĨ <s ðĶ â ( -us âðĶ) <s ( -us âðĨ))))) & âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðī <s ðĩ) â âĒ (ð â ( -us âðĩ) <s ( -us âðī)) | ||
Theorem | negsprop 27748 | Show closure and ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No ) â (( -us âðī) â No â§ (ðī <s ðĩ â ( -us âðĩ) <s ( -us âðī)))) | ||
Theorem | negscl 27749 | The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ (ðī â No â ( -us âðī) â No ) | ||
Theorem | negscld 27750 | The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ (ð â ðī â No ) â âĒ (ð â ( -us âðī) â No ) | ||
Theorem | sltnegim 27751 | The forward direction of the ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No ) â (ðī <s ðĩ â ( -us âðĩ) <s ( -us âðī))) | ||
Theorem | negscut 27752 | The cut properties of surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ (ðī â No â (( -us âðī) â No â§ ( -us â ( R âðī)) <<s {( -us âðī)} â§ {( -us âðī)} <<s ( -us â ( L âðī)))) | ||
Theorem | negscut2 27753 | The cut that defines surreal negation is legitimate. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ (ðī â No â ( -us â ( R âðī)) <<s ( -us â ( L âðī))) | ||
Theorem | negsid 27754 | Surreal addition of a number and its negative. Theorem 4(iii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ (ðī â No â (ðī +s ( -us âðī)) = 0s ) | ||
Theorem | negsidd 27755 | Surreal addition of a number and its negative. Theorem 4(iii) of [Conway] p. 17. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) â âĒ (ð â (ðī +s ( -us âðī)) = 0s ) | ||
Theorem | negsex 27756* | Every surreal has a negative. Note that this theorem, addscl 27703, addscom 27688, addsass 27727, addsrid 27686, and sltadd1im 27707 are the ordered Abelian group axioms. However, the surreals cannot be said to be an ordered Abelian group because No is a proper class. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ (ðī â No â âðĨ â No (ðī +s ðĨ) = 0s ) | ||
Theorem | negnegs 27757 | A surreal is equal to the negative of its negative. Theorem 4(ii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ (ðī â No â ( -us â( -us âðī)) = ðī) | ||
Theorem | sltneg 27758 | Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No ) â (ðī <s ðĩ â ( -us âðĩ) <s ( -us âðī))) | ||
Theorem | sleneg 27759 | Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No ) â (ðī âĪs ðĩ â ( -us âðĩ) âĪs ( -us âðī))) | ||
Theorem | sltnegd 27760 | Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 14-Mar-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) â âĒ (ð â (ðī <s ðĩ â ( -us âðĩ) <s ( -us âðī))) | ||
Theorem | slenegd 27761 | Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 14-Mar-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) â âĒ (ð â (ðī âĪs ðĩ â ( -us âðĩ) âĪs ( -us âðī))) | ||
Theorem | negs11 27762 | Surreal negation is one-to-one. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No ) â (( -us âðī) = ( -us âðĩ) â ðī = ðĩ)) | ||
Theorem | negsdi 27763 | Distribution of surreal negative over addition. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No ) â ( -us â(ðī +s ðĩ)) = (( -us âðī) +s ( -us âðĩ))) | ||
Theorem | slt0neg2d 27764 | Comparison of a surreal and its negative to zero. (Contributed by Scott Fenton, 10-Mar-2025.) |
âĒ (ð â ðī â No ) â âĒ (ð â ( 0s <s ðī â ( -us âðī) <s 0s )) | ||
Theorem | negsf 27765 | Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ -us : No âķ No | ||
Theorem | negsfo 27766 | Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ -us : No âontoâ No | ||
Theorem | negsf1o 27767 | Surreal negation is a bijection. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ -us : No â1-1-ontoâ No | ||
Theorem | negsunif 27768 | Uniformity property for surreal negation. If ðŋ and ð are any cut that represents ðī, then they may be used instead of ( L âðī) and ( R âðī) in the definition of negation. (Contributed by Scott Fenton, 14-Feb-2025.) |
âĒ (ð â ðŋ <<s ð ) & âĒ (ð â ðī = (ðŋ |s ð )) â âĒ (ð â ( -us âðī) = (( -us â ð ) |s ( -us â ðŋ))) | ||
Theorem | negsbdaylem 27769 | Lemma for negsbday 27770. Bound the birthday of the negative of a surreal number above. (Contributed by Scott Fenton, 8-Mar-2025.) |
âĒ (ðī â No â ( bday â( -us âðī)) â ( bday âðī)) | ||
Theorem | negsbday 27770 | Negation of a surreal number preserves birthday. (Contributed by Scott Fenton, 8-Mar-2025.) |
âĒ (ðī â No â ( bday â( -us âðī)) = ( bday âðī)) | ||
Theorem | subsval 27771 | The value of surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No ) â (ðī -s ðĩ) = (ðī +s ( -us âðĩ))) | ||
Theorem | subsvald 27772 | The value of surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) â âĒ (ð â (ðī -s ðĩ) = (ðī +s ( -us âðĩ))) | ||
Theorem | subscl 27773 | Closure law for surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No ) â (ðī -s ðĩ) â No ) | ||
Theorem | subscld 27774 | Closure law for surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) â âĒ (ð â (ðī -s ðĩ) â No ) | ||
Theorem | subsid1 27775 | Identity law for subtraction. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ (ðī â No â (ðī -s 0s ) = ðī) | ||
Theorem | subsid 27776 | Subtraction of a surreal from itself. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ (ðī â No â (ðī -s ðī) = 0s ) | ||
Theorem | subadds 27777 | Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 3-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â ((ðī -s ðĩ) = ðķ â (ðĩ +s ðķ) = ðī)) | ||
Theorem | subaddsd 27778 | Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðī -s ðĩ) = ðķ â (ðĩ +s ðķ) = ðī)) | ||
Theorem | pncans 27779 | Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No ) â ((ðī +s ðĩ) -s ðĩ) = ðī) | ||
Theorem | pncan3s 27780 | Subtraction and addition of equals. (Contributed by Scott Fenton, 4-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No ) â (ðī +s (ðĩ -s ðī)) = ðĩ) | ||
Theorem | npcans 27781 | Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No ) â ((ðī -s ðĩ) +s ðĩ) = ðī) | ||
Theorem | sltsub1 27782 | Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 4-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â (ðī <s ðĩ â (ðī -s ðķ) <s (ðĩ -s ðķ))) | ||
Theorem | sltsub2 27783 | Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 4-Feb-2025.) |
âĒ ((ðī â No â§ ðĩ â No â§ ðķ â No ) â (ðī <s ðĩ â (ðķ -s ðĩ) <s (ðķ -s ðī))) | ||
Theorem | sltsub1d 27784 | Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â (ðī <s ðĩ â (ðī -s ðķ) <s (ðĩ -s ðķ))) | ||
Theorem | sltsub2d 27785 | Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â (ðī <s ðĩ â (ðķ -s ðĩ) <s (ðķ -s ðī))) | ||
Theorem | negsubsdi2d 27786 | Distribution of negative over subtraction. (Contributed by Scott Fenton, 5-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) â âĒ (ð â ( -us â(ðī -s ðĩ)) = (ðĩ -s ðī)) | ||
Theorem | addsubsassd 27787 | Associative-type law for surreal addition and subtraction. (Contributed by Scott Fenton, 6-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðī +s ðĩ) -s ðķ) = (ðī +s (ðĩ -s ðķ))) | ||
Theorem | addsubsd 27788 | Law for surreal addition and subtraction. (Contributed by Scott Fenton, 4-Mar-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðī +s ðĩ) -s ðķ) = ((ðī -s ðķ) +s ðĩ)) | ||
Theorem | sltsubsubbd 27789 | Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 6-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) & âĒ (ð â ð· â No ) â âĒ (ð â ((ðī -s ðķ) <s (ðĩ -s ð·) â (ðī -s ðĩ) <s (ðķ -s ð·))) | ||
Theorem | sltsubsub2bd 27790 | Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 21-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) & âĒ (ð â ð· â No ) â âĒ (ð â ((ðī -s ðĩ) <s (ðķ -s ð·) â (ð· -s ðķ) <s (ðĩ -s ðī))) | ||
Theorem | sltsubsub3bd 27791 | Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 21-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) & âĒ (ð â ð· â No ) â âĒ (ð â ((ðī -s ðķ) <s (ðĩ -s ð·) â (ð· -s ðķ) <s (ðĩ -s ðī))) | ||
Theorem | slesubsubbd 27792 | Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) & âĒ (ð â ð· â No ) â âĒ (ð â ((ðī -s ðķ) âĪs (ðĩ -s ð·) â (ðī -s ðĩ) âĪs (ðķ -s ð·))) | ||
Theorem | slesubsub2bd 27793 | Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) & âĒ (ð â ð· â No ) â âĒ (ð â ((ðī -s ðĩ) âĪs (ðķ -s ð·) â (ð· -s ðķ) âĪs (ðĩ -s ðī))) | ||
Theorem | slesubsub3bd 27794 | Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) & âĒ (ð â ð· â No ) â âĒ (ð â ((ðī -s ðķ) âĪs (ðĩ -s ð·) â (ð· -s ðķ) âĪs (ðĩ -s ðī))) | ||
Theorem | sltsubaddd 27795 | Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 27-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðī -s ðĩ) <s ðķ â ðī <s (ðķ +s ðĩ))) | ||
Theorem | sltsubadd2d 27796 | Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 27-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðī -s ðĩ) <s ðķ â ðī <s (ðĩ +s ðķ))) | ||
Theorem | sltaddsubd 27797 | Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 28-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðī +s ðĩ) <s ðķ â ðī <s (ðķ -s ðĩ))) | ||
Theorem | sltaddsub2d 27798 | Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 28-Feb-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðī +s ðĩ) <s ðķ â ðĩ <s (ðķ -s ðī))) | ||
Theorem | subsubs4d 27799 | Law for double surreal subtraction. (Contributed by Scott Fenton, 9-Mar-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) & âĒ (ð â ðķ â No ) â âĒ (ð â ((ðī -s ðĩ) -s ðķ) = (ðī -s (ðĩ +s ðķ))) | ||
Theorem | posdifsd 27800 | Comparison of two surreals whose difference is positive. (Contributed by Scott Fenton, 10-Mar-2025.) |
âĒ (ð â ðī â No ) & âĒ (ð â ðĩ â No ) â âĒ (ð â (ðī <s ðĩ â 0s <s (ðĩ -s ðī))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |